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The early history of Lisp is described in\n        McCarthy 1978.\n      ","2":"\n    In this chapter we are going to look at more complex data.  All the\n    procedures\n    in chapter 1 operate on simple numerical data, and simple data are\n    not sufficient for many of the problems we wish to address using\n    computation.  Programs are typically designed to model complex phenomena,\n    and more often than not one must construct computational objects that have\n    several parts in order to model real-world phenomena that have several\n    aspects.  Thus, whereas our focus in chapter 1 was on building\n    abstractions by combining\n    procedures\n    to form compound\n    procedures,\n    we turn in this chapter to another key aspect of any programming language:\n    the means it provides for building abstractions by combining data objects\n    to form compound data.\n  ","3":"\n    To a large extent, then, the way we organize a large program is\n    dictated by our perception of the system to be modeled.  In this\n    chapter we will investigate two prominent organizational strategies\n    arising from two rather different \"world views\" of the\n    structure of systems.  The first organizational strategy concentrates on\n    objects, viewing a large system as a collection of distinct objects\n    whose behaviors may change over time.  An alternative organizational\n    strategy concentrates on the\n    streams of information that flow in\n    the system, much as an electrical engineer views a signal-processing\n    system.\n  ","4":"\n    In fact, we can regard almost any program as the evaluator for some\n    language.  For instance, the polynomial manipulation system of\n    section 2.5.3 embodies the rules of\n    polynomial arithmetic and implements them in terms of operations on\n    list-structured data.  If we augment this system with\n    procedures\n    to read and print polynomial expressions, we have the core of a\n    special-purpose language for dealing with problems in symbolic mathematics.\n    The digital-logic simulator of\n    section 3.3.4 and the constraint\n    propagator of section 3.3.5 are legitimate\n    languages in their own right, each with its own primitives, means of\n    combination, and means of abstraction.  Seen from this perspective, the\n    technology for coping with large-scale computer systems merges with the\n    technology for building new computer languages, and \n    \n    computer science itself becomes no more (and no less) than the discipline\n    of constructing appropriate descriptive languages.\n  ","5":"5  Computing with Register Machines\n\n    My aim is to show that the heavenly machine is not a kind of divine,\n    live being, but a kind of clockwork (and he who believes that a clock\n    has soul attributes the maker's glory to the work), insofar as nearly\n    all the manifold motions are caused by a most simple and material\n    force, just as all motions of the clock are caused by a single weight.\n\n    ","foreword02":"","foreword84":"Foreword","foreword84#p1":"\nEducators, generals, dieticians, psychologists, and parents program.\nArmies, students, and some societies are programmed.  An assault on\nlarge problems employs a succession of programs, most of which spring\ninto existence en route.  These programs are rife with issues that\nappear to be particular to the problem at hand.  To appreciate\nprogramming as an intellectual activity in its own right you must turn\nto computer programming; you must read and write computer\nprograms—many of them.  It doesn't matter much what the programs are\nabout or what applications they serve.  What does matter is how well\nthey perform and how smoothly they fit with other programs in the\ncreation of still greater programs.  The programmer must seek both\nperfection of part and adequacy of collection.  In this book the use\nof \"program\" is focused on the creation, execution, and study of\nprograms written in a dialect of Lisp for execution on a digital\ncomputer.  Using Lisp we restrict or limit not what we may program,\nbut only the notation for our program descriptions.\n    ","foreword84#p2":"\nOur traffic with the subject matter of this book involves us with\nthree foci of phenomena: the human mind, collections of computer\nprograms, and the computer.  Every computer program is a model,\nhatched in the mind, of a real or mental process.  These processes,\narising from human experience and thought, are huge in number,\nintricate in detail, and at any time only partially understood.  They\nare modeled to our permanent satisfaction rarely by our computer\nprograms.  Thus even though our programs are carefully handcrafted\ndiscrete collections of symbols, mosaics of interlocking functions,\nthey continually evolve: we change them as our perception of the model\ndeepens, enlarges, generalizes until the model ultimately attains a\nmetastable place within still another model with which we struggle.\nThe source of the exhilaration associated with computer programming is\nthe continual unfolding within the mind and on the computer of\nmechanisms expressed as programs and the explosion of perception they\ngenerate.  If art interprets our dreams, the computer executes them in\nthe guise of programs!\n    ","foreword84#p3":"\nFor all its power, the computer is a harsh taskmaster.  Its programs\nmust be correct, and what we wish to say must be said accurately in\nevery detail.  As in every other symbolic activity, we become\nconvinced of program truth through argument.  Lisp itself can be\nassigned a semantics (another model, by the way), and if a program's\nfunction can be specified, say, in the predicate calculus, the proof\nmethods of logic can be used to make an acceptable correctness\nargument.  Unfortunately, as programs get large and complicated, as\nthey almost always do, the adequacy, consistency, and correctness of\nthe specifications themselves become open to doubt, so that complete\nformal arguments of correctness seldom accompany large programs.\nSince large programs grow from small ones, it is crucial that we\ndevelop an arsenal of standard program structures of whose correctness\nwe have become sure—we call them idioms—and learn to combine them\ninto larger structures using organizational techniques of proven\nvalue.  These techniques are treated at length in this book, and\nunderstanding them is essential to participation in the Promethean\nenterprise called programming.  More than anything else, the\nuncovering and mastery of powerful organizational techniques\naccelerates our ability to create large, significant programs.\nConversely, since writing large programs is very taxing, we are\nstimulated to invent new methods of reducing the mass of function and\ndetail to be fitted into large programs.\n    ","foreword84#p4":"\nUnlike programs, computers must obey the laws of physics.  If they\nwish to perform rapidly—a few nanoseconds per state change—they\nmust transmit electrons only small distances (at most\n$1\\frac{1}{2}$\nfeet).  The heat generated by the huge number of devices so\nconcentrated in space has to be removed.  An exquisite engineering art\nhas been developed balancing between multiplicity of function and\ndensity of devices.  In any event, hardware always operates at a level\nmore primitive than that at which we care to program.  The processes\nthat transform our Lisp programs to \"machine\" programs are\nthemselves abstract models which we program.  Their study and creation\ngive a great deal of insight into the organizational programs\nassociated with programming arbitrary models.  Of course the computer\nitself can be so modeled.  Think of it: the behavior of the smallest\nphysical switching element is modeled by quantum mechanics described\nby differential equations whose detailed behavior is captured by\nnumerical approximations represented in computer programs executing on\ncomputers composed of …!\n    ","foreword84#p5":"\nIt is not merely a matter of tactical convenience to separately\nidentify the three foci.  Even though, as they say, it's all in the\nhead, this logical separation induces an acceleration of symbolic\ntraffic between these foci whose richness, vitality, and potential is\nexceeded in human experience only by the evolution of life itself.  At\nbest, relationships between the foci are metastable.  The computers\nare never large enough or fast enough.  Each breakthrough in hardware\ntechnology leads to more massive programming enterprises, new\norganizational principles, and an enrichment of abstract models.\nEvery reader should ask himself periodically \"Toward what end, toward\nwhat end?\"—but do not ask it too often lest you pass up the fun of\nprogramming for the constipation of bittersweet philosophy.\n    ","foreword84#p6":"\nAmong the programs we write, some (but never enough) perform a precise\nmathematical function such as sorting or finding the maximum of a\nsequence of numbers, determining primality, or finding the square\nroot.  We call such programs algorithms, and a great deal is known of\ntheir optimal behavior, particularly with respect to the two important\nparameters of execution time and data storage requirements.  A\nprogrammer should acquire good algorithms and idioms.  Even though\nsome programs resist precise specifications, it is the responsibility\nof the programmer to estimate, and always to attempt to improve, their\nperformance.\n    ","foreword84#p7":"\nLisp is a survivor, having been in use for about a quarter of a\ncentury.  Among the active programming languages only Fortran has had\na longer life.  Both languages have supported the programming needs of\nimportant areas of application, Fortran for scientific and engineering\ncomputation and Lisp for artificial intelligence.  These two areas\ncontinue to be important, and their programmers are so devoted to\nthese two languages that Lisp and Fortran may well continue in active\nuse for at least another quarter-century.\n    ","foreword84#p8":"\nLisp changes.  The Scheme dialect used in this text has evolved from\nthe original Lisp and differs from the latter in several important\nways, including static scoping for variable binding and permitting\nfunctions to yield functions as values.  In its semantic structure\nScheme is as closely akin to Algol 60 as to early Lisps.  Algol 60,\nnever to be an active language again, lives on in the genes of Scheme\nand Pascal.  It would be difficult to find two languages that are the\ncommunicating coin of two more different cultures than those gathered\naround these two languages.  Pascal is for building\npyramids—imposing, breathtaking, static structures built by armies\npushing heavy blocks into place.  Lisp is for building\norganisms—imposing, breathtaking, dynamic structures built by squads\nfitting fluctuating myriads of simpler organisms into place.  The\norganizing principles used are the same in both cases, except for one\nextraordinarily important difference: The discretionary exportable\nfunctionality entrusted to the individual Lisp programmer is more than\nan order of magnitude greater than that to be found within Pascal\nenterprises.  Lisp programs inflate libraries with functions whose\nutility transcends the application that produced them.  The list,\nLisp's native data structure, is largely responsible for such growth\nof utility.  The simple structure and natural applicability of lists\nare reflected in functions that are amazingly nonidiosyncratic.  In\nPascal the plethora of declarable data structures induces a\nspecialization within functions that inhibits and penalizes casual\ncooperation.  It is better to have 100 functions operate on one data\nstructure than to have 10 functions operate on 10 data structures.  As\na result the pyramid must stand unchanged for a millennium; the\norganism must evolve or perish.\n    ","foreword84#p9":"\nTo illustrate this difference, compare the treatment of material and\nexercises within this book with that in any first-course text using\nPascal.  Do not labor under the illusion that this is a text\ndigestible at MIT only, peculiar to the breed found there.  It is\nprecisely what a serious book on programming Lisp must be, no matter\nwho the student is or where it is used.\n    ","foreword84#p10":"\nNote that this is a text about programming, unlike most Lisp books,\nwhich are used as a preparation for work in artificial intelligence.\nAfter all, the critical programming concerns of software engineering\nand artificial intelligence tend to coalesce as the systems under\ninvestigation become larger.  This explains why there is such growing\ninterest in Lisp outside of artificial intelligence.\n    ","foreword84#p11":"\nAs one would expect from its goals, artificial intelligence research\ngenerates many significant programming problems.  In other\nprogramming cultures this spate of problems spawns new languages.\nIndeed, in any very large programming task a useful organizing\nprinciple is to control and isolate traffic within the task modules\nvia the invention of language.  These languages tend to become less\nprimitive as one approaches the boundaries of the system where we\nhumans interact most often.  As a result, such systems contain complex\nlanguage-processing functions replicated many times.  Lisp has such a\nsimple syntax and semantics that parsing can be treated as an\nelementary task.  Thus parsing technology plays almost no role in Lisp\nprograms, and the construction of language processors is rarely an\nimpediment to the rate of growth and change of large Lisp systems.\nFinally, it is this very simplicity of syntax and semantics that is\nresponsible for the burden and freedom borne by all Lisp programmers.\nNo Lisp program of any size beyond a few lines can be written without\nbeing saturated with discretionary functions.  Invent and fit; have\nfits and reinvent!  We toast the Lisp programmer who pens his thoughts\nwithin nests of parentheses.\n    ","prefaces03":"","prefaces96":"Prefaces\n    Is it possible that software is not like anything else, that it\n    is meant to be discarded: that the whole point is to\n    always see it as a soap bubble?\n    \n    A computer is like a violin.  You can imagine a novice trying first a\n    phonograph and then a violin.  The latter, he says, sounds terrible.\n    That is the argument we have heard from our humanists and most of our\n    computer scientists.  Computer programs are good, they say, for\n    particular purposes, but they aren't flexible.  Neither is a violin,\n    or a typewriter, until you learn how to use it.\n    ","prefaces96#h1":"\n    Preface ","prefaces96#p1":"\nThe material in this book has been the basis of MIT's entry-level\ncomputer science subject since 1980.  We had been teaching this\nmaterial for four years when the first edition was published, and\ntwelve more years have elapsed until the appearance of this second\nedition.  We are pleased that our work has been widely adopted and\nincorporated into other texts.  We have seen our students take the\nideas and programs in this book and build them in as the core of new\ncomputer systems and languages.  In literal realization of an ancient\nTalmudic pun, our students have become our builders.  We are lucky to\nhave such capable students and such accomplished builders.\n","prefaces96#p2":"\nIn preparing this edition, we have incorporated hundreds of\nclarifications suggested by our own teaching experience and the\ncomments of colleagues at MIT and elsewhere.  We have redesigned\nmost of the major programming systems in the book, including\nthe generic-arithmetic system, the interpreters, the register-machine\nsimulator, and the compiler; and we have rewritten all the program\nexamples to ensure that any Scheme implementation conforming to\nthe IEEE Scheme standard (IEEE 1990) will be able to run the code.\n","prefaces96#p3":"\nThis edition emphasizes several new themes.  The most important\nof these is the central role played by different approaches to\ndealing with time in computational models: objects with state,\nconcurrent programming, functional programming, lazy evaluation,\nand nondeterministic programming.  We have included new sections on\nconcurrency and nondeterminism, and we have tried to integrate this\ntheme throughout the book.\n","prefaces96#p4":"\nThe first edition of the book closely followed the syllabus of our MIT\none-semester subject.  With all the new material in the second\nedition, it will not be possible to cover everything in a single\nsemester, so the instructor will have to pick and choose.  In our own\nteaching, we sometimes skip the section on logic programming\n(section 4.4),\nwe have students use the\nregister-machine simulator but we do not cover its implementation\n(section 5.2),\nand we give only a cursory overview of\nthe compiler\n(section 5.5).\nEven so, this is still\nan intense course.  Some instructors may wish to cover only the first\nthree or four chapters, leaving the other material for subsequent\ncourses.\n","prefaces96#p5":"\nThe World Wide Web site of MIT Press\nprovides support for users of this book.\nThis includes programs from the book,\nsample programming assignments, supplementary materials,\nand downloadable implementations of the Scheme dialect of Lisp.\n","prefaces96#h2":"Preface to the First Edition ","prefaces96#p6":"\"The Structure and Interpretation of Computer Programs\" is the\n    entry-level subject in computer science at the Massachusetts Institute\n    of Technology.  It is required of all students at MIT who major\n    in electrical engineering or in computer science, as one-fourth of the\n    \"common core curriculum,\" which also includes two subjects on\n    circuits and linear systems and a subject on the design of digital\n    systems.  We have been involved in the development of this subject\n    since 1978, and we have taught this material in its present form since\n    the fall of 1980 to between 600 and 700 students each year.  Most of\n    these students have had little or no prior formal training in\n    computation, although many have played with computers a bit and a few\n    have had extensive programming or hardware-design experience.\n  ","prefaces96#p7":"\n    Our design of this introductory computer-science subject reflects two\n    major concerns.  First, we want to establish the idea that a computer\n    language is not just a way of getting a computer to perform operations\n    but rather that it is a novel formal medium for expressing ideas about\n    methodology.  Thus, programs must be written for people to read, and\n    only incidentally for machines to execute.  Second, we believe that\n    the essential material to be addressed by a subject at this level is\n    not the syntax of particular programming-language constructs, nor\n    clever algorithms for computing particular functions efficiently, nor\n    even the mathematical analysis of algorithms and the foundations of\n    computing, but rather the techniques used to control the intellectual\n    complexity of large software systems.\n  ","prefaces96#p8":"\n    Our goal is that students who complete this subject should have a good\n    feel for the elements of style and the aesthetics of programming.\n    They should have command of the major techniques for controlling\n    complexity in a large system. They should be capable of reading a\n    50-page-long program, if it is written in an exemplary style. They\n    should know what not to read, and what they need not understand at any\n    moment.  They should feel secure about modifying a program, retaining\n    the spirit and style of the original author.\n  ","prefaces96#p9":"\n    These skills are by no means unique to computer programming.  The\n    techniques we teach and draw upon are common to all of engineering\n    design.  We control complexity by building abstractions that hide\n    details when appropriate.  We control complexity by establishing\n    conventional interfaces that enable us to construct systems by\n    combining standard, well-understood pieces in a \"mix and match\" way.\n    We control complexity by establishing new languages for describing a\n    design, each of which emphasizes particular aspects of the design and\n    deemphasizes others.\n  ","prefaces96#p10":"\n    Underlying our approach to this subject is our conviction that\n    \"computer science\" is not a science and that its significance has\n    little to do with computers.  The computer revolution is a revolution\n    in the way we think and in the way we express what we think.  The\n    essence of this change is the emergence of what might best be called\n    procedural epistemology—the study of the structure of\n    knowledge from an imperative point of view, as opposed to the more\n    declarative point of view taken by classical mathematical subjects.\n    Mathematics provides a framework for dealing precisely with notions of\n    \"what is.\"  Computation provides a framework for dealing precisely\n    with notions of \"how to.\"","prefaces96#p11":"\n    In teaching our material we use a dialect of the programming language\n    Lisp.  We never formally teach the language, because we don't have to.\n    We just use it, and students pick it up in a few days.  This is one\n    great advantage of Lisp-like languages: They have very few ways of\n    forming compound expressions, and almost no syntactic structure.  All\n    of the formal properties can be covered in an hour, like the rules of\n    chess.  After a short time we forget about syntactic details of the\n    language (because there are none) and get on with the real\n    issues—figuring out what we want to compute, how we will decompose\n    problems into manageable parts, and how we will work on the parts.\n    Another advantage of Lisp is that it supports (but does not enforce)\n    more of the large-scale strategies for modular decomposition of\n    programs than any other language we know.  We can make procedural and\n    data abstractions, we can use higher-order functions to capture common\n    patterns of usage, we can model local state using assignment and data\n    mutation, we can link parts of a program with streams and delayed\n    evaluation, and we can easily implement embedded languages.  All of\n    this is embedded in an interactive environment with excellent support\n    for incremental program design, construction, testing, and debugging.\n    We thank all the generations of Lisp wizards, starting with John\n    McCarthy, who have fashioned a fine tool of unprecedented power and\n    elegance.\n  ","prefaces96#p12":"\n    Scheme, the dialect of Lisp that we use, is an attempt to bring\n    together the power and elegance of Lisp and Algol.  From Lisp we take\n    the metalinguistic power that derives from the simple syntax, the\n    uniform representation of programs as data objects, and the\n    garbage-collected heap-allocated data.  From Algol we take lexical\n    scoping and block structure, which are gifts from the pioneers of\n    programming-language design who were on the Algol committee.  We wish\n    to cite John Reynolds and Peter Landin for their insights into the\n    relationship of Church's lambda calculus to the structure of\n    programming languages.  We also recognize our debt to the\n    mathematicians who scouted out this territory decades before computers\n    appeared on the scene.  These pioneers include Alonzo Church, Barkley\n    Rosser, Stephen Kleene, and Haskell Curry.\n  ","acknowledgements":"Acknowledgments","acknowledgements#p1":"\nWe would like to thank the many people who have helped us develop this\nbook and this curriculum.\n","acknowledgements#p2":"\nOur subject is a clear intellectual descendant of \"6.231,\" a\nwonderful subject on programming linguistics and the lambda calculus\ntaught at MIT in the late 1960s by Jack Wozencraft and Arthur Evans,\nJr.\n","acknowledgements#p3":"\nWe owe a great debt to Robert Fano, who reorganized MIT's introductory\ncurriculum in electrical engineering and computer science to emphasize\nthe principles of engineering design.  He led us in starting out on\nthis enterprise and wrote the first set of subject notes from which\nthis book evolved.\n","acknowledgements#p4":"\nMuch of the style and aesthetics of programming that we try to teach\nwere developed in conjunction with Guy Lewis Steele Jr., who\ncollaborated with Gerald Jay Sussman in the initial development of the\nScheme language.  In addition, David Turner, Peter Henderson, Dan\nFriedman, David Wise, and Will Clinger have taught us many of the\ntechniques of the functional programming community that appear in this\nbook.\n","acknowledgements#p5":"\nJoel Moses taught us about structuring large systems.  His experience\nwith the Macsyma system for symbolic computation provided the insight\nthat one should avoid complexities of control and concentrate on\norganizing the data to reflect the real structure of the world being\nmodeled.\n","acknowledgements#p6":"\nMarvin Minsky and Seymour Papert formed many of our attitudes about\nprogramming and its place in our intellectual lives.  To them we owe\nthe understanding that computation provides a means of expression for\nexploring ideas that would otherwise be too complex to deal with\nprecisely.  They emphasize that a student's ability to write and\nmodify programs provides a powerful medium in which exploring becomes\na natural activity.\n","acknowledgements#p7":"\nWe also strongly agree with Alan Perlis that programming is lots of\nfun and we had better be careful to support the joy of programming.\nPart of this joy derives from observing great masters at work.  We are\nfortunate to have been apprentice programmers at the feet of Bill\nGosper and Richard Greenblatt.\n","acknowledgements#p8":"\nIt is difficult to identify all the people who have contributed to the\ndevelopment of our curriculum.  We thank all the lecturers, recitation\ninstructors, and tutors who have worked with us over the past fifteen\nyears and put in many extra hours on our subject, especially Bill\nSiebert, Albert Meyer, Joe Stoy, Randy Davis, Louis Braida, Eric\nGrimson, Rod Brooks, Lynn Stein, and Peter Szolovits.\nWe would like to specially acknowledge the outstanding teaching\ncontributions of Franklyn Turbak, now at Wellesley; his work\nin undergraduate instruction set a standard that we can\nall aspire to.\nWe are grateful to Jerry Saltzer and Jim Miller for\nhelping us grapple with the mysteries of concurrency, and to\nPeter Szolovits and David McAllester for their contributions\nto the exposition of nondeterministic evaluation in chapter 4.\n","acknowledgements#p9":"\nMany people have put in significant effort presenting this material at\nother universities.  Some of the people we have worked closely with\nare Jacob Katzenelson at the Technion, Hardy Mayer at the University\nof California at Irvine, Joe Stoy at Oxford, Elisha Sacks at Purdue,\nand Jan Komorowski at the Norwegian University of Science and\nTechnology.  We are exceptionally proud of our colleagues who have\nreceived major teaching awards for their adaptations of this subject\nat other universities, including Kenneth Yip at Yale, Brian Harvey at\nthe University of California at Berkeley, and Dan Huttenlocher at\nCornell.\n","acknowledgements#p10":"\nAl Moyé arranged for us to teach this material to engineers at\nHewlett-Packard, and for the production of videotapes of these\nlectures.\nWe would like to thank the talented instructors—in\nparticular Jim Miller, Bill Siebert, and Mike Eisenberg—who have\ndesigned continuing education courses incorporating these tapes and\ntaught them at universities and industry all over the world.\n","acknowledgements#p11":"\nMany educators in other countries have put in significant\nwork translating the first edition.\nMichel Briand, Pierre Chamard, and André Pic produced a French edition;\nSusanne Daniels-Herold produced a German\nedition; and Fumio Motoyoshi produced a Japanese edition.\nWe do not know who produced the Chinese edition,\nbut we consider it an honor to have been selected as the\nsubject of an \"unauthorized\" translation.\n","acknowledgements#p12":"\nIt is hard to enumerate all the people who have made technical\ncontributions to the development of the Scheme systems we use for\ninstructional purposes.  In addition to Guy Steele, principal wizards\nhave included Chris Hanson, Joe Bowbeer, Jim Miller, Guillermo Rozas,\nand Stephen Adams.  Others who have put in significant time are\nRichard Stallman, Alan Bawden, Kent Pitman, Jon Taft, Neil Mayle, John\nLamping, Gwyn Osnos, Tracy Larrabee, George Carrette, Soma\nChaudhuri, Bill Chiarchiaro, Steven Kirsch, Leigh Klotz, Wayne Noss,\nTodd Cass, Patrick O'Donnell, Kevin Theobald, Daniel Weise, Kenneth\nSinclair, Anthony Courtemanche, Henry M. Wu, Andrew Berlin, and Ruth\nShyu.\n","acknowledgements#p13":"\nBeyond the MIT implementation, we would like to thank the many people\nwho worked on the IEEE Scheme standard, including William Clinger and\nJonathan Rees, who edited the R$^4$RS,\nand Chris Haynes, David Bartley, Chris Hanson, and Jim Miller,\nwho prepared the IEEE standard.\n","acknowledgements#p14":"\nDan Friedman has been a long-time leader of the Scheme community.\nThe community's broader work goes beyond issues of language design to\nencompass significant educational innovations, such as the high-school\ncurriculum based on EdScheme by Schemer's Inc., and the wonderful\nbooks by Mike Eisenberg and by Brian Harvey and Matthew Wright.\n","acknowledgements#p15":"\nWe appreciate the work of those who contributed to making this a real\nbook, especially Terry Ehling, Larry Cohen, and Paul Bethge at the MIT\nPress.  Ella Mazel found the wonderful cover image.  For the second\nedition we are particularly grateful to Bernard and Ella Mazel for\nhelp with the book design, and to David Jones, $\\TeX$ wizard\nextraordinaire.  We also are indebted to those readers who made\npenetrating comments on the new draft: Jacob Katzenelson, Hardy\nMayer, Jim Miller, and especially Brian Harvey, who did unto this book\nas Julie did unto his book Simply Scheme.\n","acknowledgements#p16":"\nFinally, we would like to acknowledge the support of the organizations\nthat have encouraged this work over the years, including support from\nHewlett-Packard, made possible by Ira Goldstein and Joel Birnbaum, and\nsupport from DARPA, made possible by Bob Kahn.\n","see":"","1#p2":"\n    A computational process is indeed much like a sorcerer's idea of a\n    spirit.  It cannot be seen or touched.  It is not composed of matter\n    at all.  However, it is very real.  It can perform intellectual work.\n    It can answer questions.  It can affect the world by disbursing money\n    at a bank or by controlling a robot arm in a factory.  The programs we\n    use to conjure processes are like a sorcerer's spells.  They are\n    carefully composed from symbolic expressions in arcane and esoteric\n    programming languages\n    that prescribe the tasks we want our\n    processes to perform.\n  ","1#p3":"\n    A computational process, in a correctly working computer, executes\n    programs precisely and accurately.  Thus, like the sorcerer's\n    apprentice, novice programmers must learn to understand and to\n    anticipate the consequences of their conjuring.  Even small errors\n    \n    (usually called bugs or glitches)\n      \n   \n    in programs can have complex and unanticipated consequences.\n  ","1#p4":"\n    Fortunately, learning to program is considerably less dangerous than\n    learning sorcery, because the spirits we deal with are conveniently\n    contained in a secure way.  Real-world programming, however,\n    requires care, expertise, and wisdom.  A small bug in a computer-aided\n    design program, for example, can lead to the catastrophic collapse of\n    an airplane or a dam or the self-destruction of an industrial robot.\n  ","1#p5":"\n    Master software engineers have the ability to organize programs so\n    that they can be reasonably sure that the resulting processes will\n    perform the tasks intended.  They can visualize the behavior of their\n    systems in advance.  They know how to structure programs so that\n    unanticipated problems do not lead to catastrophic consequences, and\n    when problems do arise, they can\n    debug\n    their programs.  Well-designed\n    computational systems, like well-designed automobiles or nuclear\n    reactors, are designed in a modular manner, so that the parts can be\n    constructed, replaced, and debugged separately.\n  ","1#p6":"\n        We need an appropriate language for describing processes, and we will\n        use for this purpose the programming language Lisp.  Just as our\n        everyday thoughts are usually expressed in our natural language (such\n        as English, Swedish, or German), and descriptions of quantitative\n        phenomena are expressed with mathematical notations, our procedural\n        thoughts will be expressed in Lisp. \n        \n        Lisp was invented in the late\n        1950s as a formalism for reasoning about the use of certain kinds of\n        logical expressions, called \n        recursion equations, as a model for\n        computation.  The language was conceived by \n        \n        John McCarthy and is based\n        on his paper \"Recursive Functions of Symbolic Expressions and Their\n        Computation by Machine\" (McCarthy 1960).\n      ","1#p7":"\n        Despite its inception as a mathematical formalism, Lisp is a practical\n        programming language.  A Lisp \n        interpreter\n        is a machine that\n        carries out processes described in the Lisp language.  The first Lisp\n        interpreter was implemented by \n        \n        McCarthy with the help of colleagues\n        and students in the Artificial Intelligence Group of the\n        \n        MIT Research\n        Laboratory of Electronics and in the MIT Computation\n        Center.\n      Lisp, whose name is an acronym for\n      \n      LISt Processing,\n      was designed to provide symbol-manipulating capabilities for\n      attacking programming problems such as the symbolic differentiation\n      and integration of algebraic expressions.  It included for this\n      purpose new data objects known as atoms and lists, which most\n      strikingly set it apart from all other languages of the period.\n      ","1#p8":"\n        Lisp was not the product of a concerted design effort.  Instead, it\n        evolved informally in an experimental manner in response to users'\n        needs and to pragmatic implementation considerations.  Lisp's\n\tinformal evolution has continued through the years, and the community of\n\tLisp users has traditionally resisted attempts to promulgate any\n        \"official\" definition of the language.  This evolution,\n\ttogether with the flexibility and elegance of the initial conception,\n\thas enabled Lisp, which is the second oldest language in widespread use\n        today (only \n        \n        Fortran is older), to continually adapt to encompass the\n        most modern ideas about program design.  Thus, Lisp is by now a family\n        of dialects, which, while sharing most of the original features, may\n        differ from one another in significant ways.  The dialect of Lisp used\n        in this book is called \n        \n        Scheme.","1#footnote-link-2":"2","1#p9":"\n\tBecause of its experimental character and its emphasis on symbol\n\tmanipulation, \n\t\n\tLisp was at first very inefficient for numerical\n\tcomputations, at least in comparison with Fortran.  Over the years,\n\thowever, Lisp compilers have been developed that translate programs\n\tinto machine code that can perform numerical computations reasonably\n\tefficiently.  And for special applications, Lisp has been used with\n\tgreat effectiveness.\n      Although Lisp has not yet overcome its old reputation\n      as hopelessly inefficient, Lisp is now used in many applications where\n      efficiency is not the central concern.\n      \n      For example, Lisp has become\n      a language of choice for operating-system shell languages and for\n      extension languages for editors and computer-aided design systems.\n      ","1#footnote-link-3":"3","1#footnote-2":"The two dialects in which most\n        major Lisp programs of the 1970s were written are \n        \n        MacLisp \n        \n        (Moon 1978;\n        Pitman 1983), developed at the \n        \n        MIT Project MAC, and \n        \n        Interlisp\n        \n        (Teitelman 1974), developed at \n        \n        Bolt Beranek and Newman Inc. and the\n        \n        Xerox Palo Alto Research Center.  \n        \n        Portable Standard Lisp\n        \n        (Hearn 1969;\n        Griss 1981) \n        was a Lisp dialect designed to be easily portable\n        between different machines.  MacLisp spawned a number of subdialects,\n        such as \n        \n        Franz Lisp, which was developed at the \n        \n        University of\n        California at Berkeley, and\n        \n        Zetalisp (Moon 1981), which was based on a\n        special-purpose processor designed at the \n        \n        MIT Artificial Intelligence\n        Laboratory to run Lisp very efficiently.  The Lisp dialect used in\n        this book, called\n        \n        Scheme (Steele 1975), was invented in 1975 by\n        \n        Guy Lewis Steele Jr. and Gerald Jay Sussman of the MIT Artificial\n        Intelligence Laboratory and later reimplemented for instructional use\n        at MIT.  Scheme became an IEEE standard in 1990 \n        (IEEE 1990).  The\n        \n        Common Lisp dialect (Steele 1982, \n        Steele 1990) was developed by the\n        Lisp community to combine features from the earlier Lisp dialects\n        to make an industrial standard for Lisp.  Common Lisp became an ANSI\n        standard in 1994 (ANSI 1994).","1#footnote-3":"One such special application was a\n\tbreakthrough computation of scientific importance—an integration of\n\tthe motion of the \n\t\n\tSolar System that extended previous results by\n\tnearly two orders of magnitude, and demonstrated that the dynamics of\n\tthe Solar System is chaotic.  This computation was made possible by\n\tnew integration algorithms, a special-purpose compiler, and a\n\tspecial-purpose computer all implemented with the aid of software\n\ttools written in Lisp\n\t\n\t(Abelson et al. 1992; \n\tSussman and Wisdom 1992).\n      ","1.1":"1.1  The Elements of Programming","1.1#p1":"\n    A powerful programming language is more than just a means for\n    instructing a computer to perform tasks.  The language also serves as\n    a framework within which we organize our ideas about processes.  Thus,\n    when we describe a language, we should pay particular attention to the\n    means that the language provides for combining simple ideas to form\n    more complex ideas.  Every powerful language has three mechanisms for\n    accomplishing this:\n    primitive expressions,\n\t\n\twhich represent the simplest\n        entities the language is concerned with,\n      means of combination, by\n        \n\twhich compound elements are built from simpler ones, and\n      means of abstraction,\n        \n\tby which compound elements can be named and manipulated as units.\n      ","1.1#p2":"\n    In programming, we deal with two kinds of elements: \n    \n\tprocedures\n      \n    and\n    \n    data. (Later we will discover that they are really not so distinct.)\n    Informally, data is \"stuff\" that we want to manipulate, and\n    procedures\n    are descriptions of the rules for manipulating the data.\n    Thus, any powerful programming language should be able to describe\n    primitive data and primitive \n    procedures\n    and should have methods for\n    combining and abstracting \n    procedures\n    and data.\n  ","1.1#p3":"\n    In this chapter we will deal only with simple\n    \n    numerical data so that\n    we can focus on the rules for building\n    procedures.\n  In later chapters we will see that\n  these same rules allow us to build \n  procedures\n  to manipulate compound data as well.\n  ","1.1#footnote-link-1":"1","1.1#footnote-1":"The\n    \n    characterization of numbers as \"simple data\" is a barefaced\n    bluff. In fact, the treatment of numbers is one of the trickiest and most\n    confusing aspects of any programming language.  Some typical issues\n    involved are these:\n    \n    Some computer systems distinguish integers, such as 2,\n    from real numbers, such as 2.71.  Is the real number\n    2.00 different from the integer 2? Are the arithmetic operations\n    used for integers the same as the operations used for real numbers?\n    Does 6 divided by 2 produce 3, or 3.0? How large a number can we\n    represent? How many decimal places of accuracy can we represent?\n    Is the range of integers the same as the range of real numbers?\n    \n    Above and beyond these questions, of course, lies a collection of\n    issues concerning roundoff and truncation errors—the\n    entire science of numerical analysis.  Since our focus in this\n    book is on large-scale program design rather than on numerical\n    techniques, we are going to ignore these problems.  The numerical\n    examples in this chapter will exhibit the usual roundoff behavior\n    that one observes when using arithmetic operations that preserve\n    a limited number of decimal places of accuracy in noninteger\n    operations.\n  ","1.1.1":"1.1.1  Expressions","1.1.1#p1":"\n    One easy way to get started at programming is to examine some typical\n    interactions with an interpreter for the\n    Scheme dialect of Lisp.\n\tImagine that you are sitting at a\n\tcomputer terminal.\n      \n    You type \n    an expression,\n    and the interpreter responds by displaying the\n    result of its evaluating that\n    expression.","1.1.1#p2":"\n        One kind of primitive expression you might type is a number.  \n      \n    (More precisely, the expression that you type consists of the numerals that\n    represent the number in base 10.)\n    \n      If you ask our script to evaluate the expression statement\n      \n      by clicking it, it will respond by displaying a JavaScript interpreter\n      with the option\n      to evaluate the statement by pressing a \"Run\" button.\n      Click on the primitive expression statement, and see what happens!\n    ","1.1.1#p3":"\n        Expressions representing numbers may be combined with an\n        \n        expression representing a\n        \n        primitive procedure (such as \n        + or *) to\n\tform a compound expression that represents the application of the\n\tprocedure to those numbers.  For example,\n        (+ 137 349) (- 1000 334) (* 5 99) (/ 10 5) (+ 2.7 10) ","1.1.1#p4":"\n        Expressions such as these, formed by \n        \n        delimiting a list of expressions\n        within parentheses in order to denote\n        \n        procedure application,\n        are called combinations.  The leftmost\n        element in the list is called the \n        operator, and the other\n        elements are called \n        operands.  The\n        \n        value of a combination is\n        obtained by applying the procedure specified by the operator to the\n        arguments that are the values of the operands.\n      ","1.1.1#p5":"\n        The convention of placing the operator to the left of the operands is\n        known as \n        prefix notation, and it may be somewhat confusing at\n        first because it departs significantly from the customary mathematical\n        convention.  Prefix notation has several advantages, however.  One of\n        them is that it can accommodate \n        \n        procedures that may take an arbitrary\n        number of arguments, as in the following examples:\n        (+ 21 35 12 7) (* 25 4 12) \n        No ambiguity can arise, because the operator is always the leftmost \n        element and the entire combination is delimited by the\n        parentheses.\n      ","1.1.1#p6":"\n        A second advantage of prefix notation is that it extends in a\n        straightforward way to allow combinations to be\n        nested, that is, to have combinations whose elements are\n\tthemselves combinations:\n        (+ (* 3 5) (- 10 6)) ","1.1.1#p7":"\n        There is no limit (in principle) to the depth of such nesting and to the\n\toverall complexity of the expressions that the Lisp interpreter can\n\tevaluate. It is we humans who get confused by still relatively simple\n\texpressions such as\n        (+ (* 3 (+ (* 2 4) (+ 3 5))) (+ (- 10 7) 6)) \n        which the interpreter would readily evaluate to be 57.  We can help\n        ourselves by writing such an expression in the form\n        (+ (* 3\n      (+ (* 2 4)\n         (+ 3 5)))\n   (+ (- 10 7)\n      6)) \n        following a formatting convention known as  \n        pretty-printing, in which each long combination is written so\n\tthat the operands are aligned vertically.  The resulting\n\t\n\tindentations\n\tdisplay clearly the structure of the expression.","1.1.1#footnote-link-1":"1","1.1.1#p8":"\n    Even with complex expressions, the interpreter always operates in the\n    same basic cycle: It reads\n    an expression from the terminal,\n    evaluates the\n    expression,\n    and prints the result. This mode of operation is often expressed by saying\n    that the interpreter runs in a\n    read-eval-print loop.\n      \n    Observe in particular that it is not necessary to explicitly instruct the\n    interpreter to print the value of the\n    \n\texpression.","1.1.1#footnote-link-2":"2","1.1.1#footnote-1":"Lisp systems\n\ttypically provide\n        \n        features to aid the user in formatting expressions.  Two especially\n        useful features are one that automatically indents to the proper\n        pretty-print position whenever a new line is started and one that\n        highlights the matching left parenthesis whenever a right parenthesis\n        is typed.","1.1.1#footnote-2":"Lisp obeys the convention that every\n\t\n\texpression has a value. This convention, together with the old\n\treputation of Lisp as an inefficient language, is the source of the\n\tquip by Alan Perlis (paraphrasing Oscar Wilde)\n\tthat \"Lisp programmers know the value of\n\teverything but the cost of nothing.\"","1.1.2":"1.1.2  Naming and the Environment","1.1.2#p1":"\n    A critical aspect of a programming language is the means it provides\n    for using\n    \n    names to refer to computational\n    objects.\n    We say that the\n    \n    name identifies a\n    variable\n    whose\n    value is the object.\n  ","1.1.2#p2":"\n\tIn the Scheme dialect of Lisp, we name things with\n        define.\n      (define size 2) \n    causes the interpreter to associate the value 2 with the\n    name size.\n    Once the name size\n    has been associated with the number 2, we can\n    refer to the value 2 by name:\n    size (* 5 size) ","1.1.2#footnote-link-1":"1","1.1.2#p3":"\n    Here are further examples of the use of\n    define:\n      (define pi 3.14159) (define radius 10) (* pi (* radius radius)) (define circumference (* 2 pi radius)) circumference ","1.1.2#p4":"Define\n    is our language's\n    simplest means of abstraction, for it allows us to use simple names to\n    refer to the results of compound operations, such as the\n    circumference computed above.\n    In general, computational objects may have very complex\n    structures, and it would be extremely inconvenient to have to remember\n    and repeat their details each time we want to use them.  Indeed,\n    complex programs are constructed by building, step by step,\n    computational objects of increasing complexity. The\n    interpreter makes this step-by-step program construction particularly\n    convenient because name-object associations can be created\n    incrementally in successive interactions.  This feature encourages the\n    \n    incremental development and testing of programs and is largely\n    responsible for the fact that a\n    \n\tLisp\n      \n    program usually consists of a large\n    number of relatively simple\n    \n\tprocedures.\n      ","1.1.2#p5":"\n    It should be clear that the possibility of associating values with\n    names and later retrieving them means that the interpreter must\n    maintain some sort of memory that keeps track of the name-object\n    pairs.  This memory is called the\n    environment\n    (more precisely the\n    global environment,\n      \n    since we will see later that a\n    computation may involve a number of different\n    environments).","1.1.2#footnote-link-2":"2","1.1.2#footnote-1":"    \n        In this book, we do not\n        \n\tshow the interpreter's response to\n        evaluating definitions, since this is highly\n        implementation-dependent.\n      ","1.1.2#footnote-2":"Chapter 3 will show that this notion of\n    environment is crucial for understanding how the interpreter works.\n    Chapter 4 will use environments for implementing\n    interpreters.","1.1.3":"1.1.3  \n    Evaluating","1.1.3#p1":"\n    One of our goals in this chapter is to isolate issues about thinking\n    procedurally.  As a case in point, let us consider that, in evaluating\n    \n    combinations, the interpreter is itself following a procedure.\n    \n      To evaluate\n      \n\t  a combination,\n\t\n      do the following:\n      Evaluate the\n\t\n\t    subexpressions\n\t  \n\tof the combination.\n\t      Apply the\n\t      procedure\n\t      that is the value of the leftmost\n              subexpression (the operator) to the arguments that are the\n\t      values of the other subexpressions (the operands).\n\t    \n    Even this simple rule illustrates some important points about\n    processes in general. First, observe that the first step dictates\n    that in order to accomplish the evaluation process for a\n    combination we must first perform the evaluation process on each\n    operand of the combination. Thus, the evaluation rule is\n    recursive in nature;\n    that is, it includes, as one of its steps, the need to invoke the rule\n    itself.","1.1.3#footnote-link-1":"1","1.1.3#p2":"\n    Notice how succinctly the idea of recursion can be used to express\n    \n    what, in the case of a deeply nested combination, would otherwise be\n    viewed as a rather complicated process.  For example, evaluating\n    (* (+ 2 (* 4 6))\n   (+ 3 5 7)) \n    requires that the evaluation rule be applied to four different\n    combinations.  We can obtain a picture of this process by\n    representing the combination in the form of a\n    \n    tree, as shown in\n    \n\tfigure .\n      \n    Each combination is represented by a\n    \n    node with\n    \n    branches corresponding to the operator and the\n    operands of the combination stemming from it.\n    The\n    \n    terminal nodes (that is, nodes with\n    no branches stemming from them) represent either operators or numbers.\n    Viewing evaluation in terms of the tree, we can imagine that the\n    values of the operands percolate upward, starting from the terminal\n    nodes and then combining at higher and higher levels.  In general, we\n    shall see that recursion is a very powerful technique for dealing with\n    hierarchical, treelike objects.  In fact, the \"percolate values\n    upward\" form of the evaluation rule is an example of a general kind\n    of process known as\n    tree accumulation.\n\n    ","1.1.3#fig-":"","1.1.3#p3":"\n    Next, observe that the repeated application of the first step brings\n    us to the point where we need to evaluate, not combinations, but\n    primitive expressions such as \n    \n\tnumerals, built-in operators, or other names.  \n      \n    We take care of the primitive cases \n    \n    by stipulating that\n    \n        the values of numerals are the numbers that they name,\n\t\n            the values of built-in operators are the machine\n            instruction sequences that carry out the corresponding operations,\n\t    and\n          \n        the values of\n\t\n\t    other\n\t  \n\tnames are the objects associated\n        with those names in the environment.\n      \n        We may regard the second rule as a special case of the third one by\n        stipulating that symbols such as +\n        and * are also included\n        in the global environment, and are associated with the sequences of\n        machine instructions that are their \"values.\"\n    The key point to\n    notice is the role of the\n    \n    environment in determining the meaning of\n    the\n    \n\tsymbols\n      \n    in expressions.  In an interactive language such as\n    \n\tLisp,\n      \n    it is meaningless to speak of the value of an expression such as\n    (+ x 1)\n    without specifying any information about the environment\n    that would provide a meaning for the\n    \n\tsymbol x (or even for the\n        symbol +).\n      \n    As we shall see in chapter 3, the general notion of\n    the environment as providing a context in which evaluation takes place\n    will play an important role in our understanding of program execution.\n    ","1.1.3#p4":"\n    Notice that the\n    evaluation rule given above does not handle\n    \n\tdefinitions.\n      \n    For instance, evaluating\n    (define x 3)\n    does not apply\n    define\n    to two arguments, one\n    of which is the value of the\n    \n\tsymbol\n      x and the other of which is\n    3, since the purpose of the\n    define\n    is precisely to associate\n    x with a value.\n    (That is,\n    (define x 3)\n    is not\n    a combination.)\n  ","1.1.3#p5":"\n        Such exceptions to the general evaluation rule are called\n        special forms.\n        Define\n        is the only example of a special form that we\n        have seen so far, but we will meet others shortly.\n        \n        Each special form\n        has its own evaluation rule. The various kinds of expressions (each\n        with its associated evaluation rule) constitute the\n        \n        syntax of the\n        programming language.  In comparison with most other programming\n        languages, Lisp has a very simple syntax; that is, the evaluation rule\n        for expressions can be described by a simple general rule together\n        with specialized rules for a small number of special\n        forms.","1.1.3#footnote-link-2":"2","1.1.3#footnote-1":"It may seem strange that the evaluation\n    rule says, as part of the first step, that we should evaluate the leftmost\n    element of a combination, since at this point that can only be an operator\n    such as + or *\n    representing a built-in primitive procedure such as addition or\n    multiplication.  We will see later that it is useful to be able to work with\n    combinations whose operators are themselves compound expressions.\n  ","1.1.3#footnote-2":"\n        Special syntactic forms that are simply convenient\n        alternative surface structures for things that can be written in more\n        uniform ways are sometimes called syntactic sugar, to use a\n        \n        phrase coined by Peter Landin.  In comparison with users of other\n        languages, Lisp programmers, as a rule, are less concerned with\n        matters of syntax.  (By contrast, examine any Pascal manual and notice\n        how much of it is devoted to descriptions of syntax.)  This disdain\n        for syntax is due partly to the flexibility of Lisp, which makes it\n        easy to change surface syntax, and partly to the observation that many\n        \"convenient\" syntactic constructs, which make the language\n\tless uniform, end up causing more trouble than they are worth when\n\tprograms become large and complex.  In the words of Alan Perlis,\n\t\"Syntactic sugar causes cancer of the semicolon.\"","1.1.4":"1.1.4  \n    Compound","1.1.4#p1":"\n    We have identified in \n    Lisp\n    some of the elements that must appear in any powerful programming language:\n    \n        Numbers and arithmetic operations are primitive data and \n        procedures.\n        Nesting of combinations provides a means of combining operations.\n      \n        Constant declarations that associate names with values provide a\n        limited means of abstraction.\n      \n    Now we will learn about\n    procedure definitions,\n      \n    a much more powerful abstraction technique by which a compound\n    operation can be given a name and then referred to as a unit.\n  ","1.1.4#p2":"\n    We begin by examining how to express the idea of\n    \"squaring.\"\n    We might say,\n    \"To square something, multiply it by itself.\"\n    This is expressed in our language as \n    (define (square x) (* x x)) ","1.1.4#p3":"\n    We can understand this in the following way:\n    (define (square x)          (*          x      x))\n;;  ^      ^    ^            ^          ^      ^\n;;  To   square something,   multiply   it  by itself.\n    We have here a\n    compound procedure,\n      \n    which has been given the name square. The\n    procedure\n    represents the operation of multiplying something by itself. The thing to\n    be multiplied is given a local name, x, \n    which plays the same role that a pronoun plays in natural language.\n    \n    Evaluating the\n    \n\tdefinition\n      \n    creates this compound\n    \n\tprocedure\n      \n    and associates it with the name\n    square.","1.1.4#footnote-link-1":"1","1.1.4#p4":"\n        The general form of a procedure definition\n      \n    is\n    \n(define ($\\langle \\textit{name} \\rangle$ $\\langle\\textit{formal parameters}\\rangle$) $\\langle \\textit{body} \\rangle$)\n      \n    The\n    $\\langle \\textit{name}\\rangle$\n    is a symbol to be associated with the\n    \n\tprocedure\n      \n    definition in the environment.\n    The\n    $\\langle \\textit{formal parameters}\\rangle$\n    are the names used within the body of the\n    \n\tprocedure\n      \n    to refer to the\n    corresponding arguments of the\n    \n\tprocedure.\n      \n\tThe\n        $\\langle \\textit{body} \\rangle$\n        is an expression\n\tthat will yield the value of\n        the procedure\n\tapplication when the formal parameters are replaced by\n        the actual arguments to which the\n\tprocedure\n\tis applied.\n\tThe $\\langle \\textit{name} \\rangle$\n\tand the\n\t$\\langle \\textit{formal parameters} \\rangle$ \n\tare grouped within\n\t\n\tparentheses, just as they would be in an actual call to the procedure\n\tbeing defined.\n      ","1.1.4#footnote-link-2":"2","1.1.4#footnote-link-3":"3","1.1.4#p5":"\n        Having defined square, \n        we can now use it:\n      (square 21) (square (+ 2 5)) (square (square 3)) ","1.1.4#p6":"\n    We can also use square\n    as a building block in defining other\n    \n        procedures.\n      \n    For example, $x^2 +y^2$ can be expressed as\n    (+ (square x) (square y))\n    We can easily\n    define\n    a\n    \n\tprocedure\n        sum-of-squares\n    that, given any two numbers as arguments, produces the\n    sum of their squares:\n    (define (sum-of-squares x y)\n  (+ (square x) (square y)))\n\n(sum-of-squares 3 4) (sum-of-squares 3 4) \n    Now we can use\n    sum-of-squares\n    as a building block in constructing further\n    \n\tprocedures:\n      (define (f a)\n  (sum-of-squares (+ a 1) (* a 2))) (f 5) ","1.1.4#p7":"\n        Compound\n        \n\tprocedures are used in exactly the same way as primitive\n        procedures. Indeed, one could not tell by looking at the definition\n        of sum-of-squares given above whether \n        square was built into\n        the interpreter, like + and \n        *, \n        or defined as a compound procedure.\n      ","1.1.4#footnote-1":"Observe that there are two\n    different operations being combined here: we are creating the\n    \n\tprocedure,\n      \n    and we are giving\n    it the name square. It is possible, indeed\n    important, to be able to separate these two notions—to create\n    \n\tprocedures\n      \n    without naming them, and to give names to\n    \n\tprocedures\n      \n    that have already been created. We will see how to do this in\n    section 1.3.2.","1.1.4#footnote-2":"Throughout this book, we will\n    \n    describe the general syntax of expressions by using italic\n    symbols delimited by angle\n    brackets—e.g.,\n    $\\langle \\textit{name}\\rangle$—to\n      \n    denote the \"slots\" in the expression to be filled in \n    when such an expression is actually used.","1.1.4#footnote-3":"More\n        \n        generally, the body of the procedure can be a sequence of expressions.\n        In this case, the interpreter evaluates each expression in the\n        sequence in turn and returns the value of the final expression as the\n        value of the procedure application.","1.1.5":"1.1.5  \n    The Substitution Model for \n    \n      \n      \n    \n    Application","1.1.5#p1":"\n        To evaluate a combination whose operator names a compound procedure, the\n        interpreter follows much the same process as for combinations whose\n        operators name primitive procedures, which we described in\n        section 1.1.3.\n      \n    That is, the interpreter evaluates the elements of the\n    \n\tcombination\n      \n    and\tapplies the\n    \n\tprocedure\n      \n    (which is the value of the\n    \n\toperator of the combination)\n      \n    to the arguments (which are the values of the\n    \n\toperands of the combination).\n      ","1.1.5#p2":"\n    We can assume that the mechanism for applying primitive\n\tprocedures\n    to arguments is built into the interpreter.\n      \n    For compound\n    \n\tprocedures,\n      \n    the application process is as follows:\n    \n        To apply a compound\n\t\n\t    procedure\n\t  \n\tto arguments, \n\t\n\t    evaluate the body of the procedure\n\t  \n\twith each\n\tformal\n\tparameter replaced by the corresponding argument.\n    To illustrate this process, let's evaluate the\n    \n\tcombination\n      (f 5) \n    where f is the\n    \n\tprocedure defined\n      \n    in section 1.1.4.\n    We begin by retrieving the\n    \n\tbody\n      \n    of f:\n    (sum-of-squares (+ a 1) (* a 2))\n    Then we replace the parameter a \n    by the argument 5:\n    (sum-of-squares (+ 5 1) (* 5 2))\n    Thus the problem reduces to the evaluation of\n    \n\ta combination\n      \n    with two\n    \n\toperands\n      \n    and \n    \n\tan operator sum-of-squares.  \n      \n    Evaluating this\n    \n\tcombination\n      \n    involves three subproblems.  We must evaluate the\n    operator\n    to get the\n    \n\tprocedure\n      \n    to be applied, and we must evaluate the\n    \n\toperands \n      \n    to get the arguments. Now\n    (+ 5 1)\n    produces 6 and\n    (* 5 2)\n    produces 10, so we must apply the\n    sum-of-squares procedure\n      \n    to 6 and 10. These values are substituted for the\n    formal\n    parameters x and \n    y in the body of \n    sum-of-squares,\n      \n    reducing the expression to\n    (+ (square 6) (square 10))\n    If we use the\n    \n\tdefinition\n      \n    of square, this reduces to\n    (+ (* 6 6) (* 10 10))\n    which reduces by multiplication to\n    (+ 36 100)\n    and finally to\n    136","1.1.5#footnote-link-1":"1","1.1.5#p3":"\n    The process we have just described is called the substitution\n    model for \n    procedure \n    application.  It can be taken as a model that\n    determines the \"meaning\" of \n    procedure \n    application, insofar as the\n    procedures \n    in this chapter are concerned.  However, there are two\n    points that should be stressed:\n    \n        The purpose of the substitution is to help us think about\n        procedure \n        application, not to provide a description of how the interpreter\n\treally works.  Typical interpreters do not evaluate\n        procedure  \n        applications by manipulating the text of a \n        procedure to substitute values for the formal \n        parameters. In practice, the \"substitution\" is\n\taccomplished by using a local environment for the\n        formal\n\t  \n\tparameters. We will discuss this more fully in chapters 3 and\n        4 when we examine the implementation of an interpreter in detail.\n      \n        Over the course of this book, we will present a sequence of\n\tincreasingly elaborate models of how interpreters work, culminating\n        with a complete implementation of an interpreter and compiler in\n        chapter 5.  The substitution model is only the first of\n\tthese models—a way to get started thinking formally\n\tabout the evaluation process.  In general, when \n        \n        modeling phenomena in science and engineering, we begin with\n\tsimplified, incomplete models. As we examine things in greater detail,\n\tthese simple models become inadequate and must be replaced by more\n\trefined models. The substitution model is no exception.  In particular,\n\twhen we address in chapter 3 the use of\n        procedures \n        with \"mutable data,\" we will see that the substitution\n\tmodel breaks down and must be replaced by a more complicated model of\n        procedure \n        application.","1.1.5#footnote-link-2":"2","1.1.5#h1":"Applicative order versus normal order","1.1.5#p4":"\n    According to the description of evaluation given in\n    \n\tsection 1.1.3,\n      \n    the interpreter first evaluates the\n    \n\toperator\n      \n    and\n    \n\toperands\n      \n    and then applies the resulting\n    \n\tprocedure\n      \n    to the resulting arguments. This is not the only way to perform evaluation.\n    An alternative evaluation model would not evaluate the\n    \n\toperands\n      \n    until their values were needed.  Instead it would first substitute\n    \n\toperand\n      \n    expressions for parameters until it obtained an expression involving\n    only \n    primitive operators,\n    and would then perform the evaluation.  If we\n    used this method, the evaluation of\n    (f 5)\n    would proceed according to the sequence of expansions\n    (sum-of-squares (+ 5 1) (* 5 2))\n\n(+  (square (+ 5 1))     (square (* 5 2))  )\n\n(+  (* (+ 5 1) (+ 5 1))  (* (* 5 2) (* 5 2)))\n    followed by the reductions\n    (+        (* 6 6)            (* 10 10))\n\n(+          36                  100)\n\n                    136\n    This gives the same answer as our previous evaluation model, but the\n    process is different.  In particular, the evaluations of\n    (+ 5 1)\n    and \n    (* 5 2)\n    are each performed twice here, corresponding to the reduction of the\n    expression\n    (* x x)\n    with x replaced respectively by\n    (+ 5 1)\n    and\n    (* 5 2).\n      ","1.1.5#p5":"\n    This alternative \"fully expand and then reduce\"\n    evaluation method is known as \n    normal-order evaluation, in contrast to the \"evaluate\n    the arguments and then apply\" method that the interpreter actually\n    uses, which is called \n    applicative-order evaluation.  It can be shown that, for \n    procedure\n    applications that can be modeled using substitution (including all the \n    procedures\n    in the first two chapters of this book) and that yield legitimate values,\n    normal-order and applicative-order evaluation produce the same value.\n    (See exercise 1.5\n    for an instance of an \"illegitimate\" value where normal-order\n    and applicative-order evaluation do not give the same result.)\n  ","1.1.5#p6":"\n\tLisp\n        \n    uses applicative-order evaluation, partly because of the\n    additional efficiency obtained from avoiding multiple evaluations of\n    expressions such as those illustrated with\n    (+ 5 1) \n\tand (* 5 2)\n    above and, more significantly, because normal-order evaluation\n    becomes much more complicated to deal with when we leave the realm of\n    procedures\n    that can be modeled by substitution.  On the other hand,\n    normal-order evaluation can be an extremely valuable tool, and we will\n    investigate some of its implications in chapters 3 and 4.","1.1.5#footnote-link-3":"3","1.1.5#footnote-1":"If the\n\tbody of the function is a sequence of statements, the\n\tbody is evaluated with the parameters replaced, and the value of the\n\tapplication is the value of the return expression of the first\n\treturn statement encountered.","1.1.5#footnote-2":"Despite the\n        simplicity of the substitution idea, it turns out to be\n\tsurprisingly complicated to give a rigorous mathematical\n\tdefinition of the substitution process.  The problem arises\n\tfrom the possibility of confusion between the names used for the \n        formal parameters of a procedure \n        and the (possibly identical) names used in the expressions\n        to which the \n        procedure \n        may be applied.  Indeed, there is a long\n        history of erroneous definitions of substitution in the\n        literature of logic and programming semantics.  \n        \n        See Stoy 1977 for a\n        careful discussion of substitution.","1.1.5#footnote-3":"In\n    chapter 3 we will introduce stream processing, which is a\n    way of handling apparently \"infinite\" data structures\n    by incorporating a limited form of normal-order evaluation.  In\n    section 4.2 we will modify the\n    Scheme\n    interpreter to produce a normal-order variant of\n    Scheme.","1.1.6":"1.1.6  \n    Conditional Expressions and Predicates","1.1.6#p1":"\n    The expressive power of the class of\n    \n\tprocedures\n      \n    that we can define at this point is very limited, because we have no way to\n    make tests and to perform different operations depending on the result of a\n    test.\n    \n        For instance, we cannot define a procedure that computes the \n        \n        absolute\n        value of a number by testing whether the number is positive, negative,\n        or zero and taking different actions in the different cases according\n        to the rule\n        \n      \\[\\begin{array}{lll}\n          |x| & = & \\left\\{ \\begin{array}{rl}\n          x   & \\text{if $x>0$} \\\\\n          0   & \\text{if $x=0$} \\\\\n          -x  & \\text{if $x<0$}\n          \\end{array}\n          \\right.\n      \\end{array}\\]\n        \n        This construct is called a\n        case analysis, and\n        there is a special form in Lisp for notating such a case\n        analysis.  It is called \n        cond (which stands for\n        \"conditional\"), and it is used as follows:\n        (define (abs x)\n  (cond ((> x 0) x)\n        ((= x 0) 0)\n        ((< x 0) (- x)))) (abs (- 5))\n        The general form of a conditional expression is\n        \n(cond ($\\langle p_1 \\rangle $ $\\langle e_1 \\rangle$)\n      ($\\langle p_2 \\rangle $ $\\langle e_2 \\rangle$)\n      $\\vdots$\n      ($\\langle p_n \\rangle $ $\\langle e_n \\rangle$))\n          \n        consisting of the symbol cond followed by\n        \n        parenthesized pairs of expressions\n\t($\\langle p\\ e \\rangle$)\n        called \n        clauses. The first expression in each pair is a \n        predicate—that is, an expression whose value is\n\tinterpreted as either true or false.","1.1.6#footnote-link-1":"1","1.1.6#p2":"\n        Conditional expressions are\n        \n\tevaluated as follows.  The predicate\n        $\\langle p_1 \\rangle$ is evaluated first.\n\tIf its value is false, then\n        $\\langle p_2 \\rangle$ is evaluated.  \n        If $\\langle p_2 \\rangle$'s \n        value is also false, then\n        $\\langle p_3 \\rangle$ is evaluated.  \n        This process continues until a predicate is\n        found whose value is true, in which case the interpreter returns the\n        value of the corresponding \n        consequent expression$\\langle e \\rangle$ of the\n        clause as the value of the conditional expression.  If none of the\n        $\\langle p \\rangle$'s\n\tis found to be true, the value of the cond\n\tis undefined.\n      ","1.1.6#p3":"\n        The word\n        predicate is used for procedures that return true\n        or false, as well as for expressions that evaluate to true or false.\n        The absolute-value procedure abs makes use of the \n        \n        primitive predicates >, \n        <, and \n        =. These take two numbers as arguments and\n\ttest whether the first number is, respectively, greater than, less than,\n\tor equal to the second number, returning true or false accordingly.\n      ","1.1.6#footnote-link-2":"2","1.1.6#p4":"\n        Another way to write the absolute-value procedure is\n        (define (abs x)\n  (cond ((< x 0) (- x))\n    (else x))) \n        which could be expressed in English as \n        \"If $x$ is less than zero\n        return $- x$; otherwise return \n        $x$.\"Else is a special symbol that can be used\n\tin place of the $\\langle p \\rangle$ in the\n\tfinal clause of a cond.  \n        This causes the cond to return as its value \n        the value of the corresponding\n\t$\\langle e \\rangle$ \n        whenever all previous clauses have been bypassed. In fact, any\n\texpression that always evaluates to a true value could be used as the\n\t$\\langle p \\rangle$ here.\n      ","1.1.6#p5":"\n        Here is yet another way to write the absolute-value procedure:\n        (define (abs x)\n  (if (< x 0)\n    (- x)\n    x)) \n        This uses the\n        \n\tspecial form if, a restricted \n        type of conditional that can be used when there are precisely \n        \n        two cases in the case\n        analysis.  The general form of an if\n\texpression is\n        \n(if $\\langle \\textit{predicate} \\rangle$ $\\langle \\textit{consequent}\\rangle$ $\\langle \\textit{alternative}\\rangle$)\n          \n        To\n        \n\tevaluate an if expression, \n        the interpreter starts by evaluating the \n        $\\langle \\textit{predicate}\\rangle$ \n        part of the expression.  \n        If the $\\langle \\textit{predicate}\\rangle$ \n        evaluates to a true value, the interpreter then evaluates the \n        $\\langle \\textit{consequent}\\rangle$ \n        and returns its value.  Otherwise it evaluates the \n        $\\langle \\textit{alternative}\\rangle$ \n        and returns its value.","1.1.6#footnote-link-3":"3","1.1.6#p6":"\n        In addition to primitive predicates such as\n\t<,\n\t=, and >,\n\tthere are logical composition operations, which enable us to construct\n\tcompound predicates.  The three most frequently used are these:\n        (and$\\langle e_1\\rangle \\ldots \\langle e_n \\rangle$ \t    )\n            The interpreter evaluates the expressions\n\t    $\\langle e \\rangle$ one at a time, \n            in left-to-right order.  If any\n\t    $\\langle e \\rangle$ evaluates to false, \n            the value of the and expression is\n\t    false, and the rest of the\n\t    $\\langle e \\rangle$'s \n            are not evaluated. If all\n\t    $\\langle e \\rangle$'s\n\t    evaluate to true values, the value of the\n\t    and\n            expression is the value of the last one.\n          (or$\\langle e_1 \\rangle\\ldots \\langle e_n \\rangle$)\n            The interpreter evaluates the expressions \n            $\\langle e \\rangle$ one at a time, in\n\t    left-to-right order.  If any\n\t    $\\langle e \\rangle$\n\t    evaluates to a true value, that value is returned as the value of the\n\t    or expression, and the rest of the\n\t    $\\langle e \\rangle$'s are not\n\t    evaluated. If all\n\t    $\\langle e \\rangle$'s\n\t    evaluate to false, the value of the\n\t    or expression is false.\n          (not$\\langle e \\rangle$) \t    \n            The value of a not expression is true\n            when the expression $\\langle e \\rangle$ \n            evaluates to false, and false otherwise.\n          \n        Notice that and and\n\tor are special forms, not procedures,\n        because the subexpressions are not necessarily all evaluated.\n        Not is an ordinary procedure.\n      ","1.1.6#p7":"\n        As an example of how these are used, the condition that a \n        number $x$ be in the range\n\t$5 < x < 10$ may be expressed as\n        (and (> x 5) (< x 10))","1.1.6#p8":"\n    As another example, we can\n    \n\tdefine\n      \n    a predicate to test whether one number is \n    \n    greater than or equal to another as    \n    (define (>= x y)\n  (or (> x y) (= x y))) \n    or alternatively as\n    (define (>= x y)\n  (not (< x y))) ","1.1.6#ex-1.1":"\n    Below is a sequence of\n    \n\texpressions.\n      \n    What is the result printed by the interpreter in response to each\n    \n        expression?\n      \n    Assume that the sequence is to be evaluated in the order\n    in which it is presented.\n    10 (+ 5 3 4) (- 9 1) (/ 6 2) (+ (* 2 4) (- 4 6)) (define a 3) (define b (+ a 1)) (+ a b (* a b)) (= a b) (if (and (> b a) (< b (* a b)))\n  b\n  a) (cond ((= a 4) 6)\n      ((= b 4) (+ 6 7 a))\n      (else 25)) (+ 2 (if (> b a) b a)) (* (cond ((> a b) a)\n         ((< a b) b)\n         (else -1))\n   (+ a 1)) ","1.1.6#ex-1.2":"\n    Translate the following expression into \n    \n        prefix form\n      \n      $\\begin{array}{l}\n      \\quad~~\\dfrac{5+4+\\left(2-\\left(3-(6+\\frac{4}{5})\\right)\\right)}{3 (6-2) (2-7)}\n      \\end{array}$\n    ","1.1.6#ex-1.3":"\n        Define a procedure\n      \n    that takes three numbers as arguments and returns\n    the sum of the squares of the two larger numbers.\n    ","1.1.6#ex-1.4":"\n\tObserve that our model of evaluation allows for\n\t\n\tcombinations whose\n\toperators are compound expressions.  Use this observation to describe\n\tthe behavior of the following procedure:\n\t(define (a-plus-abs-b a b)\n  ((if (> b 0) + -) a b)) ","1.1.6#ex-1.5":"\n    Ben Bitdiddle has invented a test to determine whether the interpreter\n    he is faced with is using\n    \n    applicative-order evaluation or normal-order\n    evaluation. He \n    \n\tdefines the following two procedures:\n      (define (p) (p))\n\n(define (test x y)\n  (if (= x 0)\n    0\n    y)) \n      Then he evaluates the\n      expression(test 0 (p)) \n      What behavior will Ben observe with an interpreter that uses\n      applicative-order evaluation?  What behavior will he observe with an\n      interpreter that uses normal-order evaluation?  Explain your answer.\n      \n      (Assume that the evaluation rule for \n      \n          the special form if\n      is the same whether the interpreter is using normal or applicative order:\n      The predicate expression is evaluated first, and the result determines\n      whether to evaluate the consequent or the alternative expression.)\n      ","1.1.6#footnote-1":"\"Interpreted as either\n         \n\ttrue or false\"\n        means this: In Scheme, there are two distinguished values that are\n        denoted by the constants \n        #t and #f.  \n        When the interpreter checks a predicate's value, it interprets \n        #f as false.  Any other value\n        is treated as true.  (Thus, providing #t\n\tis logically unnecessary, but it is convenient.)  In this book we will\n\tuse names true and\n\tfalse, \n        which are associated with the values #t and \n        #f respectively.","1.1.6#footnote-2":"Abs also uses\n        \n        the \"minus\" operator -, \n        which, when used with a single\n        operand, as in (- x), \n        indicates negation.","1.1.6#footnote-3":"A minor \n        difference\n        \n\tbetween if and \n        cond is that the\n        $\\langle e \\rangle$ \n        part of each cond \n        clause may be a sequence of expressions.\n        If the corresponding $\\langle p \\rangle$ \n        is found to be true, \n        the expressions \n        $\\langle e \\rangle$ \n        are evaluated in sequence and the value of the final\n        expression in the sequence is returned as the value of the \n        cond.\n        In an if expression, however, \n        the $\\langle \\textit{consequent}\\rangle$ and\n        $\\langle \\textit{alternative}\\rangle$ must\n\tbe single expressions.","1.1.7":"1.1.7  Example: Square Roots by Newton's Method","1.1.7#p1":"Procedures,\n    as introduced above, are much like ordinary mathematical functions.  They\n    specify a value that is determined by one or more parameters.  But there\n    is an important difference between mathematical functions and computer\n    procedures.\n        Procedures\n      \n    must be effective.\n  ","1.1.7#p2":"\n    As a case in point, consider the problem of computing square\n    roots.  We can define the square-root function as\n    \n      \\[\n      \\sqrt{x}\\ =\\text{ the }y\\text{ such that }y \\geq 0\\text{ and }\n      y^2\\ =\\ x\n      \\]\n    \n    This describes a perfectly legitimate mathematical function.  We could\n    use it to recognize whether one number is the square root of another, or\n    to derive facts about square roots in general.  On the other hand, the\n    definition does not describe a\n    procedure.\n    Indeed, it tells us almost nothing about how to actually find the square\n    root of a given number.  It will not help matters to rephrase this\n    definition in\n    pseudo-Lisp:\n(define (sqrt x)\n  (the y (and (>= y 0)\n              (= (square y) x))))\n      \n    This only begs the question.\n  ","1.1.7#p3":"\n    The contrast between\n    \n        function and procedure\n      \n    is a reflection of the general distinction between describing properties of\n    things and describing how to do things, or, as it is sometimes referred to,\n    the distinction between\n    \n    declarative knowledge and imperative knowledge. In\n    \n    mathematics we are usually concerned with declarative (what is)\n    descriptions, whereas in computer science we are usually concerned\n    with imperative (how to) descriptions.","1.1.7#footnote-link-1":"1","1.1.7#p4":"\n    How does one compute\n    \n    square roots?  The most common way is to use\n    Newton's method of successive approximations, which says that whenever\n    we have a guess $y$ for the value of the square\n    root of a number $x$, we can perform a simple\n    manipulation to get a better guess (one closer to the actual square root)\n    by averaging $y$ with\n    $x/y$.\n    For example, we can compute the square root of 2 as follows.  Suppose our\n    initial guess is 1:\n    \n      \\[\n      \\begin{array}{lll}\n      \\textrm{Guess} & \\textrm{Quotient} & \\textrm{Average}\\\\[1em]\n      1      & {\\displaystyle \\frac{2}{1} = 2}           & {\\displaystyle \\frac{(2+1)}{2} = 1.5} \\\\[1em]\n      1.5    & {\\displaystyle \\frac{2}{1.5} = 1.3333}    & {\\displaystyle \\frac{(1.3333+1.5)}{2} = 1.4167} \\\\[1em]\n      1.4167 & {\\displaystyle \\frac{2}{1.4167} = 1.4118} & {\\displaystyle \\frac{(1.4167+1.4118)}{2} = 1.4142} \\\\[1em]\n      1.4142 & \\ldots & \\ldots\n      \\end{array}\n      \\]\n    \n    Continuing this process, we obtain better and better approximations to the\n    square root.\n  ","1.1.7#footnote-link-2":"2","1.1.7#p5":"\n    Now let's formalize the process in terms of functions.  We start with\n    a value for the\n    \n    radicand (the number whose square root we are trying to compute) and a value\n    for the guess.  If the guess is good enough for our purposes, we are done;\n    if not, we must repeat the process with an improved guess.  We write this\n    basic strategy as a\n    \n        procedure:\n      (define (sqrt-iter guess x)\n  (if (good-enough? guess x)\n      guess\n      (sqrt-iter (improve guess x) x))) \n    A guess is improved by averaging it with the quotient of the radicand and\n    the old guess:\n    (define (improve guess x)\n  (average guess (/ x guess))) \n    where\n    (define (average x y)\n  (/ (+ x y) 2)) \n    We also have to say what we mean by \"good enough.\"  The\n    following will do for illustration, but it is not really a very good\n    test.  (See exercise 1.7.)\n    The idea is to improve the answer until it is close enough so that its\n    square differs from the radicand by less than a predetermined\n    tolerance (here 0.001):(define (good-enough? guess x)\n  (< (abs (- (square guess) x)) 0.001)) \n    Finally, we need a way to get started.  For instance, we can always guess\n    that the square root of any number\n    is 1:(define (sqrt x)\n  (sqrt-iter 1.0 x)) \n    If we type these\n    \n\tdefinitions\n      \n    to the interpreter, we can use sqrt\n    just as we can use any\n    \n        procedure:\n      (sqrt 9) (sqrt (+ 100 37)) (sqrt (+ (sqrt 2) (sqrt 3))) (square (sqrt 1000)) ","1.1.7#footnote-link-3":"3","1.1.7#footnote-link-4":"4","1.1.7#p6":"\n    The sqrt program also illustrates that the\n    simple\n    \n\tprocedural\n      \n    language we have introduced so far is sufficient for writing any purely\n    numerical program that one could write in, say, C or Pascal.  This might\n    seem surprising, since we have not included in our language any iterative\n    \n    (looping) constructs that direct the computer to do something over and over\n    again.\n    Sqrt-iter,\n      \n    on the other hand, demonstrates how iteration can be accomplished using no\n    special construct other than the ordinary ability to call a\n    procedure.","1.1.7#footnote-link-5":"5","1.1.7#ex-1.6":"\n        Alyssa P. Hacker doesn't see why if\n\tneeds to be provided as a\n        \n\tspecial form.  \"Why can't I just\n\tdefine it as an ordinary procedure in terms of\n        cond?\" she asks.\n        Alyssa's friend Eva Lu Ator claims this can indeed be done, and\n        she defines a new version of if:\n        (define (new-if predicate then-clause else-clause)\n  (cond (predicate then-clause)\n        (else else-clause))) \n        Eva demonstrates the program for Alyssa:\n        (new-if (= 2 3) 0 5) (new-if (= 1 1) 0 5) \n        Delighted, Alyssa uses new-if to rewrite\n\tthe square-root program:\n        (define (sqrt-iter guess x)\n  (new-if (good-enough? guess x)\n          guess\n          (sqrt-iter (improve guess x)\n                     x))) \n        What happens when Alyssa attempts to use this to compute square roots?\n        Explain.\n        ","1.1.7#ex-1.7":"\n    The\n    good-enough?\n    test used in computing square roots will not be very effective for finding\n    the square roots of very small numbers. Also, in real computers, arithmetic\n    operations are almost always performed with limited precision.  This makes\n    our test inadequate for very large numbers.  Explain these statements, with\n    examples showing how the test fails for small and large numbers.  An\n    alternative strategy for implementing\n    good-enough?\n    is to watch how guess changes from one\n    iteration to the next and to stop when the change is a very small fraction\n    of the guess.  Design a square-root\n    \n        procedure\n      \n    that uses this kind of end test.  Does this work better for small and\n    large numbers?\n    ","1.1.7#ex-1.8":"\n    Newton's method for\n    \n    cube roots is based on the fact that if\n    $y$ is an\n    approximation to the cube root of $x$, then a better approximation is\n    given by the value\n    \n      \\[\n      \\begin{array}{lll}\n      \\dfrac{x/y^{2}+2y} {3}\n      \\end{array}\n      \\]\n    \n    Use this formula to implement a cube-root\n    \n\tprocedure\n      \n    analogous to the square-root\n    procedure.\n    (In section 1.3.4 we will see how to\n    implement Newton's method in general as an abstraction of these\n    square-root and cube-root\n    procedures.)","1.1.7#footnote-1":"Declarative and\n    imperative descriptions are intimately related, as indeed are\n    mathematics and computer science.  For instance, to say that the\n    answer produced by a program is\n    \"correct\" is to make a declarative statement about the program.\n    There is a large amount of research aimed at establishing techniques for\n    \n    proving that programs are correct, and much of the technical difficulty of\n    this subject has to do with negotiating the transition between imperative\n    statements (from which programs are constructed) and declarative statements\n    (which can be used to deduce things).\n    \n\tIn a related vein, an important\n\tcurrent area in programming-language design is the exploration of so-called\n\t\n\tvery high-level languages, in which one actually programs in terms of\n\tdeclarative statements.\n      \n    The idea is to make interpreters sophisticated\n    enough so that, given \"what is\" knowledge specified by the\n    programmer, they can generate \"how to\" knowledge automatically.\n    This cannot be done in general, but there are important areas where progress\n    has been made.  We shall revisit this idea in chapter 4.","1.1.7#footnote-2":"This square-root algorithm is\n    actually a special case of Newton's method, which is a general\n    technique for finding roots of equations.  The square-root algorithm itself\n    was developed by Heron of\n    \n    Alexandria in the first century CE.  We will see how to express\n    the general Newton's method as a\n    \n        Lisp procedure\n      \n    in section 1.3.4.","1.1.7#footnote-3":"We will usually give\n     \n    predicates names ending with question marks, to help us remember that they\n    are predicates.  This is just a stylistic convention.  As far as the\n    interpreter is concerned, the question mark is just an ordinary\n    character.","1.1.7#footnote-4":"Observe that we express\n    our initial guess as 1.0 rather than 1.  This would not make any difference\n    in many Lisp implementations.\n    \n    MIT Scheme, however, distinguishes between exact integers and decimal values,\n    and dividing two integers produces a rational number rather than a decimal.\n    For example, dividing 10 by 6 yields 5/3, while dividing 10.0 by 6.0 yields\n    1.6666666666666667.  (We will learn how to implement arithmetic on rational\n    numbers in section 2.1.1.)  If we start with an\n    initial guess of 1 in our square-root program, and\n    $x$ is an exact integer, all subsequent values\n    produced in the square-root computation will be rational numbers rather than\n    decimals.  Mixed operations on rational numbers and decimals always yield\n    decimals, so starting with an initial guess of 1.0 forces all subsequent\n    values to be decimals.","1.1.7#footnote-5":"Readers who are worried about the efficiency issues involved in using\n    \n\tprocedure\n      \n    calls to implement iteration should note the remarks on \"tail\n    recursion\" in\n    section 1.2.1.","1.1.8":"1.1.8  \n    \n      \n      \n    \n    as Black-Box Abstractions","1.1.8#p1":"Sqrt\n    is our first example of a process defined by a set of mutually \n    defined procedures.\n    Notice that the \n    definition of sqrt-iter\n    is\n    recursive; that is, the\n    \n        procedure\n      \n    is defined in terms of itself.  The idea of being able to define a\n    \n        procedure\n      \n    in terms of itself may be disturbing; it may seem unclear how such a\n    \"circular\" definition could make sense at all, much less\n    specify a well-defined process to be carried out by a computer.  This will\n    be addressed more carefully in\n    section 1.2.  But first\n    let's consider some other important points illustrated by the\n    sqrt example.\n  ","1.1.8#p2":"\n    Observe that the problem of computing square roots breaks up naturally\n    into a number of subproblems:\n    \n    how to tell whether a guess is good\n    enough, how to improve a guess, and so on.  Each of these tasks is\n    accomplished by a separate\n    procedure.\n    The entire sqrt program can be viewed as a\n    cluster of\n    \n        procedures\n\t(shown in figure )\n      \n    that mirrors the decomposition of the problem into subproblems.\n  ","1.1.8#fig-":"","1.1.8#p3":"\n    The importance of this\n    \n    decomposition strategy is not simply that one\n    is dividing the program into parts.  After all, we could take any\n    large program and divide it into parts—the first ten lines, the next\n    ten lines, the next ten lines, and so on.  Rather, it is crucial that\n    each\n    \n        procedure\n      \n    accomplishes an identifiable task that can be used as a module in defining\n    other\n    procedures.\n    For example, when we define the\t      \n    good-enough? procedure\n      \n    in terms of square, we are able to\n    regard the square\n        procedure\n      \n    as a\n    \"black box.\"  We are not at that moment concerned with\n    how the\n    \n        procedure\n      \n    computes its result, only with the fact that it computes the\n    square.  The details of how the square is computed can be suppressed,\n    to be considered at a later time.  Indeed, as far as the\n    good-enough? procedure\n      \n    is concerned, square is not quite a\n    \n        procedure\n      \n    but rather an abstraction of a\n    procedure,\n    a so-called\n    procedural abstraction.\n      \n    At this level of abstraction, any\n    \n        procedure\n      \n    that computes the square is equally good.\n  ","1.1.8#p4":"\n    Thus, considering only the values they return, the following two\n    \n        procedures\n      \n    squaring a number should be indistinguishable. Each takes a numerical\n    argument and produces the square of that number as the value.(define (square x) (* x x)) (define (square x)\n  (exp (double (log x))))\n\n(define (double x) (+ x x)) ","1.1.8#footnote-link-1":"1","1.1.8#p5":"\n    So a\n    \n        procedure\n      \n    should be able to suppress detail.  The users of the\n    \n        procedure\n      \n    may not have written the\n    \n        procedure\n      \n    themselves, but may have obtained it from another programmer as a\n    black box. A user should not need to know how the\n    \n        procedure\n      \n    is implemented in order to use it.\n    ","1.1.8#h1":"Local names","1.1.8#p6":"\n    One detail of a\n    procedure's\n    implementation that should not matter to the user of the\n    \n        procedure\n      \n    is the implementer's choice of names for the\n    procedure's formal parameters.\n    Thus, the following\n    \n        procedures\n      \n    should not be distinguishable:\n    (define (square x) (* x x)) (define (square y) (* y y)) \n    This principle—that the meaning of a\n    \n        procedure\n      \n    should be independent of the parameter names used by its\n    author—seems on the surface to be self-evident, but its\n    consequences are profound.  The simplest consequence is that the\n    parameter names of a\n    \n        procedure\n      \n    must be local to the body of the\n    \n        procedure.\n      \n    For example, we used square\n\tin the definition of\t      \n        good-enough?\n    in our square-root\n    \n        procedure:\n      (define (good-enough? guess x)\n  (< (abs (- (square guess) x)) 0.001)) \n    The intention of the author of\n    good-enough?\n    is to determine if the square of the first argument is within a given\n    tolerance of the second argument.  We see that the author of\n    good-enough?\n    used the name guess to refer to the\n    first argument and x to refer to the\n    second argument.  The argument of square\n    is guess.  If the author of\n    square used x\n    (as above) to refer to that argument, we see that the\n    x in\n    good-enough?\n    must be a different x than the one\n    in  square. Running the\n    \n        procedure\n      square must not affect the value\n    of x that is used by\n    good-enough?,\n      \n    because that value of x may be needed by\n    good-enough?\n    after square is done computing.\n  ","1.1.8#p7":"\n    If the parameters were not local to the bodies of their respective\n    procedures,\n    then the parameter x in\n    square could be confused with the parameter\n    x in\n    good-enough?,\n      \n    and the behavior of\n    good-enough?\n    would depend upon which version of square\n    we used.  Thus, square would not be the\n    black box we desired.\n  ","1.1.8#p8":"\n    A     \n    \n\tformal parameter of a procedure\n      \n    has a very special role in the\n    \n        procedure definition, \n      \n    in that it doesn't matter what name the\n    \n\tformal\n      \n    parameter has.  Such a name is called\n    \n\ta bound variable, and we say that the procedure definition\n      binds its\n    \n\tformal parameters.\n      \n    The meaning of a\n    \n        procedure definition is unchanged if a bound variable\n      \n    is consistently renamed throughout the\n    definition.\n    If a\n    variable\n    is not bound, we say that it is\n    free.  The set of\n    expressions\n    for which a binding\n    defines\n    a name is called the\n    scope of that name. In a\n    \n        procedure definition, the bound variables\n      \n    declared as the\n    \n        formal parameters of the procedure\n      \n    have the body of the\n    \n        procedure\n      \n    as their scope.\n  ","1.1.8#footnote-link-2":"2","1.1.8#p9":"\n    In the\n    \n\tdefinition of good-enough?\n    above,\n    guess and\n    x are\n    bound\n    \n\tvariables\n      \n    but\n    <,\n        -,\n      abs\n    and square are free.\n    The meaning of\n    good-enough?\n    should be independent of the names we choose for\n    guess and\n    x so long as they are distinct and\n    different from\n    <,\n        -,\n      abs\n    and square.  (If we renamed\n    guess to\n    abs we would have introduced a bug by\n    capturing the\n    \n\tvariable\n      abs.\n    It would have changed from free to bound.)  The meaning of\n    good-enough?\n    is not independent of the \n    \n\tnames of its free variables,\n      \n    however. It surely depends upon the fact\n    \n\t(external to this definition)\n      \n\tthat the symbol abs names a procedure\n      \n    for computing the absolute value of a number.\n    Good-enough?\n    will compute a different function if we substitute\n    cos\n    for abs in its\n    \n\tdefinition.\n      ","1.1.8#h2":"Internal\n    definitions\n    and block structure","1.1.8#p10":"\n    We have one kind of name isolation available to us so far:\n    \n\tThe formal parameters of a procedure\n      \n    are local to the body of the\n    procedure.\n    The square-root program illustrates another way in which we would like to\n    control the use of names.\n    \n    The existing program consists of separate\n    \n        procedures:\n      (define (sqrt x)\n  (sqrt-iter 1.0 x))\n\n(define (sqrt-iter guess x)\n  (if (good-enough? guess x)\n      guess\n      (sqrt-iter (improve guess x) x)))\n\n(define (good-enough? guess x)\n  (< (abs (- (square guess) x)) 0.001))\n\n(define (improve guess x)\n  (average guess (/ x guess))) ","1.1.8#p11":"\n    The problem with this program is that the only\n    \n        procedure\n      \n    that is important to users of sqrt is\n    sqrt.  The other\n    \n        procedures\n      \n\t(sqrt-iter,\n\tgood-enough?,\n      \n    and improve) only clutter up their minds.\n    They may not \n    \n\tdefine any other procedure\n      \n    called\n    good-enough?\n    as part of another program to work together\n    with the square-root program, because sqrt\n    needs it.  The problem is especially severe in the construction of large\n    systems by many separate programmers.  For example, in the construction\n    of a large library of numerical\n    procedures,\n    many numerical functions are computed as successive approximations and\n    thus might have\n    \n        procedures\n      \n    named\n    good-enough?\n    and improve as auxiliary\n    procedures.\n    We would like to localize the\n    subprocedures,\n    hiding them inside sqrt so that\n    sqrt could coexist with other\n    successive approximations, each having its own private\n    good-enough? procedure.\n      \n    To make this possible, we allow a\n    \n        procedure\n      \n    to have\n    \n\tinternal definitions\t\n      \n    that are local to that\n    procedure.\n    For example, in the square-root problem we can write\n    (define (sqrt x)\n  (define (good-enough? guess x)\n    (< (abs (- (square guess) x)) 0.001))\n  (define (improve guess x)\n    (average guess (/ x guess)))\n  (define (sqrt-iter guess x)\n    (if (good-enough? guess x)\n        guess\n        (sqrt-iter (improve guess x) x)))\n  (sqrt-iter 1.0 x)) ","1.1.8#p12":"\n    Such nesting of\n    definitions,\n    called block structure, is basically the right solution to the\n    simplest name-packaging problem.  But there is a better idea lurking here.\n    In addition to internalizing the \n    definitions of the auxiliary procedures,\n    we can simplify them.  Since x is bound in the\n    definition\n    of sqrt, the\n    \n        procedures\n      good-enough?,improve, and\n    sqrt-iter,\n      which are defined internally tosqrt, are in the scope of\n    x. Thus, it is not necessary to pass\n    x explicitly to each of these\n    procedures.\n    Instead, we allow x to be a free\n    variable\n    in the internal\n    definitions,\n    as shown below. Then x gets its value from\n    the argument with which the enclosing\n    \n        procedure\n      sqrt is called.  This discipline is called\n    lexical scoping.(define (sqrt x)\n  (define (good-enough? guess)\n    (< (abs (- (square guess) x)) 0.001))\n  (define (improve guess)\n    (average guess (/ x guess)))\n  (define (sqrt-iter guess)\n    (if (good-enough? guess)\n        guess\n        (sqrt-iter (improve guess))))\n  (sqrt-iter 1.0)) ","1.1.8#footnote-link-3":"3","1.1.8#p13":"\n    We will use block structure extensively to help us break up large programs\n    into tractable pieces.\n    The idea of block structure originated with the programming language\n    \n    Algol 60. It appears in most advanced programming languages and is an\n    important tool for helping to organize the construction of large programs. \n    ","1.1.8#footnote-link-4":"4","1.1.8#footnote-1":"It\n    is not even clear which of these\n    \n        procedures\n      \n    is a more efficient implementation.  This depends upon the hardware\n    available.  There are machines for which the \"obvious\"\n    implementation is the less efficient one.  Consider a machine that has\n    extensive tables of logarithms and antilogarithms stored in a very\n    efficient manner.","1.1.8#footnote-2":"The\n    concept of consistent renaming is actually subtle and difficult to\n    define formally.  Famous logicians have made embarrassing errors\n    here.","1.1.8#footnote-3":"Lexical scoping dictates that free\n    \n\tvariables in a procedure\n      \n    are taken to refer to bindings made by enclosing\n    \n        procedure definitions;\n      \n    that is, they are looked up in\n    \n    the environment in which the\n    \n        procedure was defined.  \n      \n    We will see how this works in detail in chapter 3 when we\n    study environments and the detailed behavior of the interpreter.","1.1.8#footnote-4":"Embedded \n    \n        definitions must come first in a procedure\n      \n    body.\n    \n    The management is not responsible for the consequences of running programs\n    that intertwine\n    definition\n    and use; see also\n    footnotes 2\n    and 4\n    in section 1.3.2.\n    ","1.2":"1.2  \n    \n      \n      \n    \n    and the Processes They Generate","1.2#p1":"\n    We have now considered the elements of programming: We have used\n    primitive arithmetic operations, we have combined these operations, and\n    we have abstracted these composite operations by \n    defining them as compound procedures.\n    But that is not enough to enable us to say that we know\n    how to program.  Our situation is analogous to that of someone who has\n    learned the rules for how the pieces move in chess but knows nothing\n    of typical openings, tactics, or strategy.  Like the novice chess\n    player, we don't yet know the common patterns of usage in the domain.\n    We lack the knowledge of which moves are worth making \n    (which procedures are worth defining).\n    We lack the experience to predict the consequences of making a move\n    (executing a procedure).","1.2#p2":"\n    The ability to visualize the consequences of the actions under\n    consideration is crucial to becoming an expert programmer, just as it\n    is in any synthetic, creative activity.  In becoming an expert\n    photographer, for example, one must learn how to look at a scene and\n    know how dark each region will appear on a print for each possible\n    choice of exposure and\n    development.\n    Only then can one reason backward, planning framing, lighting,\n    exposure, and\n    development\n    to obtain the desired effects.  So it is with programming, where we are\n    planning the course of action to be taken by a process and where we control\n    the process by means of a program.  To become experts, we must learn to\n    visualize the processes generated by various types of\n    procedures.\n    Only after we have developed such a skill can we learn\n    to reliably construct programs that exhibit the desired behavior.\n  ","1.2#p3":"\n    A\n    procedure \n    is a\n    \n    pattern for the local evolution of a\n    computational process.  It specifies how each stage of the process is\n    built upon the previous stage.  We would like to be able to make\n    statements about the overall, or global, behavior of a\n    process whose local\n    \n    evolution has been specified by a \n    procedure.\n    This is very difficult to do in general, but we can at least try to\n    describe some typical patterns of process evolution.\n  ","1.2#p4":"\n    In this section we will examine some common \"shapes\" for\n    processes generated by simple \n    procedures.\n    We will also investigate the rates at which these processes consume the\n    important computational resources of time and space. The \n    procedures \n    we will consider are very simple.  Their role is like that played by test\n    patterns in photography: as oversimplified prototypical patterns, rather\n    than practical examples in their own right.\n  ","1.2.1":"1.2.1  Linear Recursion and Iteration","1.2.1#p1":"\n    We begin by considering the\n    \n    factorial function, defined by\n    \n      \\[\n      \\begin{array}{lll}\n      n! &=& n\\cdot(n-1)\\cdot(n-2)\\cdots3\\cdot2\\cdot1\n      \\end{array}\n      \\]\n    \n    There are many ways to compute factorials.  One way is to make use of\n    the observation that $n!$ is equal to \n    $n$ times $(n-1)!$ for\n    any positive integer $n$:\n    \n      \\[\n      \\begin{array}{lll}\n      n! &=& n\\cdot\\left[(n-1)\\cdot(n-2)\\cdots3\\cdot2\\cdot1\\right] \\quad = \\quad n \\cdot(n-1)!\n      \\end{array}\n      \\]\n    \n    Thus, we can compute $n!$ by computing \n    $(n-1)!$ and multiplying the\n    result by $n$.  If we add the stipulation that 1! \n    is equal to 1,\n    this observation translates directly into a \n    procedure:(define (factorial n)\n  (if (= n 1)\n      1\n      (* n (factorial (- n 1))))) \n          We can use the substitution model of\n          section 1.1.5 to watch this \n          \n\t        procedure in action computing 6!, as shown in\n\t        figure .\n\t","1.2.1#fig-":"","1.2.1#p2":"\n    Now let's take a different perspective on computing factorials.  We\n    could describe a rule for computing $n!$ by \n    specifying that we first multiply 1 by 2, then multiply the result by 3,\n    then by 4, and so on until we reach $n$.\n    More formally, we maintain a running product, together with a counter\n    that counts from 1 up to $n$.  We can describe\n    the computation by saying that the counter and the product simultaneously\n    change from one step to the next according to the rule\n    \n      \\[\n      \\begin{array}{lll}\n      \\textrm{product} & \\leftarrow & \\textrm{counter} \\cdot \\textrm{product}\\\\\n      \\textrm{counter} & \\leftarrow & \\textrm{counter} + 1\n      \\end{array}\n      \\]\n    \nand stipulating that $n!$ is the value of the\n    product when the counter exceeds $n$.\n  ","1.2.1#p3":"\n    Once again, we can recast our description as a \n    procedure \n    for computing\n    factorials:(define (factorial n)\n  (fact-iter 1 1 n))\n\n(define (fact-iter product counter max-count)\n  (if (> counter max-count)\n      product\n      (fact-iter (* counter product)\n                 (+ counter 1)\n                 max-count))) \n        As before, we can use the substitution model to visualize the process\n  \n        of computing $6!$, as shown in\n\tfigure .\n    ","1.2.1#footnote-link-1":"1","1.2.1#p4":"\n    Compare the two processes.  From one point of view, they seem hardly\n    different at all.  Both compute the same mathematical function on the\n    same domain, and each requires a number of steps proportional to\n    $n$\n    to compute $n!$.  Indeed, both processes even\n    carry out the same sequence of multiplications, obtaining the same sequence\n    of partial products.  On the other hand, when we consider the\n    \"shapes\" of the two processes, we find that they evolve quite\n    differently.\n  ","1.2.1#p5":"\n    Consider the first process.  The substitution model reveals a shape of\n    expansion followed by contraction, indicated by the arrow in\n    figure 1.3.\n    The expansion occurs as the process builds up a chain of \n    deferred operations (in this case, a chain of multiplications).\n    The contraction occurs as the operations are actually performed.  This\n    type of process, characterized by a chain of deferred operations, is called a\n    recursive process.  Carrying out this process requires that the\n    interpreter keep track of the operations to be performed later on.  In the\n    computation of $n!$, the length of the chain of\n    deferred multiplications, and hence the amount of information needed to\n    keep track of it, \n    \n    grows linearly with $n$ (is proportional to\n    $n$), just like the number of steps.\n    Such a process is called a\n    linear recursive process.\n  ","1.2.1#p6":"\n    By contrast, the second process does not grow and shrink.  At each\n    step, all we need to keep track of, for any $n$,\n    are the current values of the\n    variablesproduct, counter,\n    and\n    max-count.\n    We call this an \n    iterative process.  In general, an iterative process is one whose\n    state can be summarized by a fixed number of\n    state variables, together with a fixed rule that describes how\n    the state variables should be updated as the process moves from state to\n    state and an (optional) end test that specifies conditions under which the\n    process should terminate.  In computing $n!$, the\n    number of steps required grows linearly with $n$.\n    Such a process is called a \n    linear iterative process.\n  ","1.2.1#p7":"\n    The contrast between the two processes can be seen in another way.\n    In the iterative case, the state variables provide a complete description of\n    the state of the process at any point. If we stopped the computation between\n    steps, all we would need to do to resume the computation is to supply the\n    interpreter with the values of the three state variables. Not so with the\n    recursive process.  In this case there is some additional\n    \"hidden\" information, maintained by the interpreter and not\n    contained in the state variables, which indicates \"where the process\n    is\" in negotiating the chain of deferred operations.  The longer the\n    chain, the more information must be maintained.","1.2.1#footnote-link-2":"2","1.2.1#p8":"\n    In contrasting iteration and recursion, we must be careful not to\n    confuse the notion of a \n    \n    recursive process with the notion of a recursive \n    procedure.\n      \n    When we describe a\n    procedure\n    as recursive, we are referring to the syntactic fact that the\n    procedure definition \n    refers (either directly or indirectly) to the\n    procedure\n    itself.  But when we describe a process as following a pattern that is, say,\n    linearly recursive, we are speaking about how the process evolves, not\n    about the syntax of how a\n    procedure\n    is written.  It may seem disturbing that we refer to a recursive\n    procedure\n    such as \n    fact-iter\n    as generating an iterative process.  However, the process really is\n    iterative: Its state is captured completely by its three state variables,\n    and an interpreter need keep track of only three\n    variables\n    in order to execute the process.\n  ","1.2.1#p9":"\n    One reason that the distinction between process and\n    procedure\n    may be confusing is that most implementations of common languages\n    \n\t(including\n\t\n\tAda, Pascal, and C)\n      \n    are designed in such a way that the interpretation of\n    any recursive\n    procedure\n    consumes an amount of memory that grows with the number of\n    procedure\n    calls, even when the process described is, in principle, iterative.\n    As a consequence, these languages can describe iterative processes only\n    by resorting to special-purpose \n    \"looping constructs\" such as\n    $\\texttt{do}$,\n    $\\texttt{repeat}$,\n    $\\texttt{until}$,\n    $\\texttt{for}$, and\n    $\\texttt{while}$.\n    The implementation of\n    Scheme\n    we shall consider in chapter 5 does not share this defect.  It will\n    execute an iterative process in constant space, even if the iterative\n    process is described by a recursive\n    procedure.\n\tAn implementation with this property is called \n\ttail-recursive.  With a tail-recursive implementation, \n\titeration can be expressed using the ordinary procedure\n\t\n\tcall mechanism, so that special iteration constructs are useful only as \n\t\n\tsyntactic sugar.","1.2.1#footnote-link-3":"3","1.2.1#ex-1.9":"\n    Each of the following two\n    procedures\n    defines a method for adding two positive integers in terms of the\n    proceduresinc, which increments its argument by 1,\n    and dec, which decrements its argument by 1.\n    (define (+ a b)\n  (if (= a 0)\n      b\n      (inc (+ (dec a) b)))) (define (+ a b)\n  (if (= a 0)\n      b\n      (+ (dec a) (inc b)))) \n\n    Using the substitution model, illustrate the process generated by each\n    procedure\n    in evaluating\n    (+ 4 5).\n    Are these processes iterative or recursive?\n\n    ","1.2.1#ex-1.10":"\n    The following\n    procedure\n    computes a mathematical function called\n    \n    Ackermann's function.\n\n    (define (A x y)\n  (cond ((= y 0) 0)\n        ((= x 0) (* 2 y))\n        ((= y 1) 2)\n        (else (A (- x 1)\n                 (A x (- y 1)))))) \n\n    What are the values of the following\n    \n\texpressions?\n      (A 1 10) (A 2 4) (A 3 3) \n\n    Consider the following\n    procedures,\n    where A is the\n    procedure defined  \n    above:\n    (define (f n) (A 0 n))\n\n(define (g n) (A 1 n))\n\n(define (h n) (A 2 n))\n\n(define (k n) (* 5 n n)) \n    Give concise mathematical definitions for the functions computed by\n    the\n    proceduresf, g, and\n    h for positive integer values of\n    $n$.  For example,\n    $k(n)$ computes\n    $5n^2$.\n\n    ","1.2.1#footnote-1":"In a real program we would probably use the\n    block structure introduced in the last section to hide the \n    \n\tdefinition of fact-iter:\n      (define (factorial n)\n  (define (iter product counter)\n    (if (> counter n)\n        product\n        (iter (* counter product)\n              (+ counter 1))))\n  (iter 1 1)) \n    We avoided doing this here so as to minimize the number of things to\n    think about at\n    once.","1.2.1#footnote-2":"When we discuss the\n    implementation of\n    procedures \n    on register machines in chapter 5, we will see that any iterative\n    process can be realized \"in hardware\" as a machine that has a\n    fixed set of registers and no auxiliary memory.  In contrast, realizing a\n    recursive process requires a machine that uses an\n    auxiliary data structure known as a\n    stack.","1.2.1#footnote-3":"Tail recursion has long been\n\tknown as a compiler optimization trick.  A coherent semantic basis for\n\ttail recursion was provided by\n\t\n\tCarl Hewitt (1977), who explained it in\n\t\n\tterms of the \"message-passing\" model of computation that we\n\tshall discuss in chapter 3. Inspired by this, Gerald Jay Sussman\n\tand\n\t\n\tGuy Lewis Steele Jr. (see Steele 1975) \n\tconstructed a tail-recursive interpreter for Scheme.  Steele later showed\n\thow tail recursion is a consequence of the natural way to compile\n\tprocedure\n\t\n\tcalls (Steele 1977).\n\tThe IEEE standard for Scheme requires that Scheme implementations\n\t\n\tbe tail-recursive.","1.2.2":"1.2.2  Tree Recursion","1.2.2#p1":"\n    Another common pattern of computation is called tree recursion.\n    As an example, consider computing the sequence of\n    \n    Fibonacci numbers,\n    in which each number is the sum of the preceding two: \n    \n      \\[\\begin{array}{l}\n      0, 1, 1, 2, 3, 5, 8, 13, 21, \\ldots\n      \\end{array}\\]\n    \n    In general, the Fibonacci numbers can be defined by the rule\n    \n      \\[\\begin{array}{lll}\n      \\textrm{Fib}(n) & = & \\left\\{ \\begin{array}{ll}\n      0   &  \\text{if $n=0$}\\\\\n      1   &  \\text{if $n=1$}\\\\\n      \\textrm{Fib}(n-1)+\\textrm{Fib}(n-2) & \\text{otherwise}\n      \\end{array}\n      \\right.\n      \\end{array}\\]\n    \n    We can immediately translate this definition into a recursive\n    procedure\n    for computing Fibonacci numbers:\n    (define (fib n)\n  (cond ((= n 0) 0)\n        ((= n 1) 1)\n        (else (+ (fib (- n 1))\n                 (fib (- n 2)))))) ","1.2.2#fig-":"","1.2.2#p2":"\n    Consider the pattern of this computation.  To compute\n    (fib 5),\n    we compute\n    (fib 4)\n    and\n    (fib 3).\n    To compute\n    (fib 4),\n    we compute\n    (fib 3)\n    and\n    (fib 2).\n    In general, the evolved process looks like a tree, as shown in\n    \n\tfigure .\n      \n    Notice that the branches split into\n    two at each level (except at the bottom); this reflects the fact that the\n    fibprocedure\n    calls itself twice each time it is invoked.\n  ","1.2.2#p3":"\n    This\n    procedure\n    is instructive as a prototypical tree recursion, but it is a terrible way to\n    compute Fibonacci numbers because it does so much redundant computation.\n    Notice in\n    \n\tfigure \n    that the entire\n    computation of\n    (fib 3)—almost\n\thalf the work—is\n      \n    duplicated.  In fact, it is not hard to show that the number of times the\n    procedure\n    will compute\n    (fib 1)\n    or\n    (fib 0)\n    (the number of leaves in the above tree, in general) is precisely\n    $\\textrm{Fib}(n+1)$.  To get an idea of how\n    bad this is, one can show that the value of\n    $\\textrm{Fib}(n)$\n    grows exponentially with $n$.  More precisely\n    (see exercise 1.13),\n    $\\textrm{Fib}(n)$ is the closest integer to\n    $\\phi^{n} /\\sqrt{5}$, where\n    \n      \\[\\begin{array}{lllll}\n      \\phi&=&(1+\\sqrt{5})/2 & \\approx & 1.6180\n      \\end{array}\\]\n    \n    is the\n    golden ratio, which satisfies the equation\n    \n      \\[\\begin{array}{lll}\n      \\phi^{2} &=&\\phi + 1\n      \\end{array}\\]\n    \n    Thus, the process uses a number of steps that grows exponentially with the\n    input.  On the other hand, the space required grows only linearly with the\n    input, because we need keep track only of which nodes are above us in the\n    tree at any point in the computation.  In general, the number of steps\n    required by a tree-recursive process will be proportional to the number of\n    nodes in the tree, while the space required will be proportional to the\n    maximum depth of the tree.\n  ","1.2.2#p4":"\n    We can also formulate an iterative process for computing the Fibonacci\n    numbers. The idea is to use a pair of integers $a$\n    and $b$, initialized to\n    $\\textrm{Fib}(1)=1$ and\n    $\\textrm{Fib}(0)=0$, and to repeatedly apply the\n    simultaneous transformations\n    \n      \\[\\begin{array}{lll}\n      a & \\leftarrow & a+b \\\\\n      b & \\leftarrow & a\n      \\end{array}\\]\n    \n    It is not hard to show that, after applying this transformation\n    $n$ times, $a$ and\n    $b$ will be equal, respectively, to\n    $\\textrm{Fib}(n+1)$ and\n    $\\textrm{Fib}(n)$.  Thus, we can compute\n    Fibonacci numbers iteratively using the\n    procedure(define (fib n)\n  (fib-iter 1 0 n))\n\n(define (fib-iter a b count)\n  (if (= count 0)\n      b\n      (fib-iter (+ a b) a (- count 1)))) \n    This second method for computing $\\textrm{Fib}(n)$\n    is a linear iteration. The difference in number of steps required by the two\n    methods—one linear in $n$, one growing as\n    fast as $\\textrm{Fib}(n)$ itself—is\n    enormous, even for small inputs.\n  ","1.2.2#p5":"\n    One should not conclude from this that tree-recursive processes are useless.\n    When we consider processes that operate on hierarchically structured data\n    rather than numbers, we will find that tree recursion is a natural and\n    powerful tool. But\n    even in numerical operations, tree-recursive processes can be useful in\n    helping us to understand and design programs.  For instance, although the\n    first\n    fibprocedure\n    is much less efficient than the second one, it is more straightforward,\n    being little more than a translation into\n    \n        Lisp\n      \n    of the definition of the Fibonacci sequence.  To formulate the iterative\n    algorithm required noticing that the computation could be recast as an\n    iteration with three state variables.\n  ","1.2.2#footnote-link-1":"1","1.2.2#h1":"Example: Counting change","1.2.2#p6":"\n    It takes only a bit of cleverness to come up with the iterative Fibonacci\n    algorithm.  In contrast, consider the following problem:\n    How many different ways can we make change of\n    \\$1.00,\n      \n    given half-dollars, quarters, dimes, nickels, and pennies\n    (50 cents, 25 cents, 10 cents, 5 cents, and 1 cent, respectively)?\n    More generally, can\n    we write a\n    procedure\n    to compute the number of ways to change any given amount of money?\n  ","1.2.2#p7":"\n    This problem has a simple solution as a recursive\n    procedure.\n    Suppose we think of the types of coins available as arranged in some order.\n    Then the following relation holds:\n    \n    The number of ways to change amount $a$ using\n    $n$ kinds of coins equals\n    \n        the number of ways to change amount $a$\n\tusing all but the first kind of coin, plus\n      \n        the number of ways to change amount $a-d$\n\tusing all $n$ kinds of coins, where\n\t$d$ is the denomination of the first kind\n\tof coin.\n      ","1.2.2#p8":"\n    To see why this is true, observe that the ways to make change can be divided\n    into two groups: those that do not use any of the first kind of coin, and\n    those that do.  Therefore, the total number of ways to make change for some\n    amount is equal to the number of ways to make change for the amount without\n    using any of the first kind of coin, plus the number of ways to make change\n    assuming that we do use the first kind of coin.  But the latter number is\n    equal to the number of ways to make change for the amount that remains after\n    using a coin of the first kind.\n  ","1.2.2#p9":"\n    Thus, we can recursively reduce the problem of changing a given amount to\n    problems of changing smaller amounts or using fewer kinds of coins. Consider\n    this reduction rule carefully, and convince yourself that we can use it to\n    describe an algorithm if we specify the following degenerate\n    cases:\n        If $a$ is exactly 0, we should count that\n\tas 1 way to make change.\n      \n        If $a$ is less than 0, we should count\n\tthat as 0 ways to make change.\n       If $n$ is 0, we should count that\n      as 0 ways to make change.\n      \n    We can easily translate this description into a recursive\n    procedure:(define (count-change amount)\n  (cc amount 5))\n\n(define (cc amount kinds-of-coins)\n  (cond ((= amount 0) 1)\n        ((or (< amount 0)\n             (= kinds-of-coins 0)) 0)\n        (else (+ (cc amount\n                     (- kinds-of-coins 1))\n                 (cc (- amount\n                        (first-denomination\n                         kinds-of-coins))\n                     kinds-of-coins)))))\n\n(define (first-denomination kinds-of-coins)\n  (cond ((= kinds-of-coins 1) 1)\n        ((= kinds-of-coins 2) 5)\n        ((= kinds-of-coins 3) 10)\n        ((= kinds-of-coins 4) 25)\n        ((= kinds-of-coins 5) 50))) \n    (The\n    first-denomination procedure\n      \n    takes as input the number of kinds of coins available and returns the\n    denomination of the first kind.  Here we are thinking of the coins as\n    arranged in order from largest to smallest, but any order would do as well.)\n    We can now answer our original question about changing a dollar:\n\n    (count-change 100) ","1.2.2#footnote-link-2":"2","1.2.2#p10":"Count-change\n    generates a tree-recursive process with redundancies similar to those in\n    our first implementation of fib.\n    \n\t(It will take\n\tquite a while for that\n\t292\n\tto be computed.)\n      \n    On the other hand, it is not\n    obvious how to design a better algorithm for computing the result, and we\n    leave this problem as a challenge. The observation that a\n    \n    tree-recursive process may be highly inefficient but often easy to specify\n    and understand has led people to propose that one could get the best of both\n    worlds by designing a \"smart compiler\" that could transform\n    tree-recursive\n    procedures\n    into more efficient\n    procedures\n    that compute the same result.","1.2.2#footnote-link-3":"3","1.2.2#ex-1.11":"\n    A function $f$ is defined by the\n    \n\trule that\n      $f(n)=n$ if $n < 3$\n    and $f(n)={f(n-1)}+2f(n-2)+3f(n-3)$ if\n    $n\\ge 3$.  Write a\n    procedure\n    that computes $f$ by means of a recursive process.\n    Write a\n    procedure\n    that computes $f$ by means of an iterative\n    process.\n    ","1.2.2#ex-1.12":"\n    The following pattern of numbers is called\n    Pascal's triangle.\n    \n      \\[\n      {\n      \\begin{array}{rrrrcrrrr}\n      &   &   &   & 1 &   &   &   &     \\\\\n      &   &   &1  &   &1  &   &   &     \\\\\n      &   &1  &   & 2 &   &1  &   &     \\\\\n      &1  &   &3  &   &3  &   &1  &     \\\\\n      1  &   & 4 &   & 6 &   & 4 &   & 1   \\\\\n      &   &   &   & \\ldots & & & & \n      \\end{array}}\n      \\]\n    \n    The numbers at the edge of the triangle are all 1, and each number inside\n    the triangle is the sum of the two numbers above it.\n    Write a\n    procedure\n    that computes elements of Pascal's triangle by means of a recursive\n    process.\n    ","1.2.2#footnote-link-4":"4","1.2.2#ex-1.13":"\n    Prove that $\\textrm{Fib}(n)$ is the closest\n    integer to $\\phi^n/\\sqrt{5}$, where\n    $\\phi= (1+\\sqrt{5})/2$.\n    \n\tHint: Let\n\t$\\psi= (1-\\sqrt{5})/2$.  Use induction and the\n\tdefinition of the Fibonacci numbers (see\n\tsection 1.2.2) to prove that\n\t$\\textrm{Fib}(n)=(\\phi^n-\\psi^n)/\\sqrt{5}$.\n      ","1.2.2#footnote-1":"An example of this was hinted at in\n    section 1.1.3: The interpreter\n    itself evaluates expressions using a tree-recursive process.","1.2.2#footnote-2":"For example, work through in detail how the reduction rule\n    applies to the problem of making change for 10 cents using pennies and\n    nickels.","1.2.2#footnote-3":"One approach to coping with redundant\n    computations is to arrange matters so that we automatically construct a\n    table of values as they are computed.  Each time we are asked to apply the\n    procedure\n    to some argument, we first look to see if the value is already stored in the\n    table, in which case we avoid performing the redundant computation. This\n    strategy, known as\n    tabulation or\n    memoization, can be implemented in a\n    straightforward way.  Tabulation can sometimes be used to transform processes\n    that require an exponential number of steps\n    \n\t(such as count-change)\n      \n    into processes whose space and time requirements grow linearly with the\n    input.  See exercise 3.27.","1.2.2#footnote-4":"The elements\n    of Pascal's triangle are called the binomial coefficients,\n    because the $n$th row consists of\n    \n    the coefficients of the terms in the expansion of\n    $(x+y)^n$.  This pattern for computing the\n    coefficients\n    appeared in\n    \n    Blaise Pascal's 1653 seminal work on probability theory,\n    Traité du triangle arithmétique.\n    According to\n    \n    Edwards (2019), the same pattern appears\n    in the works of\n    the eleventh-century Persian mathematician\n    \n    Al-Karaji,\n    in the works of the twelfth-century Hindu mathematician\n    \n    Bhaskara, and\n    in the works of the\n    thirteenth-century Chinese mathematician\n    \n    Yang Hui.\n  ","1.2.3":"1.2.3  Orders of Growth","1.2.3#p1":"\n    The previous examples illustrate that processes can differ\n    considerably in the rates at which they consume computational\n    resources.  One convenient way to describe this difference is to use\n    the notion of \n    order of growth to obtain a gross measure of the\n    \n    resources required by a process as the inputs become larger.\n  ","1.2.3#p2":"\n    Let $n$ be a parameter that measures the size of\n    the problem,  and let $R(n)$ be the amount \n    of resources the process requires for a problem of size\n    $n$.  In our previous examples we took \n    $n$ to be the number for which a given\n    function is to be computed, but there are other possibilities.\n    For instance, if our goal is to compute an approximation to the\n    square root of a number, we might take \n    $n$ to be the number of digits accuracy required.\n    For matrix multiplication we might take $n$ to\n    be the number of rows in the matrices. In general there are a number of\n    properties of the problem with respect to which it will be desirable to\n    analyze a given process. Similarly, $R(n)$\n    might measure the number of internal storage registers used, the\n    number of elementary machine operations performed, and so on.  In\n    computers that do only a fixed number of operations at a time, the\n    time required will be proportional to the number of elementary machine\n    operations performed.\n  ","1.2.3#p3":"\n    We say that $R(n)$ has order of growth\n    $\\Theta(f(n))$, written\n    $R(n)=\\Theta(f(n))$ (pronounced\n    \"theta of $f(n)$\"), if there are\n    positive constants $k_1$ and\n    $k_2$ independent of\n    $n$ such that\n    \n      \\[\n      \\begin{array}{lllll}\n      k_1\\,f(n) & \\leq & R(n) & \\leq & k_2\\,f(n)\n      \\end{array}\n      \\]\n    \n    for any sufficiently large value of $n$.\n    (In other words, for large $n$, \n    the value $R(n)$ is sandwiched between \n    $k_1f(n)$ and\n    $k_2f(n)$.)\n  ","1.2.3#p4":"\n    For instance, with the linear recursive process for computing factorial\n    described in section 1.2.1 the\n    number of steps grows proportionally to the input\n    $n$.  Thus, the steps required for this process\n    grows as $\\Theta(n)$.  We also saw that the space\n    required grows as $\\Theta(n)$. For the \n    \n    iterative factorial, the number of steps is still\n    $\\Theta(n)$ but the space is\n    $\\Theta(1)$—that is, \n    constant. \n    The \n    \n    tree-recursive Fibonacci computation requires\n    $\\Theta(\\phi^{n})$ steps and space \n    $\\Theta(n)$, where\n    $\\phi$ is the golden ratio described in\n    section 1.2.2.\n  ","1.2.3#footnote-link-1":"1","1.2.3#p5":"\n    Orders of growth provide only a crude description of the behavior of a\n    process.  For example, a process requiring $n^2$\n    steps and a process requiring $1000n^2$ steps and\n    a process requiring $3n^2+10n+17$ steps all have\n    $\\Theta(n^2)$ order of growth.  On the other hand,\n    order of growth provides a useful indication of how we may expect the\n    behavior of the process to change as we change the size of the problem.\n    For a\n    $\\Theta(n)$ (linear) process, doubling the size\n    will roughly double the amount of resources used.  For an \n    \n    exponential process, each increment in problem size will multiply the\n    resource utilization by a constant factor.  In the remainder of\n    section 1.2\n    we will examine two algorithms whose order of growth is \n    \n    logarithmic, so that doubling the problem size increases the resource\n    requirement by a constant amount.\n    ","1.2.3#ex-1.14":"\n    Draw the tree illustrating the process generated by the \n    count-change procedure\n       \n    of section 1.2.2 in making change for\n    11 cents.  What are the orders of growth of the space and number of steps\n    used by this process as the amount to be changed increases?\n    ","1.2.3#ex-1.15":"\n    The sine of an angle (specified in radians) can be computed by making use\n    of the approximation $\\sin x\\approx x$\n    if $x$ is sufficiently small, and the\n    trigonometric identity \n    \n      \\[\n      \\begin{array}{lll}\n      \\sin x &=& 3\\sin {\\dfrac{x}{3}}-4\\sin^3{\\dfrac{x}{3}}\n      \\end{array}\n      \\]\n    \n    to reduce the size of the argument of $\\sin$.\n    (For purposes of this exercise an angle is considered \"sufficiently\n    small\" if its magnitude is not greater than 0.1 radians.) These\n    ideas are incorporated in the following \n    procedures:(define (cube x) (* x x x))\n\n(define (p x)\n  (- (* 3 x) (* 4 (cube x))))\n\n(define (sine angle)\n  (if (not (> (abs angle) 0.1))\n      angle\n      (p (sine (/ angle 3.0))))) How many times is the\n      procedurep \n      applied when\n      (sine 12.15)\n      is evaluated?\n      \n        What is the order of growth in space and number of steps (as a function\n\tof $a$) used by the process generated\n\tby the sineprocedure\n\twhen\n\t(sine a)\n\tis evaluated?\n      ","1.2.3#footnote-1":"These statements mask a great deal of oversimplification.\n    For instance, if we count process steps as \"machine operations\"\n    we are making the assumption that the number of machine operations needed to\n    perform, say, a multiplication is independent of the size of the numbers to\n    be multiplied, which is false if the numbers are sufficiently large.\n    Similar remarks hold for the estimates of space.  Like the design and\n    description of a process, the analysis of a process can be carried out at\n    various levels of abstraction.","1.2.4":"1.2.4  Exponentiation","1.2.4#p1":"\n    Consider the problem of computing the exponential of a given number.\n    We would like a\n    procedure\n    that takes as arguments a base $b$ and a positive\n    integer exponent $n$ and\n    computes $b^n$.  One way to do this is via\n    the recursive definition\n    \n      \\[\n      \\begin{array}{lll}\n      b^{n} &=& b\\cdot b^{n-1}\\\\\n      b^{0} &=& 1\n      \\end{array}\n      \\]\n    \n    which translates readily into the\n    procedure(define (expt b n)\n  (if (= n 0)\n      1\n      (* b (expt b (- n 1))))) \n    This is a linear recursive process, which requires\n    $\\Theta(n)$ steps and\n    $\\Theta(n)$ space.  Just as with factorial, we\n    can readily formulate an equivalent linear iteration:\n    (define (expt b n)\n  (expt-iter b n 1))\n\n(define (expt-iter b counter product)\n  (if (= counter 0)\n      product\n      (expt-iter b\n                (- counter 1)\n                (* b product)))) \n\n    This version requires $\\Theta(n)$ steps and\n    $\\Theta(1)$ space.\n  ","1.2.4#p2":"\n    We can compute exponentials in fewer steps by using\n    \n    successive squaring.\n    For instance, rather than computing $b^8$ as\n    \n      \\[\n      \\begin{array}{l}\n      b\\cdot(b\\cdot(b\\cdot(b\\cdot(b\\cdot(b\\cdot(b\\cdot b))))))\n      \\end{array}\n      \\]\n    \n    we can compute it using three multiplications:\n    \n      \\[\n      \\begin{array}{lll}\n      b^{2} &= & b\\cdot b\\\\\n      b^{4} &= & b^{2}\\cdot b^{2}\\\\\n      b^{8} &= & b^{4}\\cdot b^{4}\n      \\end{array}\n      \\]\n    ","1.2.4#p3":"\n    This method works fine for exponents that are powers of 2.  We can also take\n    advantage of successive squaring in computing exponentials in general if we\n    use the rule\n    \n      \\[\n      \\begin{array}{llll}\n      b^{n} &=& (b^{n/2})^{2}  &\\qquad\\,\\text{if}\\ n\\ \\text{is even}\\\\\n      b^{n} &=& b\\cdot b^{n-1} &\\qquad\\text{if}\\ n\\ \\text{is odd}\n      \\end{array}\n      \\]\n    \n    We can express this method as a\n    procedure:(define (fast-expt b n)\n  (cond ((= n 0) 1)\n        ((even? n) \n         (square (fast-expt b (/ n 2))))\n        (else \n         (* b (fast-expt b (- n 1)))))) \n    where the predicate to test whether an integer is even is defined in terms\n    of the \n    \n\tprimitive procedure\n\tremainder,\n      \n    by\n    (define (even? n)\n  (= (remainder n 2) 0)) \n    The process evolved by \n    fast-expt\n    grows logarithmically with $n$ in both space and\n    number of steps.  To see this, observe that computing\n    $b^{2n}$ using \n    fast-expt\n    requires only one more multiplication than computing\n    $b^n$.  The size of the exponent we can compute\n    therefore doubles (approximately) with every new multiplication we are\n    allowed.  Thus, the number of multiplications required for an exponent of\n    $n$ grows about as fast as the logarithm of\n    $n$ to the base 2.  The process has\n    $\\Theta(\\log n)$ growth.","1.2.4#footnote-link-1":"1","1.2.4#p4":"\n    The difference between $\\Theta(\\log n)$ growth\n    and $\\Theta(n)$ growth becomes striking as\n    $n$ becomes large.  For example, \n    fast-expt\n    for $n=1000$ requires only 14\n    multiplications. \n    It is also possible to use the idea of successive squaring to devise an\n    iterative algorithm that computes exponentials with a logarithmic number of\n    steps (see exercise 1.16), although, as is\n    often the case with iterative algorithms, this is not written down so\n    straightforwardly as the recursive algorithm.","1.2.4#footnote-link-2":"2","1.2.4#footnote-link-3":"3","1.2.4#ex-1.16":"\n    Design a\n    procedure\n    that evolves an iterative exponentiation process that uses successive\n    squaring and uses a logarithmic number of steps, as does   \n    fast-expt.\n    (Hint: Using the observation that\n    $(b^{n/2})^2 =(b^2)^{n/2}$, keep, along with the\n    exponent $n$ and the base\n    $b$, an additional state variable\n    $a$, and define the state transformation in such\n    a way that the product $a b^n$ is unchanged from\n    state to state.  At the beginning of the process\n    $a$ is taken to be 1, and the answer is given by\n    the value of $a$ at the end of the process.  In\n    general, the technique of defining an\n    invariant quantity that remains unchanged from state to state is a\n    powerful way to think about the\n    design of\n    \n    iterative algorithms.)\n    ","1.2.4#ex-1.17":" \n      The exponentiation algorithms in this section are based on performing\n      exponentiation by means of repeated multiplication.  In a similar way,\n      one can perform integer multiplication by means of repeated addition.\n      The following multiplication\n      procedure\n      (in which it is assumed that our language can only add, not multiply) is\n      analogous to the exptprocedure:(define (* a b)\n  (if (= b 0)\n      0\n      (+ a (* a (- b 1))))) \n      This algorithm takes a number of steps that is linear in\n      b. Now suppose we include, together with\n      addition,\n      operationsdouble, which doubles an\n      integer, and halve, which divides an (even)\n      integer by 2.  Using these, design a multiplication\n      procedure\n      analogous to \n      fast-expt\n      that uses a logarithmic number of steps.\n      ","1.2.4#ex-1.18":" \n      Using the results of exercises 1.16\n      and 1.17, devise a\n      procedure\n      that generates an iterative process for multiplying two integers in terms\n      of adding, doubling, and halving and uses a logarithmic number of\n      steps.","1.2.4#footnote-link-4":"4","1.2.4#ex-1.19":"\n      There is a clever algorithm for computing the Fibonacci numbers in a\n      \n      logarithmic number of steps. Recall the transformation of the state\n      variables $a$ and\n      $b$ in the\n      fib-iter\n      process of section 1.2.2:\n      $a\\leftarrow a+b$ and\n      $b\\leftarrow a$.  Call this transformation\n      $T$, and observe that applying\n      $T$ over\n      and over again $n$ times, starting with 1 and 0,\n      produces the pair $\\textrm{Fib}(n+1)$ and\n      $\\textrm{Fib}(n)$.  In other words, the\n      Fibonacci numbers are produced by applying\n      $T^n$, the $n$th\n      power of the transformation $T$, starting with\n      the pair $(1,0)$.  Now consider\n      $T$ to be the special case of\n      $p=0$ and $q=1$ in\n      a family of transformations $T_{pq}$, where\n      $T_{pq}$ transforms the pair\n      $(a,b)$ according to\n      $a\\leftarrow bq+aq+ap$ and\n      $b\\leftarrow bp+aq$.  Show that if we apply such\n      a transformation $T_{pq}$ twice, the effect is\n      the same as using a single transformation\n      $T_{p'q'}$ of the same form, and compute\n      $p'$ and $q'$ in\n      terms of $p$\n      and $q$.  This gives us an explicit way\n      to square these transformations, and thus we can compute\n      $T^n$ using successive squaring, as in the \n      fast-exptprocedure.\n      Put this all together to complete the following\n      procedure,\n      which runs in a logarithmic number of steps:\n(define (fib n)\n  (fib-iter 1 0 0 1 n))\n(define (fib-iter a b p q count)\n  (cond ((= count 0) b)\n        ((even? count)\n         (fib-iter a\n                   b\n                   ??      ; compute p'\n                   ??      ; compute q'\n                   (/ count 2)))\n        (else (fib-iter (+ (* b q) (* a q) (* a p))\n                        (+ (* b p) (* a q))\n                        p\n                        q\n                        (- count 1)))))\n        ","1.2.4#footnote-link-5":"5","1.2.4#footnote-1":"More precisely,\n    the number of multiplications required is equal to 1 less than the log\n    base 2 of $n$, plus the number of ones in the\n    binary representation of $n$.  This total is\n    always less than twice the log base 2 of $n$.\n    The arbitrary constants $k_1$ and\n    $k_2$ in the definition of order notation imply\n    that, for a logarithmic process, the base to which logarithms are taken does\n    not matter, so all such processes are described as\n    $\\Theta(\\log n)$.","1.2.4#footnote-2":"You may wonder why anyone would care about raising\n    numbers to the 1000th power. See\n    section 1.2.6.","1.2.4#footnote-3":"This iterative\n    algorithm is ancient.  It appears in the\n    Chandah-sutra by\n    Áchárya, written before 200 BCE.\n    See\n    Knuth 1997b, section 4.6.3, for a full discussion\n    and analysis of this and other methods of exponentiation.","1.2.4#footnote-4":"This\n      algorithm, which is sometimes known as the\n      \"Russian peasant\n      method\" of multiplication, is ancient.  Examples of its use are\n      found in the\n      \n      Rhind Papyrus, one of the two oldest mathematical documents in existence,\n      written about 1700 BCE (and copied from an even\n      older document) by an Egyptian scribe named\n      \n      A'h-mose.","1.2.4#footnote-5":"This exercise was\n      suggested\n      \n\t  to us\n\t\n      by\n      \n      Joe Stoy, based on an example in \n      Kaldewaij 1990.","1.2.5":"1.2.5  Greatest Common Divisors","1.2.5#p1":"\n    The greatest common divisor (GCD) of two integers\n    $a$ and $b$ is defined\n    to be the largest integer that divides both $a$\n    and $b$ with no remainder. For example, the GCD\n    of 16 and 28 is 4.  In chapter 2, when we investigate how to\n    implement rational-number arithmetic, we will need to be able to compute\n    GCDs in order to reduce rational numbers to lowest terms.  (To reduce a\n    rational number to lowest terms, we must divide both the numerator and the\n    denominator by their GCD.  For example, 16/28 reduces to 4/7.)  One way to\n    find the GCD of two integers is to factor them and search for common\n    factors, but there is a famous algorithm that is much more efficient.\n  ","1.2.5#p2":"\n    The idea of the algorithm is based on the observation that, if\n    $r$ is the remainder when\n    $a$ is divided by \n    $b$, then the common divisors of\n    $a$ and $b$ are \n    precisely the same as the common divisors of $b$\n    and $r$.  Thus, we can use the equation\n    \n      \\[\\begin{array}{lll}\n      \\textrm{GCD} (a, b) &=& \\textrm{GCD}(b, r)\n      \\end{array}\\]\n    \n    to successively reduce the problem of computing a GCD to the problem of\n    computing the GCD of smaller and smaller pairs of integers.  For example,\n    \n      \\[\\begin{array}{lll}\n      \\textrm{GCD}(206,40) & = & \\textrm{GCD}(40,6) \\\\\n                           & = & \\textrm{GCD}(6,4) \\\\\n                           & = & \\textrm{GCD}(4,2) \\\\\n                           & = & \\textrm{GCD}(2,0) \\\\\n                           & = & 2\n      \\end{array}\\]\n    \n    reduces $\\textrm{GCD}(206, 40)$ to\n    $\\textrm{GCD}(2, 0)$, which is 2.  It is\n    possible to show that starting with any two positive integers and\n    performing repeated reductions will always eventually produce a pair\n    where the second number is 0.  Then the GCD is the other\n    number in the pair.  This method for computing the GCD is\n    known as Euclid's Algorithm.","1.2.5#footnote-link-1":"1","1.2.5#p3":"\n    It is easy to express Euclid's Algorithm as a \n    procedure:(define (gcd a b)\n  (if (= b 0)\n      a\n      (gcd b (remainder a b)))) \n    This generates an iterative process, whose number of steps grows as\n    the logarithm of the numbers involved.\n  ","1.2.5#p4":"\n    The fact that the number of steps required by Euclid's Algorithm has\n    \n    logarithmic growth bears an interesting relation to the\n    \n    Fibonacci numbers:\n    Lamé's Theorem:\n      If Euclid's Algorithm\n      requires $k$ steps to compute the GCD of some\n      pair, then the smaller number in the pair must be greater than or equal\n      to the $k$th Fibonacci number.","1.2.5#footnote-link-2":"2","1.2.5#p5":"\n    We can use this theorem to get an order-of-growth estimate for Euclid's\n    Algorithm.  Let $n$ be the smaller of the two\n    inputs to the\n    procedure.\n    If the process takes $k$ steps, then we must have \n    $n\\geq {\\textrm{Fib}} (k)\\approx\\phi^k/\\sqrt{5}$.\n    Therefore the number of steps $k$ grows as the\n    logarithm (to the base $\\phi$) of\n    $n$. Hence, the order of growth is\n    $\\Theta(\\log n)$.\n    ","1.2.5#ex-1.20":" \n    The process that a \n    procedure\n    generates is of course dependent on the rules used by the interpreter.\n    As an example, consider the iterative gcdprocedure\n    given above. Suppose we were to interpret this \n    procedure\n    using\n    \n    normal-order evaluation, as discussed in\n    section 1.1.5. (The\n    normal-order-evaluation rule for\n    if\n    is described\n    in exercise 1.5.)  \n    Using the substitution method (for normal order), illustrate the process\n    generated in evaluating \n    (gcd 206 40)\n    and indicate the remainder operations that are\n    actually performed. How many remainder\n    operations are actually performed in the normal-order evaluation of \n    (gcd 206 40)?\n    In the applicative-order evaluation?\n    ","1.2.5#footnote-1":"Euclid's \n    Algorithm is so\n    called because it appears in Euclid's\n    Elements (Book 7,\n    ca. 300 BCE).  According to\n    \n    Knuth (1997a), it can be considered the\n    oldest known nontrivial algorithm.  The ancient Egyptian method of\n    multiplication (exercise 1.18) is surely\n    older, but, as Knuth explains, Euclid's Algorithm is the oldest known\n    to have been presented as a general algorithm, rather than as a set of\n    illustrative examples.","1.2.5#footnote-2":"This\n      theorem was proved in 1845 by\n      \n      Gabriel Lamé, a\n      French mathematician and engineer known chiefly for his contributions\n      to mathematical physics.  To prove the theorem, we consider pairs\n      $(a_k ,b_k)$, where \n      $a_k\\geq b_k$, for which Euclid's\n      Algorithm terminates in $k$ steps. The proof is\n      based on the claim that, if\n      $(a_{k+1},\\ b_{k+1}) \\rightarrow (a_{k},\\ b_{k})       \\rightarrow (a_{k-1},\\ b_{k-1})$ are three successive pairs\n      in the reduction process, then we must have \n      $b_{k+1}\\geq b_{k} + b_{k-1}$.\n      To verify the claim, consider that a reduction step is defined by applying\n      the transformation $a_{k-1} = b_{k}$, \n      $b_{k-1} =       \\textrm{remainder of}\\ a_{k}\\ \\textrm{divided by}\\ b_{k}$.  \n      The second equation means that\n      $a_{k} = qb_{k} + b_{k-1}$ for some positive\n      integer $q$. And since\n      $q$ must be at least 1 we have \n      $a_{k} = qb_{k} + b_{k-1} \\geq b_{k} + b_{k-1}$.\n      But in the previous reduction step we have\n      $b_{k+1}= a_{k}$. Therefore,\n      $b_{k+1} = a_{k}\\geq b_{k} + b_{k-1}$.\n      This verifies the claim.  Now we can prove the theorem by induction on\n      $k$, the number of steps that the algorithm\n      requires to terminate. The result is true for\n      $k=1$, since this merely requires that\n      $b$ be at least as large as \n      $\\text{Fib}(1)=1$. Now, assume that the result\n      is true for all integers less than or equal to\n      $k$ and establish the result for \n      $k+1$.  Let\n      $(a_{k+1},\\ b_{k+1})\\rightarrow(a_{k},\\ b_{k})       \\rightarrow(a_{k-1},\\ b_{k-1})$ be successive pairs in the\n      reduction process.  By our induction hypotheses, we have\n      $b_{k-1}\\geq {\\textrm{Fib}}(k-1)$ and \n      $b_{k}\\geq {\\textrm{Fib}}(k)$.  Thus, applying\n      the claim we just proved together with the definition of the Fibonacci\n      numbers gives\n      $b_{k+1} \\geq b_{k} + b_{k-1}\\geq {\\textrm{Fib}}(k) +       {\\textrm{Fib}}(k-1) = {\\textrm{Fib}}(k+1)$, which completes\n      the proof of Lamé's Theorem.","1.2.6":"1.2.6  Example: Testing for Primality","1.2.6#p1":"\n    This section describes two methods for checking the primality of an\n    integer $n$, one with order of growth\n    $\\Theta(\\sqrt{n})$, and a\n    \"probabilistic\" algorithm with order of growth\n    $\\Theta(\\log n)$.  The exercises at the end of\n    this section suggest programming projects based on these algorithms.\n  ","1.2.6#h1":"Searching for divisors","1.2.6#p2":"\n    Since ancient times, mathematicians have been fascinated by problems\n    concerning prime numbers, and many people have worked on the problem\n    of determining ways to test if numbers are prime.  One way\n    to test if a number is prime is to find the number's divisors.  The\n    following program finds the smallest integral divisor (greater than 1)\n    of a given number $n$.  It does this in a\n    straightforward way, by testing $n$ for\n    divisibility by successive integers starting with 2.\n    (define (smallest-divisor n)\n  (find-divisor n 2))\n\n(define (find-divisor n test-divisor)\n  (cond ((> (square test-divisor) n) n)\n        ((divides? test-divisor n) test-divisor)\n        (else (find-divisor n (+ test-divisor 1)))))\n\n(define (divides? a b)\n  (= (remainder b a) 0)) ","1.2.6#p3":"\n    We can test whether a number is prime as follows:\n    $n$ is prime if and only if\n    $n$ is its own smallest divisor.\n    (define (prime? n)\n  (= n (smallest-divisor n))) ","1.2.6#p4":"\n    The end test for \n    find-divisor\n    is based on the fact that if $n$ is not prime it\n    must have a divisor less than or equal to\n    $\\sqrt{n}$.\n    This means that the algorithm need only test divisors between 1 and\n    $\\sqrt{n}$.  Consequently, the number of steps\n    required to identify $n$ as prime will have order\n    of growth $\\Theta(\\sqrt{n})$.\n  ","1.2.6#footnote-link-1":"1","1.2.6#h2":"The Fermat test","1.2.6#p5":"\n    The $\\Theta(\\log n)$ primality test is based on\n    a result from number theory known as\n    \n    Fermat's Little\n    Theorem.Fermat's Little Theorem:\n      If $n$ is a prime number and \n      $a$ is any positive integer less than \n      $n$, then $a$ raised\n      to the $n$th power is congruent to\n      $a$ modulo $n$.\n    \n    (Two numbers are said to be\n    congruent modulo$n$ if they both have the same remainder when\n    divided by $n$.  The remainder of a number\n    $a$ when divided by \n    $n$ is also referred to as the \n    remainder of$a$modulo$n$, or simply as $a$modulo$n$.)\n  ","1.2.6#footnote-link-2":"2","1.2.6#p6":"\n    If $n$ is not prime, then, in general, most of\n    the numbers $a < n$ will not satisfy the above\n    relation.  This leads to the following algorithm for testing primality:\n    Given a number $n$, pick a \n    \n    random number $a < n$ and compute the\n    remainder of $a^n$ modulo\n    $n$.  If the result is not equal to\n    $a$, then $n$ is\n    certainly not prime.  If it is $a$, then chances\n    are good that $n$ is prime.  Now pick another\n    random number $a$ and test it with the same\n    method.  If it also satisfies the equation, then we can be even more\n    confident that $n$ is prime.  By trying more and\n    more values of $a$, we can increase our\n    confidence in the result.  This algorithm is known as the Fermat test.\n  ","1.2.6#p7":"\n    To implement the Fermat test, we need a\n    procedure\n    that computes the\n    \n    exponential of a number modulo another number:\n    (define (expmod base exp m)\n  (cond ((= exp 0) 1)\n        ((even? exp)\n         (remainder \n           (square (expmod base (/ exp 2) m))\n           m))\n        (else\n         (remainder \n           (* base (expmod base (- exp 1) m))\n           m)))) \n    This is very similar to the \n    fast-exptprocedure\n    of section 1.2.4.  It uses successive\n    squaring, so that the number of steps grows logarithmically with the\n    exponent.","1.2.6#footnote-link-3":"3","1.2.6#p8":"\n    The Fermat test is performed by choosing at random a number\n    $a$  between 1 and\n    $n-1$ inclusive and checking whether the remainder\n    modulo $n$ of the\n    $n$th power of $a$ is\n    equal to $a$.  The random number\n    $a$ is chosen using the\n    \n\tprocedure\n\trandom, \n      which we assume is included as a primitive in Scheme.Random \n\treturns a nonnegative integer less than its integer input.  Hence, to obtain\n\ta random number between 1 and $n-1$, we call\n\trandom with an input of\n\t$n-1$ and add 1 to the result:\t\n      (define (fermat-test n)\n  (define (try-it a)\n    (= (expmod a n n) a))\n  (try-it (+ 1 (random (- n 1))))) ","1.2.6#p9":"\n    The following \n    procedure \n    runs the test a given number of times, as specified by a parameter.  Its\n    value is true if the test succeeds every time, and false otherwise.\n    (define (fast-prime? n times)\n  (cond ((= times 0) true)\n        ((fermat-test n) \n         (fast-prime? n (- times 1)))\n        (else false))) ","1.2.6#h3":"Probabilistic methods","1.2.6#p10":"\n    The Fermat test differs in character from most familiar algorithms, in which\n    one computes an answer that is guaranteed to be correct.  Here, the answer\n    obtained is only probably correct.  More precisely, if\n    $n$ ever fails the Fermat test, we can be certain\n    that $n$ is not prime. But the fact that\n    $n$ passes the test, while an extremely strong\n    indication, is still not a guarantee that $n$ is\n    prime.  What we would like to say is that for any number\n    $n$, if we perform the test enough times and find\n    that $n$ always passes the test, then the\n    probability of error in our primality test can be made as small as we like.\n  ","1.2.6#p11":"\n    Unfortunately, this assertion is not quite correct.  There do exist numbers\n    that fool the Fermat test: numbers $n$ that are\n    not prime and yet have the property that $a^n$ is\n    congruent to $a$ modulo\n    $n$ for all integers\n    $a < n$.  Such numbers are extremely rare, so\n    the Fermat test is quite reliable in practice.\n    There are variations of the Fermat test that cannot be fooled.  In these\n    tests, as with the Fermat method, one tests the primality of an integer\n    $n$ by choosing a random integer\n    $a < n$ and checking some condition that\n    depends upon $n$ and\n    $a$.  (See\n    exercise 1.28 for an example of such a test.)\n    On the other hand, in contrast to the Fermat test, one can prove that, for\n    any $n$, the condition does not hold for most of\n    the integers $a < n$ unless\n    $n$ is prime.  Thus, if\n    $n$ passes the test for some random choice\n    of $a$, the chances are better than even\n    that $n$ is prime.  If\n    $n$ passes the test for two random choices of\n    $a$, the chances are better than 3 out of 4 that\n    $n$ is prime. By running the test with more and\n    more randomly chosen values of $a$ we can make\n    the probability of error as small as we like.\n    ","1.2.6#footnote-link-4":"4","1.2.6#p12":"\n    The existence of tests for which one can prove that the chance of error\n    becomes arbitrarily small has sparked interest in algorithms of this type,\n    which have come to be known as probabilistic algorithms.  There is\n\n    a great deal of research activity in this area, and probabilistic algorithms\n    have been fruitfully applied to many fields.","1.2.6#footnote-link-5":"5","1.2.6#ex-1.21":" \n    Use the \n    smallest-divisorprocedure \n    to find the smallest divisor of each of the following numbers: 199, 1999,\n    19999.\n    ","1.2.6#ex-1.22":"\n        Most Lisp implementations include a primitive called \n        runtime\n        that returns an integer that specifies the amount of time the system has\n\tbeen running (measured, for example, in microseconds).  The following \n        timed-prime-test\n\tprocedure,\n      \n    when called with an integer $n$, prints\n    $n$ and checks to see if\n    $n$ is prime.  If $n$\n    is prime, the \n    procedure\n    prints three asterisks followed by the amount of\n    time used in performing the test.\n    (define (timed-prime-test n)\n  (newline)\n  (display n)\n  (start-prime-test n (runtime)))\n\n(define (start-prime-test n start-time)\n  (if (prime? n)\n      (report-prime (- (runtime) start-time))))\n\n(define (report-prime elapsed-time)\n  (display \" *** \")\n  (display elapsed-time)) \n    Using this \n    procedure,\n    write a \n    proceduresearch-for-primes\n    that checks the primality of consecutive odd integers in a specified range.\n    Use your \n    \n        procedure\n      \n    to find the three smallest primes larger than 1000; larger than 10,000;\n    larger than 100,000; larger than 1,000,000.  Note the time needed to test\n    each prime.  Since the testing algorithm has order of growth of\n    $\\Theta(\\sqrt{n})$, you should expect that testing\n    for primes around 10,000 should take about\n    $\\sqrt{10}$ times as long as testing for primes\n    around 1000.  Do your timing data bear this out? How well do the data for\n    100,000 and 1,000,000 support the $\\sqrt{n}$\n    prediction?  Is your result compatible with the notion that programs on\n    your machine run in time proportional to the number of steps required for\n    the computation?\n    ","1.2.6#ex-1.23":" \n    The \n    smallest-divisorprocedure \n    shown at the start of this section does lots of needless testing: After it\n    checks to see if the number is divisible by 2 there is no point in checking\n    to see if it is divisible by any larger even numbers.  This suggests that\n    the values used for \n    test-divisor\n    should not be 2, 3, 4, 5, 6, … but rather 2, 3, 5, 7, 9,\n    …. To implement this change, \n    define a procedurenext that returns 3 if its input is equal to 2\n    and otherwise returns its input plus 2. Modify the \n    smallest-divisorprocedure \n    to use \n    (next test-divisor)\n    instead of \n    (+ test-divisor 1).\n    With \n    timed-prime-test\n    incorporating this modified version of \n    smallest-divisor,\n    run the test for each of the 12 primes found in\n    exercise 1.22.  \n    Since this modification halves the number of test steps, you should expect\n    it to run about twice as fast. Is this expectation confirmed?  If not, what\n    is the observed ratio of the speeds of the two algorithms, and how do you\n    explain the fact that it is different from 2?\n    ","1.2.6#ex-1.24":" \n    Modify the \n    timed-prime-testprocedure \n    of exercise 1.22 to use \n    fast-prime?\n    (the Fermat method), and test each of the 12 primes you found in that\n    exercise.  Since the Fermat test has\n    $\\Theta(\\log n)$ growth, how would you expect\n    the time to test primes near 1,000,000 to compare with the time needed to\n    test primes near 1000?  Do your data bear this out?  Can you explain any\n    discrepancy you find?\n    ","1.2.6#ex-1.25":" \n    Alyssa P. Hacker complains that we went to a lot of extra work in writing\n    expmod. After all, she says, since we already\n    know how to compute exponentials, we could have simply written\n    (define (expmod base exp m)\n  (remainder (fast-expt base exp) m)) \n    Is she correct?  \n    Would this \n    procedure \n    serve as well for our fast prime tester?  Explain.\n    ","1.2.6#ex-1.26":"\n    Louis Reasoner is having great difficulty doing\n    exercise 1.24.\n    His \n    fast-prime?\n    test seems to run more slowly than his \n    prime?\n    test. Louis calls his friend Eva Lu Ator over to help.  When they examine\n    Louis's code, they find that he has rewritten the \n    expmodprocedure\n    to use an explicit multiplication, rather than calling\n    square:\n    (define (expmod base exp m)\n  (cond ((= exp 0) 1)\n        ((even? exp)\n         (remainder \n           (* (expmod base (/ exp 2) m)\n              (expmod base (/ exp 2) m))\n           m))\n        (else\n         (remainder \n           (* base (expmod base (- exp 1) m))\n           m)))) \"I don't see what difference that could make,\"\n    says Louis. \"I do.\"  says Eva. \"By writing the \n    procedure \n    like that, you have transformed the\n    $\\Theta(\\log n)$ process into a\n    $\\Theta(n)$ process.\" Explain.\n\n    ","1.2.6#ex-1.27":"\n    Demonstrate that the\n    \n    Carmichael numbers listed in\n    footnote 4 really do fool the Fermat\n    test.  That is, write a \n    procedure \n    that takes an integer $n$ and tests whether\n    $a^n$ is congruent to \n    $a$ modulo $n$ for\n    every $a < n$, and try your \n    procedure \n    on the given Carmichael numbers.\n    ","1.2.6#ex-1.28":"\n    One variant of the Fermat test that cannot be fooled is called the\n    Miller–Rabin test (Miller 1976;\n    Rabin 1980).  This starts from\n    an alternate form of\n    \n    Fermat's Little Theorem, which states that if\n    $n$ is a prime number and\n    $a$ is any positive integer less than\n    $n$, then $a$ raised\n    to the $(n-1)$st power is congruent to 1\n    modulo $n$.  To test the primality of a\n    number $n$ by the Miller–Rabin test, we pick a\n    random number $a < n$ and raise \n    $a$ to the $(n-1)$st\n    power modulo $n$ using the\n    expmodprocedure.\n    However, whenever we perform the squaring step in\n    expmod, we check to see if we have discovered a\n    \"nontrivial square root of 1\n    modulo $n$,\" \n    that is, a number not equal to 1 or $n-1$ whose\n    square is equal to 1 modulo $n$.  It is\n    possible to prove that if such a nontrivial square root of 1 exists, then\n    $n$ is not prime. It is also possible to prove\n    that if $n$ is an odd number that is not prime,\n    then, for at least half the numbers $a < n$,\n    computing $a^{n-1}$ in this way will reveal a\n    nontrivial square root of 1 modulo $n$.  \n    (This is why the Miller–Rabin test cannot be fooled.)  Modify the\n    expmodprocedure \n    to signal if it discovers a nontrivial square root of 1, and use this to\n    implement the Miller–Rabin test with a\n    procedure \n    analogous to \n    fermat-test.\n    Check your\n    procedure \n    by testing various known primes and non-primes. Hint: One convenient way to\n    make expmod signal is to have it return 0.\n    ","1.2.6#footnote-1":"If \n    $d$ is a divisor of \n    $n$, then so is $n/d$.\n    But $d$ and $n/d$\n    cannot both be greater than $\\sqrt{n}$.","1.2.6#footnote-2":"Pierre\n    \n    de Fermat (1601–1665) is considered to be\n    the founder of modern\n    \n    number theory.  He obtained many important number-theoretic results,\n    but he usually announced just the results, without providing his proofs.  \n    \n    Fermat's Little Theorem was stated in a letter he wrote in 1640.\n    The first published proof was given by \n    \n    Euler in 1736 (and an\n    earlier, identical proof was discovered in the unpublished manuscripts\n    of\n    \n    Leibniz).  The most famous of Fermat's results—known as\n    Fermat's Last Theorem—was jotted down in 1637 in his copy of\n    the book Arithmetic (by the third-century Greek mathematician \n    \n    Diophantus) with the remark \"I have discovered a truly remarkable\n    proof, but this margin is too small to contain it.\"  Finding a proof\n    of Fermat's Last Theorem became one of the most famous challenges in\n    number theory.  A complete\n    solution was finally given in 1995 by\n    \n    Andrew Wiles of Princeton\n    University.","1.2.6#footnote-3":"The reduction steps in the cases where the exponent\n    $e$ is greater than 1 are based on the fact that,\n    for any integers $x$,\n    $y$, and $m$, we can\n    find the remainder of $x$ times\n    $y$ modulo $m$ by\n    computing separately the remainders of $x$ modulo\n    $m$ and $y$ modulo\n    $m$, multiplying these, and then taking the\n    remainder of the result modulo $m$.  For\n    instance, in the case where $e$ is even, we\n    compute the remainder of $b^{e/2}$ modulo\n    $m$, square this, and take the remainder modulo\n    $m$.  This technique is useful because it means\n    we can perform our computation without ever having to deal with numbers much\n    larger than $m$.  (Compare\n    exercise 1.25.)","1.2.6#footnote-4":"\n    Numbers that fool the\n    Fermat test are called\n    Carmichael numbers, and little is known\n    about them other than that they are extremely rare.  There are 255\n    Carmichael numbers below 100,000,000.  The smallest few are 561, 1105,\n    1729, 2465, 2821, and 6601.  In testing primality of very large\n    numbers chosen at random, the chance of stumbling upon a value that\n    fools the Fermat test is less than the chance that \n    \n    cosmic radiation will cause the computer to make an error in carrying out a\n    \"correct\" algorithm.  Considering an algorithm to be inadequate\n    for the first reason but not for the second illustrates the difference\n    between\n    \n    mathematics and engineering.\n  ","1.2.6#footnote-5":"One of the most\n    striking applications of\n    probabilistic prime testing has been to the field of\n    \n    cryptography.\n    \n\tAlthough it is now computationally infeasible to factor an arbitrary 200-digit\n\tnumber, the primality of such a number can be checked in a few seconds with the\n\tFermat test.\n      \n    This fact forms the basis of a technique for constructing\n    \"unbreakable codes\" suggested by \n    \n    Rivest,\n    \n    Shamir, and \n    \n    Adleman (1977).  The resulting \n    RSA algorithm has become a widely used technique for enhancing the\n    security of electronic communications.  Because of this and related\n    developments, the study of \n    \n    prime numbers, once considered the epitome of a topic in \"pure\"\n    mathematics to be studied only for its own sake, now turns out to have\n    important practical applications to cryptography, electronic funds transfer,\n    and information retrieval.","1.3":"1.3  \n        Formulating Abstractions with Higher-Order","1.3#p1":"\n        We have seen that\n        procedures\n        are, in effect, abstractions that describe compound operations on\n\tnumbers independent of the particular numbers. For example, when we\n\t(define (cube x) (* x x x)) \n        we are not talking about the cube of a particular number, but rather\n        about a method for obtaining the cube of any number.  Of course we could\n\tget along without ever \n        defining this procedure,\n\tby always writing expressions such as\n        (* 3 3 3)\n(* x x x)\n(* y y y)\n        and never mentioning cube explicitly. This\n\twould place us at a serious disadvantage, forcing us to work always at\n\tthe level of the particular operations that happen to be primitives in\n\tthe language (multiplication, in this case) rather than in terms of\n\thigher-level operations.  Our programs would be able to compute cubes,\n\tbut our language would lack the ability to express the concept of cubing.\n\tOne of the things we should demand from a powerful programming language\n\tis the ability to build abstractions by assigning names to common\n        patterns and then to work in terms of the abstractions directly.\n        Procedures\n        provide this ability.  This is why all but the most primitive\n\tprogramming languages include mechanisms for \n        defining procedures.","1.3#p2":"\n        Yet even in numerical processing we will be severely limited in our\n        ability to create abstractions if we are restricted to\n        procedures\n        whose parameters must be numbers.  Often the same programming pattern\n        will be used with a number of different\n        procedures.\n\tTo express such patterns as concepts, we will need to construct\n        procedures\n        that can accept\n        procedures\n        as arguments or return\n        procedures\n        as values.\n        Procedures\n        that manipulate\n        procedures\n        are called \n        higher-order procedures.\n\tThis section shows how higher-order\n        procedures\n        can serve as powerful abstraction mechanisms, vastly increasing the\n\texpressive power of our language.\n      ","1.3.1":"1.3.1  \n    \n      \n      \n    \n    as Arguments","1.3.1#p1":"\n    Consider the following three\n    procedures.\n    The first computes the sum of the integers from\n    a through b:\n    (define (sum-integers a b)\n  (if (> a b)\n      0\n      (+ a (sum-integers (+ a 1) b)))) \n    The second computes the sum of the cubes of the integers in the given range:\n    (define (sum-cubes a b)\n  (if (> a b)\n      0\n      (+ (cube a) (sum-cubes (+ a 1) b)))) \n    The third computes the sum of a sequence of terms in the series\n    \n      \\[ \\frac{1}{1\\cdot3}+\\frac{1}{5\\cdot7}+\\frac{1}{9\\cdot11}+\\cdots \\]\n    \n    which converges to $\\pi/8$ (very\n    slowly):(define (pi-sum a b)\n  (if (> a b)\n      0\n      (+ (/ 1.0 (* a (+ a 2))) \n         (pi-sum (+ a 4) b)))) ","1.3.1#footnote-link-1":"1","1.3.1#p2":"\n    These three\n    procedures\n    clearly share a common underlying pattern. They are for the most part\n    identical, differing only in the name of the\n    procedure,\n    the function of a used to compute the term to\n    be added, and the function that provides the next value of\n    a. We could generate each of the\n    procedures\n    by filling in slots in the same template:\n\n    \n(define ($\\langle name \\rangle$ a b)\n  (if (> a b)\n      0\n      (+ ($\\langle term \\rangle$ a)\n         ($\\langle name \\rangle$ ($\\langle next \\rangle$ a) b))))\n      ","1.3.1#p3":"\n    The presence of such a common pattern is strong evidence that there is a\n    useful\n    \n    abstraction waiting to be brought to the surface.  Indeed,\n    mathematicians long ago identified the abstraction of\n    summation of a series and invented \"sigma\n    notation,\" for example\n    \n      \\[\\begin{array}{lll}\n      \\displaystyle\\sum_{n=a}^{b}\\ f(n)&=&f(a)+\\cdots+f(b)\n      \\end{array}\\]\n    \n    to express this concept.  The power of sigma notation is that it allows\n    mathematicians to deal with the concept of summation itself rather than only\n    with particular sums—for example, to formulate general results about\n    sums that are independent of the particular series being summed.\n  ","1.3.1#p4":"\n    Similarly, as program designers, we would like our language to be powerful\n    enough so that we can write a\n    procedure\n    that expresses the concept of summation itself rather than only\n    procedures\n    that compute particular sums.  We can do so readily in our\n    procedural\n    language by taking the common template shown above and transforming the\n    \"slots\" into\n    formal parameters:\n      (define (sum term a next b)\n  (if (> a b)\n      0\n      (+ (term a)\n         (sum term (next a) next b))))\n    Notice that sum takes as its arguments the\n    lower and upper bounds a and\n    b together with the\n    proceduresterm and next.\n    We can use sum just as we would any\n    procedure.\n    For example, we can use it (along with a\n    procedureinc that increments its argument by 1) to define\n    sum-cubes:(define (inc n) (+ n 1))\n\n(define (sum-cubes a b)\n  (sum cube a inc b)) \n    Using this, we can compute the sum of the cubes of the integers from 1 to 10:\n    (sum-cubes 1 10) \n    With the aid of an identity\n    procedure\n    to compute the term, we can define\n    sum-integers\n    in terms of sum:\n    (define (identity x) x) (define (sum-integers a b)\n  (sum identity a inc b)) \n    Then we can add up the integers from 1 to 10:\n    (sum-integers 1 10) \n    We can also \n    define pi-sum\n    in the same way:(define (pi-sum a b)\n  (define (pi-term x)\n    (/ 1.0 (* x (+ x 2))))\n  (define (pi-next x)\n    (+ x 4))\n  (sum pi-term a pi-next b)) \n    Using these\n    procedures,\n    we can compute an approximation to $\\pi$:\n    (* 8 (pi-sum 1 1000)) ","1.3.1#footnote-link-2":"2","1.3.1#p5":"\n    Once we have sum, we can use it as a building\n    block in formulating further concepts.  For instance, the \n    \n    definite integral of a function $f$ between the\n    limits $a$ and $b$ can\n    be approximated numerically using the formula\n    \n      \\[\n      \\begin{array}{lll}\n      \\displaystyle\\int_{a}^{b}f & = &\n      \\left[\\,f\\!\\left( a+\\dfrac{dx}{2} \\right)\\,+\\,f\\!\\left(a+dx+\\dfrac{dx}{2}\n      \\right)\\,+\\,f\\!\\left( a+2dx+\\dfrac{dx}{2}\\right)\\,+\\,\\cdots\n      \\right] dx\n      \\end{array}\n      \\]\n    \n    for small values of $dx$. We can express this\n    directly as a\n    procedure:(define (integral f a b dx)\n  (define (add-dx x) (+ x dx))\n  (* (sum f (+ a (/ dx 2)) add-dx b)\n     dx)) (integral cube 0 1 0.01) (integral cube 0 1 0.001) \n\n    (The exact value of the integral of cube between\n    0 and 1 is 1/4.)\n  ","1.3.1#ex-1.29":"\n    Simpson's Rule is a more accurate method of numerical integration than\n\n    the method illustrated above.  Using Simpson's Rule, the integral of a\n    function $f$ between\n    $a$ and $b$ is\n    approximated as\n    \n      \\[\n      \\frac{h}{3}[ y_0 +4y_1 +2y_2 +4y_3 +2y_4 +\\cdots+2y_{n-2}\n      +4y_{n-1}+y_n ]\n      \\]\n    \n    where $h=(b-a)/n$, for some even integer\n    $n$, and\n    $y_k =f(a+kh)$. (Increasing\n    $n$ increases the accuracy of the approximation.)\n    Define a procedure\n    that takes as arguments $f$,\n    $a$, $b$, and\n    $n$ and returns the value of the integral,\n    computed using Simpson's Rule. Use your\n    procedure\n    to integrate cube between 0 and 1 (with\n    $n=100$ and $n=1000$),\n    and compare the results to those of the integralprocedure\n    shown above.\n    ","1.3.1#ex-1.30":" \n    The\n    sumprocedure\n    above generates a linear recursion.  The\n    procedure\n    can be rewritten so that the sum is performed iteratively. Show how to do\n    this by filling in the missing expressions in the following\n    definition:\n(define (sum term a next b)\n  (define (iter a result)\n    (if ??\n        ??\n        (iter ?? ??)))\n  (iter ?? ??))\n      ","1.3.1#ex-1.31":"\n\tThe\n\tsumprocedure\n\tis only the simplest of a vast number of similar abstractions that can\n\tbe captured as higher-order\n\tprocedures.  \n\tWrite an analogous\n\tprocedure\n\tcalled\n\tproduct that returns the product of\n\tthe values of a function at points over a given range. Show how to\n\tdefine \n\tfactorial in terms of\n\tproduct.  Also use\n\tproduct to compute approximations to\n\t$\\pi$ using the formula\n          \\[\n\t  \\begin{array}{lll}\n\t  \\dfrac{\\pi}{4} & = & \\dfrac{2 \\cdot 4\\cdot 4\\cdot 6\\cdot 6\\cdot 8\\cdots}{3\\cdot\n          3\\cdot 5\\cdot 5\\cdot 7\\cdot 7\\cdots}\n\t  \\end{array}\n\t  \\]\n\t\n        If your productprocedure\n        generates a recursive process, write one that generates an iterative\n\tprocess. If it generates an iterative process, write one that generates\n        a recursive process.\n      ","1.3.1#footnote-link-3":"3","1.3.1#footnote-link-4":"4","1.3.1#ex-1.32":"\n        Show that sum and\n\tproduct\n        (exercise 1.31) are both special cases of a\n\tstill more general notion called\n\taccumulate\n\tthat combines a collection of terms, using some general accumulation\n\tfunction:\n        (accumulate combiner null-value term a next b)Accumulate\n        takes as arguments the same term and range specifications as\n\tsum and\n\tproduct, together with a\n\tcombinerprocedure\n        (of two arguments) that specifies how the current term is to be combined\n\twith the accumulation of the preceding terms and a \n        null-value\n        that specifies what base value to use when the terms run out.  Write\n\taccumulate and show how\n\tsum and product\n\tcan both be\n\tdefined\n\tas simple calls to accumulate.\n      \n        If your accumulateprocedure\n        generates a recursive process, write one that generates an iterative\n\tprocess. If it generates an iterative process, write one that generates\n        a recursive process.\n      ","1.3.1#ex-1.33":" \n    You can\n    obtain an even more general version of\n    accumulate\n    (exercise 1.32)\n    by introducing the notion of a\n    filter on the terms to be combined.  That is, combine only those\n    terms derived from values in the range that satisfy a specified condition.\n    The resulting \n    filtered-accumulate\n    abstraction takes the same arguments as accumulate, together with an\n    additional predicate of one argument that specifies the filter. Write\n    filtered-accumulate\n    as a\n    procedure.\n    Show how to express the following using\n    filtered-accumulate:\n      \n        the sum of the squares of the prime numbers in the interval \n        $a$ to $b$ \n        (assuming that you have an \n        prime? \n        predicate already written)\n      \n        the product of all the positive integers less than\n\t$n$\n        that are\n        \n\trelatively prime to $n$ (i.e.,\n\tall positive integers $i < n$ such that \n        $\\textrm{GCD}(i,n)=1$).\n      ","1.3.1#footnote-1":"This series,\n    \n    usually written in the equivalent form \n    $\\frac {\\pi}{4} = 1-\\frac{1}     {3}+\\frac{1}{5}-\\frac{1}{7}+\\cdots$, \n    is due to Leibniz.  We'll see how to use this as the basis for some\n    fancy numerical tricks in\n    section 3.5.3.","1.3.1#footnote-2":"Notice that we have used block structure\n    (section 1.1.8) to embed the \n    definitions of pi-next\n    and \n    pi-term\n    within\n    pi-sum,\n    since these\n    procedures\n    are unlikely to be useful for any other purpose.  We will see how to get rid\n    of them altogether in section 1.3.2.","1.3.1#footnote-3":"The intent of exercises 1.31–1.33 is to demonstrate the expressive\n\tpower that is attained by using an appropriate abstraction to\n\tconsolidate many seemingly disparate operations.  However, though\n\taccumulation and filtering are elegant ideas, our hands are somewhat\n\ttied in using them at this point since we do not yet have data\n\tstructures to provide suitable means of combination for these\n\tabstractions.  We will return to these ideas in\n\tsection 2.2.3 when\n\twe show how to use sequences as interfaces for combining\n\tfilters and accumulators to build even more powerful abstractions.  We\n\twill see there how these methods really come into their own as a\n\tpowerful and elegant approach to designing programs.","1.3.1#footnote-4":"This formula\n\twas discovered by the seventeenth-century\n\tEnglish mathematician\n\t\n\tJohn Wallis.","1.3.2":"1.3.2","1.3.2#p1":"\n        In using sum as in\n\tsection 1.3.1,\n        it seems terribly awkward to have to define trivial procedures such as\n        pi-term and\n\tpi-next just so we can use them as\n\targuments to our higher-order procedure.\n        Rather than define pi-next and \n        pi-term, it would be more convenient to\n\thave a way to directly specify \"the procedure that returns its\n        input incremented by 4\" and \"the procedure that returns the\n        reciprocal of its input times its input plus 2.\"  We can do this\n\tby introducing the special form lambda,\n\twhich creates procedures. Using lambda we\n\tcan describe what we want as\n        (lambda (x) (+ x 4))\n        and \n        (lambda (x) (/ 1.0 (* x (+ x 2))))\n\tThen our \n\tpi-sum\n\tprocedure\n\tcan be expressed\n      \n    without \n    defining any auxiliary procedures as(define (pi-sum a b)\n  (sum (lambda (x) (/ 1.0 (* x (+ x 2))))\n       a\n       (lambda (x) (+ x 4))\n       b)) ","1.3.2#p2":"\n    Again using \n    lambda, \n    we can write the integralprocedure\n    without having to \n    define the auxiliary procedureadd-dx:(define (integral f a b dx)\n  (* (sum f\n          (+ a (/ dx 2.0))\n          (lambda (x) (+ x dx))\n          b)\n     dx)) ","1.3.2#p3":"\n        In general, lambda is used to create\n\t  \n        procedures in the same way as define,\n\texcept that \n        \n        no name is specified for the procedure:\n        (lambda ($formal-parameters$) $body$)\n\tThe resulting procedure\tis just as much a procedure\n\tas one that is created using define.\n\tThe only difference is that it has not been associated with any name in the\n\tenvironment.\n    In fact,(define (plus4 x) (+ x 4)) is equivalent to(define plus4 (lambda (x) (+ x 4))) \n        We can read a lambda expression as follows:\n        \n(         lambda           (x)           (+   x     4))\n// read as: ^               ^             ^   ^     ^\n//   the procedure of an argument x that adds x and 4\n          ","1.3.2#p4":"\n    Like any expression that has a\n    procedure\n    as its value, a\n    lambda\n    expression can be used as the function expression in an application\n    combination such as\n    ((lambda (x y z) (+ x y (square z))) \n 1 2 3) \n    or, more generally, in any context where we would normally use a\n    procedure\n    name.","1.3.2#footnote-link-1":"1","1.3.2#h1":"Using \n    let to create local variables","1.3.2#p5":"\n    Another use of \n    lambda is in creating local variables.\n      \n\tWe often need local variables in our procedures\n\tother than those that have been bound as formal parameters.\n      \n    For example, suppose we wish to compute the function\n    \n      \\[\\begin{array}{lll}\n      f(x, y)&=&x(1 + x y)^2 +y (1 - y) + (1 + x y)(1 - y)\n      \\end{array}\\]\n    \n    which we could also express as\n    \n      \\[\\begin{array}{rll}\n      a &=& 1+xy\\\\\n      b &=& 1-y\\\\\n      f(x, y) &= &x a^2 +y b + a b\n      \\end{array}\\]\n    \n    In writing a\n    procedure\n    to compute $f$, we would like to include as\n    \n\tlocal variables\n      \n    not only $x$ and $y$\n    but also the names of intermediate quantities like\n    $a$ and $b$.  One way\n    to accomplish this is to use an auxiliary\n    procedure to bind the local variables:(define (f x y)\n  (define (f-helper a b)\n    (+ (* x (square a))\n       (* y b)\n       (* a b)))\n  (f-helper (+ 1 (* x y)) \n            (- 1 y))) ","1.3.2#p6":"\n    Of course, we could use a \n    lambda\n    expression to specify an anonymous\n    procedure for binding our local variables.\n    The\n    \n\tbody of\n\tf\n    then becomes a single call to that\n    procedure:(define (f x y)\n  ((lambda (a b)\n     (+ (* x (square a))\n        (* y b)\n        (* a b)))\n   (+ 1 (* x y))\n   (- 1 y))) \n        This construct is so useful that there is a special form called\n        let to make its use more convenient.  \n        Using let, the\n\tf procedure could be written as\n        (define (f x y)\n  (let ((a (+ 1 (* x y)))\n       (b (- 1 y)))\n    (+ (* x (square a))\n       (* y b)\n       (* a b)))) ","1.3.2#p7":"\n        The general form of a let expression is\n        (let (($\\textit{var}_1$ $\\textit{exp}_1$)\n      ($var_2$ $exp_2$)\n      $\\vdots$\n      ($var_n$ $exp_n$))\n   $body$)\n        which can be thought of as saying\n        \n\t  \\[\\begin{array}{ll}\n          \\text{let}\\ &\\textit{var}_1\\ \\text{have the value}\\ \\textit{exp}_1\\ \\text{and}\\\\\n                      &\\textit{var}_2\\ \\text{have the value}\\ \\textit{exp}_2\\ \\text{and}\\\\\n                      &\\vdots\\\\\n                      &\\textit{var}_n\\ \\text{have the value}\\ \\textit{exp}_n\\\\\n          \\text{in}\\  & \\textit{body}\n          \\end{array}\\]\n\t","1.3.2#p8":"\n        The first part of the let expression is a\n\tlist of name-expression pairs.  When the let\n\tis evaluated, each name is associated with the value of the\n\tcorresponding expression.  The body of the\n\tlet is evaluated with these names bound as\n\tlocal variables. The way this happens is that the\n\tlet expression is interpreted as an\n\talternate syntax for\n        ((lambda ($var_1$ $\\ldots$ $var_n$)\n   $body$)\n $exp_1$\n $\\vdots$\n $exp_n$)\n        No new mechanism is required in the interpreter in order to\n        provide local variables.  A \n        let expression is simply syntactic sugar for\n        the underlying lambda application.\n        \n        We can see from this equivalence that the scope of a variable specified\n\tby a let expression is the body of the\n\tlet. This implies that:\n        Let allows one to bind variables as\n\t    locally as possible to where they are to be used.  For example, if\n\t    the value of x is 5, the value of the\n\t    expression\n            (+ (let ((x 3))\n     (+ x (* x 10)))\n   x)  \n            is 38. Here, the x in the body of the\n\t    let is 3, so the value of the\n\t    let expression is 33.  On the other\n\t    hand, the x that is the second argument\n\t    to the outermost + is still 5.\n          \n            The variables' values are computed outside the\n\t    let. This matters when the expressions\n\t    that provide the values for the local variables depend upon\n\t    variables having the same names as the local variables themselves.\n            For example, if the value of x is 2,\n\t    the expression\n            (let ((x 3)\n      (y (+ x 2)))\n  (* x y))  \n            will have the value 12 because, inside the body of the\n\t    let, x\n\t    will be 3 and y will be 4 (which is the\n            outer x plus 2).\n          ","1.3.2#p9":"\n        Sometimes we can use internal definitions to get the same effect as with\n        let.  For example, we could have defined\n\tthe procedure f above as\n        (define (f x y)\n  (define a (+ 1 (* x y)))\n  (define b (- 1 y))\n  (+ (* x (square a))\n     (* y b)\n     (* a b))) \n        We prefer, however, to use let in\n\tsituations like this and to use internal\n\tdefine \n        only for internal procedures.","1.3.2#footnote-link-2":"2","1.3.2#ex-1.34":" \n    Suppose we \n    define the procedure(define (f g)\n  (g 2)) \n    Then we have\n    (f square) (f (lambda (z) (* z (+ z 1)))) \n    What happens if we (perversely) ask the interpreter to evaluate the\n    combination (f f)?\n    Explain.\n    ","1.3.2#footnote-1":"It would be clearer and less intimidating to people learning\n    Lisp\n    if a\n    name\n    more obvious than\n    lambda, such as\n        make-procedure,\n      \n    were used. But the convention is\n    \n\tfirmly entrenched.\n      \n    The notation is adopted from the\n    $\\lambda$ calculus, a\n    mathematical formalism introduced by the mathematical logician\n    \n    Alonzo Church (1941).  Church developed the\n    $\\lambda$ calculus to provide a rigorous\n    foundation for studying the notions of\n    function and function application.  The\n    $\\lambda$ calculus has become a basic\n    tool for mathematical investigations of the\n    semantics of programming languages.\n  ","1.3.2#footnote-2":"Understanding internal\n\tdefinitions well enough to be sure a program means what we intend it to\n\tmean requires a more elaborate model of the evaluation process than we\n\thave presented in this chapter. The subtleties do not arise with\n\tinternal definitions of procedures, however.  We will return to this\n\tissue in sections 3.2.4\n\tand 4.1.6, after\n\twe learn more about the evaluation of blocks.","1.3.3":"1.3.3  \n    \n      \n    \n    as General Methods","1.3.3#p1":"\n    We introduced compound\n    procedures\n    in section 1.1.4 as a mechanism for\n    abstracting patterns of numerical operations so as to make them independent\n    of the particular numbers involved.  With higher-order\n    procedures,\n    such as\n    the integralprocedure\n    of section 1.3.1, we began to\n    see a more powerful kind of abstraction:\n    procedures\n    used to express general methods of computation, independent of the\n    particular functions involved.  In this section we discuss two more elaborate\n    examples—general methods for finding zeros and fixed points of\n    functions—and show how these methods can be expressed directly as\n    procedures.","1.3.3#h1":"Finding roots of equations by the half-interval method","1.3.3#p2":"\n    The\n    half-interval method is a simple but powerful technique for\n    finding roots of an equation $f(x)=0$, where\n    $f$ is a continuous function.  The idea is that,\n    if we are given points $a$ and\n    $b$ such that\n    $f(a) < 0 < f(b)$, then\n    $f$ must have at least one zero between\n    $a$ and $b$. To locate\n    a zero, let $x$ be the average of\n    $a$ and $b$ and\n    compute $f(x)$.  If\n    $f(x) > 0$, then\n    $f$ must have a zero between\n    $a$ and $x$. If\n    $f(x) < 0$, then\n    $f$ must have a zero between\n    $x$ and $b$.\n    Continuing in this way, we can identify smaller and smaller intervals on\n    which $f$ must have a zero.  When we reach a\n    point where the interval is small enough, the process stops.  Since the\n    interval of uncertainty is reduced by half at each step of the process, the\n    maximal number of steps required grows as\n    $\\Theta(\\log( L/T))$, where\n    $L$ is the length of the original interval and\n    $T$ is the error tolerance (that is, the size of\n    the interval we will consider \"small enough\").\n    Here is a\n    procedure that implements this strategy:(define (search f neg-point pos-point)\n  (let ((midpoint (average neg-point pos-point)))\n    (if (close-enough? neg-point pos-point)\n        midpoint\n        (let ((test-value (f midpoint)))\n          (cond ((positive? test-value)\n                 (search f neg-point midpoint))\n                ((negative? test-value)\n                 (search f midpoint pos-point))\n                (else midpoint)))))) ","1.3.3#p3":"\n    We assume that we are initially given the function\n    $f$ together with points at which its values are\n    negative and positive.  We first compute the midpoint of the two given\n    points.  Next we check to see if the given interval is small enough, and if\n    so we simply return the midpoint as our answer.  Otherwise, we compute as a\n    test value the value of $f$ at the midpoint.  If\n    the test value is positive, then we continue the process with a new interval\n    running from the original negative point to the midpoint.  If the test value\n    is negative, we continue with the interval from the midpoint to the positive\n    point. Finally, there is the possibility that the test value is 0, in\n    which case the midpoint is itself the root we are searching for.\n\n    To test whether the endpoints are \"close enough\" we can use a\n    procedure\n    similar to the one used in section 1.1.7 for\n    computing square roots:(define (close-enough? x y)\n  (< (abs (- x y)) 0.001)) ","1.3.3#footnote-link-1":"1","1.3.3#p4":"Search\n    is awkward to use directly, because we can accidentally give it points at\n    which $f$'s values do not have the required\n    sign, in which case we get a wrong answer. Instead we will use\n    search via the following\n    procedure,\n    which checks to see which of the endpoints has a negative function value and\n    which has a positive value, and calls the searchprocedure\n    accordingly.  If the function has the same sign on the two given points, the\n    half-interval method cannot be used, in which case the\n    procedure\n    signals an error.(define (half-interval-method f a b)\n  (let ((a-value (f a))\n        (b-value (f b)))\n    (cond ((and (negative? a-value) (positive? b-value))\n           (search f a b))\n          ((and (negative? b-value) (positive? a-value))\n           (search f b a))\n          (else\n           (error \"Values are not of opposite sign\" a b))))) ","1.3.3#footnote-link-2":"2","1.3.3#p5":"\n    The following example uses the\n    \n    half-interval method to approximate\n    $\\pi$ as the root between 2 and 4 of\n    $\\sin\\, x = 0$:\n    (half-interval-method sin 2.0 4.0) ","1.3.3#p6":"\n    Here is another example, using the half-interval method to search for a root\n    of the equation $x^3 - 2x - 3 = 0$ between 1\n    and 2:\n    (half-interval-method (lambda (x) (- (* x x x) (* 2 x) 3))\n                      1.0\n                      2.0) ","1.3.3#h2":"Finding fixed points of functions","1.3.3#p7":"\n    A number $x$ is called a\n    fixed point of a\n    function $f$ if $x$\n    satisfies the equation $f(x)=x$. For some\n    functions $f$ we can locate a fixed point by\n    beginning with an initial guess and applying $f$\n    repeatedly,\n    \n      \\[\n      \\begin{array}{l}\n      f(x), \\ f(f(x)), \\ f(f(f(x))), \\ \\ldots\n      \\end{array}\n      \\]\n    \n    until the value does not change very much.  Using this idea, we can devise a\n    procedurefixed-point\n    that takes as inputs a function and an initial guess and produces an\n    approximation to a fixed point of the function.  We apply the function\n    repeatedly until we find two successive values whose difference is less\n    than some prescribed tolerance:\n\n    (define tolerance 0.00001)\n\n(define (fixed-point f first-guess)\n  (define (close-enough? v1 v2)\n    (< (abs (- v1 v2)) tolerance))\n  (define (try guess)\n    (let ((next (f guess)))\n      (if (close-enough? guess next)\n          next\n          (try next))))\n  (try first-guess)) \n    For example, we can use this method to approximate the fixed point of the\n    \n    cosine function, starting with 1 as an initial approximation:(fixed-point cos 1.0) \n    Similarly, we can find a solution to the equation\n    $y=\\sin y + \\cos y$:\n    (fixed-point (lambda (y) (+ (sin y) (cos y)))\n             1.0) ","1.3.3#footnote-link-3":"3","1.3.3#p8":"\n    The fixed-point process is reminiscent of the process we used for finding\n    square roots in section 1.1.7.  Both are based on\n    the idea of repeatedly improving a guess until the result satisfies some\n    criterion.  In fact, we can readily formulate the\n    \n    square-root computation as a fixed-point search.  Computing the square root\n    of some number $x$ requires finding a\n    $y$ such that\n    $y^2 = x$.  Putting this equation into the\n    equivalent form $y = x/y$, we recognize that we\n    are looking for a fixed point of the function$y \\mapsto x/y$, and we can therefore try to\n    compute square roots as\n    (define (sqrt x)\n  (fixed-point (lambda (y) (/ x y))\n               1.0))               \n    Unfortunately, this fixed-point search does not converge.  Consider an\n    initial guess $y_1$.  The next guess is\n    $y_2 = x/y_1$ and the next guess is\n    $y_3 = x/y_2 = x/(x/y_1) = y_1$.  This results\n    in an infinite loop in which the two guesses\n    $y_1$ and $y_2$ repeat\n    over and over, oscillating about the answer.\n  ","1.3.3#footnote-link-4":"4","1.3.3#p9":"\n    One way to control such oscillations is to prevent the guesses from changing\n    so much. Since the answer is always between our guess\n    $y$\n    and $x/y$, we can make a new guess that is not as\n    far from $y$ as $x/y$\n    by averaging $y$ with\n    $x/y$, so that the next guess after\n    $y$ is\n    $\\frac{1}{2}(y+x/y)$ instead of\n    $x/y$. The process of making such a sequence of\n    guesses is simply the process of looking for a fixed point of\n    $y \\mapsto \\frac{1}{2}(y+x/y)$:\n\n    (define (sqrt x)\n  (fixed-point (lambda (y) (average y (/ x y)))\n               1.0)) \n    (Note that $y=\\frac{1}{2}(y+x/y)$ is a simple\n    transformation of the equation $y=x/y$; to derive\n    it, add $y$ to both sides of the equation and\n    divide by 2.)\n  ","1.3.3#p10":"\n    With this modification, the square-root\n    procedure\n    works.  In fact, if we unravel the definitions, we can see that the sequence\n    of approximations to the square root generated here is precisely the same as\n    the one generated by our original square-root\n    procedure\n    of section 1.1.7.  This approach of averaging\n    successive approximations to a solution, a technique we call\n    average damping, often aids the convergence of fixed-point searches.\n    ","1.3.3#ex-1.35":"\n    Show that the\n    \n    golden ratio $\\phi$\n    (section 1.2.2) is a fixed point of the\n    transformation $x \\mapsto 1 + 1/x$, and use this\n    fact to compute $\\phi$ by means of the\n    fixed-point procedure.\n      ","1.3.3#ex-1.36":"\n    Modify\n    fixed-point\n    so that it prints the sequence of approximations it generates, using the\n    primitive function display shown in\n    exercise 1.22. Then find a solution to\n    $x^x = 1000$ by finding a fixed point of\n    $x \\mapsto \\log(1000)/\\log(x)$.  \n    \n\t(Use Scheme's primitive log procedure\n\twhich computes natural logarithms.)\n      \n    Compare the number of steps this takes with and without average damping.\n    (Note that you cannot start\n    fixed-point with a guess of 1, as this would cause division by\n      $\\log(1)=0$.)\n    ","1.3.3#ex-1.37":"\n    An infinite\n    continued fraction is an expression of the form\n    \n      \\[\n      \\begin{array}{lll}\n      f & = & {\\dfrac{N_1}{D_1+\n      \\dfrac{N_2}{D_2+\n      \\dfrac{N_3}{D_3+\\cdots }}}}\n      \\end{array}\n      \\]\n    \n    As an example, one can show that the infinite continued fraction\n    expansion with the $N_i$ and the\n    $D_i$ all equal to 1 produces\n    $1/\\phi$, where\n    $\\phi$ is the\n    \n    golden ratio (described in\n    section 1.2.2). One way to approximate\n    an infinite continued fraction is to truncate the expansion after a given\n    number of terms.  Such a truncation—a so-called\n    $k$-term finite continued\n    fraction—has the form\n    \n      \\[\n      {\\dfrac{N_1}{D_1 +\n      \\dfrac{N_2}{\\ddots +\n      \\dfrac{N_K}{D_K}}}}\n      \\]\n    \n        Suppose that n and\n\td are\n        procedures\n        of one argument (the term index $i$) that\n\treturn the $N_i$ and\n\t$D_i$ of the terms of the continued fraction.\n        Define a procedurecont-frac\n        such that evaluating\n        (cont-frac n d k)\n        computes the value of the $k$-term finite\n        continued fraction.  Check your\n        procedure\n        by approximating $1/\\phi$ using\n        (cont-frac (lambda (i) 1.0)\n           (lambda (i) 1.0)\n           k) \n        for successive values of k. How large must\n\tyou make k in order to get an approximation\n\tthat is accurate to 4 decimal places?\n      \n        If your\n\tcont-fracprocedure\n        generates a recursive process, write one that generates an iterative\n\tprocess. If it generates an iterative process, write one that generates\n        a recursive process.\n      ","1.3.3#ex-1.38":"\n    In 1737, the Swiss mathematician\n    \n    Leonhard Euler published a memoir\n    De Fractionibus Continuis, which included a\n    \n    continued fraction expansion for $e-2$, where\n    $e$ is the base of the natural logarithms. In\n    this fraction, the $N_i$ are all 1, and the\n    $D_i$ are successively\n    1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, ….  Write a program that uses your\n    cont-frac procedure\n    from exercise 1.37 to approximate\n    $e$, based on Euler's expansion.\n    ","1.3.3#ex-1.39":"\n    A continued fraction representation of the tangent function was\n    published in 1770 by the German mathematician\n    \n    J.H. Lambert:\n    \n      \\[\n      \\begin{array}{lll}\n      \\tan x & = & {\\dfrac{x}{1-\n      \\dfrac{x^2}{3-\n      \\dfrac{x^2}{5-\n      \\dfrac{x^2}{ \\ddots }}}}}\n      \\end{array}\n      \\]\n    \n    where $x$ is in radians.\n    Define a procedure (tan-cf x k)\n    that computes an approximation to the tangent function based on\n    Lambert's formula.\n    K specifies the number of terms to compute,\n\tas in exercise 1.37.\n      ","1.3.3#footnote-1":"We have used 0.001 as a representative\n    \"small\" number to indicate a tolerance for the acceptable error\n    in a calculation. The appropriate tolerance for a real calculation depends\n    upon the problem to be solved and the limitations of the computer and the\n    algorithm.  This is often\n    a very subtle consideration, requiring help from a\n    \n    numerical analyst or some\n    other kind of magician.","1.3.3#footnote-2":"This\n    can be accomplished using\n    error,\n    which takes as\n      arguments a number of items that are printed as error messages.\n      ","1.3.3#footnote-3":"\n\tTry this during a boring lecture: Set your calculator to\n\t\n\tradians mode and then repeatedly press the\n\t$\\cos$\n\tbutton until you obtain the fixed point.\n      ","1.3.3#footnote-4":"$\\mapsto$\n    (pronounced \"maps to\") is the mathematician's way of\n    writing\n    lambda.$y \\mapsto x/y$ means\n    (lambda(y) (/ x y)),\n    that is, the function whose value at $y$ is\n    $x/y$.","1.3.4":"1.3.4  \n    \n      \n      \n    \n    as Returned Values","1.3.4#p1":"\n    The above examples demonstrate how the ability to pass\n    procedures\n    as arguments significantly enhances the expressive power of our programming\n    language.  We can achieve even more expressive power by creating\n    procedures\n    whose returned values are themselves\n    procedures.","1.3.4#p2":"\n    We can illustrate this idea by looking again at the fixed-point example\n    described at the end of\n    section 1.3.3.  We formulated a new\n    version of the square-root\n    procedure\n    as a fixed-point search, starting with the observation that\n    $\\sqrt{x}$ is a fixed-point of the function\n    $y\\mapsto x/y$.  Then we used average damping to\n    make the approximations converge.  Average damping is a useful general\n    technique in itself.  Namely, given a\n    function $f$, we consider the function\n    whose value at $x$ is equal to the average of\n    $x$ and $f(x)$.\n  ","1.3.4#p3":"\n    We can express the idea of average damping by means of the following\n    procedure:(define (average-damp f)\n  (lambda (x) (average x (f x)))) Average-damp\n\tis a procedure that\n      \n    takes as its argument a\n    proceduref and returns as its value a\n    procedure(produced by the lambda)\n    that, when applied to a number x, produces the\n    average of x and \n    (f x).\n    For example, applying \n    average-damp\n    to the squareprocedure\n    produces a\n    procedure\n    whose value at some number $x$ is the average of \n    $x$ and $x^2$.  \n    Applying this resulting\n    procedure\n    to 10 returns the average of 10 and 100, or 55:((average-damp square) 10) ","1.3.4#footnote-link-1":"1","1.3.4#p4":"\n    Using \n    average-damp,\n    we can reformulate the\n    \n    square-root\n    procedure\n    as follows:\n    (define (sqrt x)\n  (fixed-point (average-damp (lambda (y) (/ x y)))\n               1.0)) \n    Notice how this formulation makes explicit the three ideas in the method:\n    fixed-point search, average damping, and the function\n    $y\\mapsto x/y$.  It is instructive to compare\n    this formulation of the square-root method with the original version given\n    in section 1.1.7.  Bear in mind that these\n    procedures\n    express the same process, and notice how much clearer the idea becomes when\n    we express the process in terms of these abstractions.  In general, there\n    are many ways to formulate a process as a\n    procedure.\n    Experienced programmers know how to choose\n    procedural\n    formulations that are particularly perspicuous, and where useful elements of\n    the process are exposed as separate entities that can be reused in other\n    applications. As a simple example of reuse, notice that the cube root of\n    $x$ is a fixed point of the function\n    $y\\mapsto x/y^2$, so we can immediately\n    generalize our square-root\n    procedure\n    to one that extracts \n    \n    cube roots:(define (cube-root x)\n  (fixed-point (average-damp (lambda (y) (/ x (square y))))\n               1.0)) ","1.3.4#footnote-link-2":"2","1.3.4#h1":"Newton's method","1.3.4#p5":"\n    When we first introduced the square-root\n    procedure,\n    in section 1.1.7, we mentioned that this was a\n    special case of\n    Newton's method. If\n    $x\\mapsto g(x)$ is a differentiable function,\n    then a solution of the equation $g(x)=0$ is a\n    fixed point of the function $x\\mapsto f(x)$ where\n    \n      \\[\n      \\begin{array}{lll}\n      f(x) & = & x - \\dfrac{g(x)}{Dg(x)}\n      \\end{array}\n      \\]\n    \n    and $Dg(x)$ is the derivative of\n    $g$ evaluated at $x$.  \n    \n    Newton's method is the use of the fixed-point method we saw above to\n    approximate a solution of the equation by finding a fixed point of the\n    function $f$.\n    For many functions $g$ and for sufficiently good\n    initial guesses for $x$, Newton's method\n    converges very rapidly to a solution of\n    $g(x)=0$.","1.3.4#footnote-link-3":"3","1.3.4#footnote-link-4":"4","1.3.4#p6":"\n    In order to implement Newton's method as a\n    procedure,\n    we must first express the idea of\n    \n    derivative.  Note that\n    \"derivative,\" like average damping, is something that\n    transforms a function into another function.  For instance, the derivative\n    of the function $x\\mapsto x^3$ is the function\n    $x \\mapsto 3x^2$.  In general, if\n    $g$ is a function and\n    $dx$ is a small number, then the derivative\n    $Dg$ of $g$ is the\n    function whose value at any number $x$ is given\n    (in the limit of small $dx$) by\n    \n      \\[\n      \\begin{array}{lll}\n      Dg(x) & = & \\dfrac {g(x+dx) - g(x)}{dx}\n      \\end{array}\n      \\]\n    \n    Thus, we can express the idea of derivative (taking\n    $dx$ to be, say, 0.00001) as the\n    procedure(define (deriv g)\n  (lambda (x)\n    (/ (- (g (+ x dx)) (g x))\n       dx))) \n    along with the\n    definition(define dx 0.00001) ","1.3.4#p7":"\n    Like \naverage-damp,deriv is a\n    procedure\n    that takes a\n    procedure\n    as argument and returns a\n    procedure\n    as value.  For example, to approximate the derivative of\n    $x \\mapsto x^3$ at 5 (whose exact value is 75)\n    we can evaluate\n    (define (cube x) (* x x x))\n\n((deriv cube) 5) ","1.3.4#p8":"\n    With the aid of deriv, we can express\n    Newton's method as a fixed-point process:\n\n    (define (newton-transform g)\n  (lambda (x)\n    (- x (/ (g x) ((deriv g) x)))))\n\n(define (newtons-method g guess)\n  (fixed-point \n    (newton-transform g) guess)) \n    The\n    newton-transformprocedure\n    expresses the formula at the beginning of this section, and\n    newtons-method\n    is readily defined in terms of this.  It takes as arguments a\n    procedure\n    that computes the function for which we want to find a zero, together with\n    an initial guess.  For instance, to find the \n    square root of $x$, we can use\n    \n    Newton's\n    method to find a zero of the function\n    $y\\mapsto y^2-x$ starting with an initial guess\n    of 1.\n    This provides yet another form of the square-root\n    procedure:(define (sqrt x)\n  (newtons-method (lambda (y) (- (square y) x))\n                  1.0)) ","1.3.4#footnote-link-5":"5","1.3.4#h2":"\n      Abstractions and first-class\n      procedures","1.3.4#p9":"\n    We've seen two ways to express the square-root computation as an\n    instance of a more general method, once as a fixed-point search and once\n    using Newton's method.  Since Newton's method was itself\n    expressed as a fixed-point process, we actually saw two ways to compute\n    square roots as fixed points. Each method begins with a function and finds a \n    \n    fixed point of some transformation of the function.  We can express this\n    general idea itself as a\n    procedure:(define (fixed-point-of-transform g transform guess)\n  (fixed-point (transform g) guess)) \n    This very general\n    procedure\n    takes as its arguments a\n    procedureg\n    that computes some function, a\n    procedure\n    that transforms g, and an initial guess.\n    The returned result is a fixed point of the transformed function.\n  ","1.3.4#p10":"\n    Using this abstraction, we can recast the first square-root computation\n    \n    from this section (where we look for a fixed point of the average-damped\n    version of $y \\mapsto x/y$) as an instance of\n    this general method:\n    (define (sqrt x)\n  (fixed-point-of-transform (lambda (y) (/ x y))\n                            average-damp\n                            1.0)) \n    Similarly, we can express the second square-root computation from this\n    section (an instance of\n    \n    Newton's method that finds a fixed point of\n    the Newton transform of $y\\mapsto y^2-x$) as\n    (define (sqrt x)\n  (fixed-point-of-transform (lambda (y) (- (square y) x))\n                            newton-transform\n                            1.0)) ","1.3.4#p11":"\n    We began section 1.3 with the\n    observation that compound\n    procedures\n    are a crucial abstraction mechanism, because they permit us to express\n    general methods of computing as explicit elements in our programming\n    language.  Now we've seen how higher-order\n    procedures\n    permit us to manipulate these general methods to create further abstractions.\n  ","1.3.4#p12":"\n    As programmers, we should be alert to opportunities to identify the\n    underlying abstractions in our programs and to build upon them and\n    generalize them to create more powerful abstractions.  This is not to say\n    that one should always write programs in the most abstract way possible;\n    expert programmers know how to choose the level of abstraction appropriate\n    to their task.  But it is important to be able to think in terms of these\n    abstractions, so that we can be ready to apply them in new contexts.  The\n    significance of higher-order\n    procedures\n    is that they enable us to represent these abstractions explicitly as\n    elements in our programming language, so that they can be handled just\n    like other computational elements.\n  ","1.3.4#p13":"\n    In general, programming languages impose restrictions on the ways in which\n    computational elements can be manipulated.  Elements with the fewest\n    restrictions are said to have \n    first-class status.  Some of the \"rights and\n    privileges\" of first-class elements are:They may be named by variables.\n\tThey may be passed as arguments to\n\tprocedures.\n\tThey may be returned as the results of\n\tprocedures.\n\tThey may be included in data structures.Lisp,\n\tunlike other\n      \n    programming languages, awards\n    procedures\n    full first-class status.  This poses challenges for efficient\n    implementation, but the resulting gain in expressive power is\n    enormous.","1.3.4#footnote-link-6":"6","1.3.4#footnote-link-7":"7","1.3.4#footnote-link-8":"8","1.3.4#ex-1.40":"Define a procedurecubic that can be used together with the \n    newtons-methodprocedure\n    in expressions of the form\n    (newtons-method (cubic a b c) 1)\n    to approximate zeros of the cubic\n    $x^3 +ax^2 +bx +c$.\n    ","1.3.4#ex-1.41":"Define a proceduredouble that takes a\n    procedure\n    of one argument as argument and returns a\n    procedure\n    that applies the original\n    procedure\n    twice.  For example, if inc is a\n    procedure\n    that adds 1 to its argument, then\n    (double inc)\n    should be a\n    procedure\n    that adds 2.  What value is returned by\n    (((double (double double)) inc) 5) ","1.3.4#ex-1.42":"\n    Let $f$ and $g$ be\n    two one-argument functions.  The\n    composition$f$ after $g$ is\n    defined to be the function $x\\mapsto f(g(x))$.\n    Define a procedurecompose that implements composition.  For\n    example, if inc is a\n    procedure\n    that adds 1 to its argument,\n    ((compose square inc) 6) returns 49.","1.3.4#ex-1.43":" \n    If $f$ is a numerical function and\n$n$ is a positive integer, then we can form the\n    $n$th\n    \n    repeated application of\n    $f$, which is defined to be the function whose\n    value at $x$ is\n    $f(f(\\ldots(f(x))\\ldots))$.  For example, if\n    $f$ is the function\n    $x \\mapsto x+1$, then the\n    $n$th repeated application of\n    $f$ is the function\n    $x \\mapsto x+n$. If\n    $f$ is the operation of squaring a number, then\n    the $n$th repeated application of\n    $f$ is the function that raises its argument to\n    the $2^n$th power.  Write a\n    procedure\n    that takes as inputs a\n    procedure\n    that computes $f$ and a positive integer\n    $n$ and returns the\n    procedure\n    that computes the $n$th repeated application of\n    $f$.  Your\n    procedure\n    should be able to be used as follows:\n    ((repeated square 2) 5) \n    Hint: You may find it convenient to use\n    compose from\n    exercise 1.42.\n    ","1.3.4#ex-1.44":" \n  The idea of\n  smoothing a function is an important concept in\n  \n  signal processing.  If $f$ is a function and\n  $dx$ is some small number, then the smoothed\n  version of $f$ is the function whose value at a\n  point $x$ is the average of\n  $f(x-dx)$, $f(x)$, and\n  $f(x+dx)$.  Write a\n  proceduresmooth that takes as input a\n  procedure\n  that computes $f$ and returns a\n  procedure\n  that computes the smoothed $f$.  It is sometimes\n  valuable to repeatedly smooth a function (that is, smooth the smoothed\n  function, and so on) to obtained the $n$-fold\n  smoothed function.  Show how to generate the\n  $n$-fold smoothed function of any given function\n  using smooth and\n  repeated from\n  exercise 1.43.\n  ","1.3.4#ex-1.45":"\n    We saw in section 1.3.3 that\n    attempting to compute square roots by naively finding a fixed point of\n    $y\\mapsto x/y$ does not converge, and that this\n    can be fixed by average damping.  The same method works for finding cube\n    roots as fixed points of the average-damped\n    $y\\mapsto x/y^2$. Unfortunately, the process\n    does not work for\n    \n    fourth roots—a single average damp is not enough to make a\n    fixed-point search for $y\\mapsto x/y^3$\n    converge.  On the other hand, if we average-damp twice (i.e., use the\n    average damp of the average damp of\n    $y\\mapsto x/y^3$) the fixed-point search does\n    converge.  Do some experiments to determine how many average damps are\n    required to compute \n    $n$th roots as a fixed-point search based upon\n    repeated average damping of $y\\mapsto x/y^{n-1}$.\n    Use this to implement a simple\n    procedure\n    for computing $n$th roots using \n    fixed-point,average-damp,\n    and the repeatedprocedure\n    of exercise 1.43. Assume that any arithmetic\n    operations you need are available as primitives.\n    ","1.3.4#ex-1.46":"\n    Several of the numerical methods described in this chapter are instances\n    of an extremely general computational strategy known as\n    iterative improvement.  Iterative improvement says that, to compute something,\n    we start with an initial guess for the answer, test if the guess is good\n    enough, and otherwise improve the guess and continue the process using the\n    improved guess as the new guess.  Write a\n    procedureiterative-improve\n    that takes two\n    procedures\n    as arguments: a method for telling whether a guess is good enough and a\n    method for improving a guess.\n    Iterative-improve\n    should return as its value a\n    procedure\n    that takes a guess as argument and keeps improving the guess until it is\n    good enough.  Rewrite the sqrtprocedure\n    of section 1.1.7 and the \n    fixed-pointprocedure\n    of section 1.3.3 in terms of\n    iterative-improve.","1.3.4#footnote-1":"\n\tObserve that this is a combination whose operator is itself\n\t\n\ta combination. Exercise 1.4 already\n\tdemonstrated the ability to form such combinations, but that was only a toy\n\texample.  Here we begin to see the real need for such\n\tcombinations—when applying a procedure\n\tthat is obtained as the value returned by a higher-order procedure.\n      ","1.3.4#footnote-2":"See exercise 1.45 \n    for a further generalization.","1.3.4#footnote-3":"Elementary calculus books\n    usually describe Newton's method in terms of the sequence of\n    approximations $x_{n+1}=x_n-g(x_n)/Dg(x_n)$.\n    Having language for talking about processes and using the idea of fixed\n    points simplifies the description of the method.","1.3.4#footnote-4":"Newton's method does not\n    always converge to an answer, but it can be shown that in favorable cases\n    each iteration doubles the number-of-digits accuracy of the approximation\n    to the solution. In such cases, \n    \n    Newton's method will converge much more rapidly than the half-interval\n    method.","1.3.4#footnote-5":"For finding square roots, Newton's method converges\n    rapidly to the correct solution from any starting point.","1.3.4#footnote-6":"The notion of\n    first-class status of programming-language\n    elements is due to the British computer scientist\n    \n    Christopher Strachey (1916–1975).","1.3.4#footnote-7":"We'll see\n\texamples of this after we introduce data structures in\n\tchapter 2.","1.3.4#footnote-8":"The major implementation cost of first-class\n    procedures\n    is that allowing\n    procedures\n    to be returned as values requires reserving storage for a\n    procedure's free variables\n    even while the\n    procedure\n    is not executing.  \n    \n\tIn the Scheme implementation we will study in\n\tsection 4.1, these variables are stored in\n\tthe procedure's\n       environment.","2#p1":"\n    We concentrated in chapter 1 on computational processes and on the\n    role of\n    procedures\n    in program design.  We saw how to use primitive data (numbers) and primitive\n    operations (arithmetic operations), how to combine\n    procedures\n    to form compound\n    procedures\n    through composition, conditionals, and the use of parameters, and how to\n    abstract\n    procedures\n    by using\n    define.\n    We saw that a\n    procedure\n    can be regarded as a pattern for the local evolution of a process, and we\n    classified, reasoned about, and performed simple algorithmic analyses of\n    some common patterns for processes as embodied in\n    procedures.\n    We also saw that higher-order\n    procedures\n    enhance the power of our language by enabling us to manipulate, and thereby\n    to reason in terms of, general methods of computation.  This is much of the\n    essence of programming.\n  ","2#p3":"\n    Why do we want compound data in a programming language?  For the same\n    reasons that we want compound\n    procedures:\n    to elevate the conceptual level at which we can design our programs, to\n    increase the modularity of our designs, and to enhance the expressive power\n    of our language. Just as the ability to \n    define procedures\n    enables us to deal with processes at a higher conceptual level than that of\n    the primitive operations of the language, the ability to construct compound\n    data objects enables us to deal with data at a higher conceptual level than\n    that of the primitive data objects of the language.\n  ","2#p4":"\n    Consider the task of designing a system to perform\n    \n    arithmetic with rational\n    numbers.  We could imagine an operation\n    add-rat\n    that takes two rational numbers and produces their sum.  In terms of\n    simple data, a rational number can be thought of as two integers: a\n    numerator and a denominator.  Thus, we could design a program in which\n    each rational number would be represented by two integers (a numerator\n    and a denominator) and where\n    add-rat\n    would be implemented by two\n    procedures\n    (one producing the numerator of the sum and one producing\n    the denominator).  But this would be awkward, because we would then\n    need to explicitly keep track of which numerators corresponded to\n    which denominators.  In a system intended to perform many operations\n    on many rational numbers, such bookkeeping details would clutter the\n    programs substantially, to say nothing of what they would do to our\n    minds.  It would be much better if we could \"glue together\"\n    a numerator and denominator to form a pair—a compound data\n    object—that our programs could manipulate in a way that would\n    be consistent with regarding a rational number as a single conceptual\n    unit.\n  ","2#p5":"\n    The use of compound data also enables us to increase the modularity of\n    our programs.  If we can manipulate rational numbers directly as\n    objects in their own right, then we can separate the part of our\n    program that deals with rational numbers per se from the details of\n    how rational numbers may be represented as pairs of integers.  The\n    general technique of isolating the parts of a program that deal with\n    how data objects are represented from the parts of a program that deal\n    with how data objects are used is a powerful design methodology called\n    data abstraction.  We will see how data abstraction makes\n    programs much easier to design, maintain, and modify.\n  ","2#p6":"\n    The use of compound data leads to a real increase in the expressive power\n    of our programming language.  Consider the idea of forming a\n    \"linear combination\"$ax+by$.  We\n    might like to write a\n    procedure\n    that would accept $a$,\n    $b$, $x$, and\n    $y$ as arguments and return the value of\n    $ax+by$.  This presents no difficulty if the\n    arguments are to be numbers, because we can readily \n    define the procedure(define (linear-combination a b x y) \n  (+ (* a x) (* b y))) \n    But suppose we are not concerned only with numbers. Suppose we would like to\n    express, in procedural terms, the idea that one can form\n    linear combinations whenever addition and multiplication are\n    defined—for rational numbers, complex numbers, polynomials, or\n    whatever.  We could express this as a\n    procedure\n    of the form\n    (define (linear-combination a b x y)\n  (add (mul a x) (mul b y)))\n    where add and mul\n    are not the primitive\n    procedures+ and * but rather\n    more complex things that will perform the appropriate operations for\n    whatever kinds of data we pass in as the arguments\n    a, b,\n    x, and y. The key\n    point is that the only thing\n    linear-combination\n    should need to know about a,\n    b, x, and\n    y is that the\n    proceduresadd and mul will\n    perform the appropriate manipulations.  From the perspective of the\n    procedurelinear-combination,\n    it is irrelevant what a,\n    b, x, and\n    y are and even more irrelevant how they might\n    happen to be represented in terms of more primitive data.  This same example\n    shows why it is important that our programming language provide the ability\n    to manipulate compound objects directly: Without this, there is no way for a\n    procedure\n    such as\n    linear-combination\n    to pass its arguments along to add and\n    mul without having to know their detailed\n    structure.","2#footnote-link-1":"1","2#p7":"\n    We begin this chapter by implementing the rational-number arithmetic system\n    mentioned above.  This will form the background for our discussion of\n    compound data and data abstraction.  As with compound\n    procedures,\n    the main issue to be addressed is that of abstraction as a technique for\n    coping with complexity, and we will see how data abstraction enables us to\n    erect suitable \n    abstraction barriers\n    between different parts of a program.\n  ","2#p8":"\n    We will see that the key to forming compound data is that a programming\n    language should provide some kind of \"glue\" so that data\n    objects can be combined to form more complex data objects.  There are\n    many possible kinds of glue.  Indeed, we will discover how to form compound\n    data using no special \"data\" operations at all, only\n    procedures.\n    This will further blur the distinction between\n    \"procedure\"\n    and \"data,\" which was already becoming tenuous toward the end\n    of chapter 1.  We will also explore some conventional techniques for\n    representing sequences and trees.  One key idea in dealing with compound\n    data is the notion of \n    closure—that the\n    glue we use for combining data objects should allow us to combine not only\n    primitive data objects, but compound data objects as well. Another key idea\n    is that compound data objects can serve as \n    conventional interfaces for combining program modules in\n    mix-and-match ways.  We illustrate some of these ideas by presenting a\n    simple graphics language that exploits closure.\n  ","2#p9":"\n    We will then augment the representational power of our language by\n    introducing \n    symbolic expressions—data whose elementary parts\n    can be arbitrary symbols rather than only numbers.  We explore various\n    alternatives for representing sets of objects.  We will find that,\n    just as a given numerical function can be computed by many different\n    computational processes, there are many ways in which a given data\n    structure can be represented in terms of simpler objects, and the\n    choice of representation can have significant impact on the time and\n    space requirements of processes that manipulate the data.  We will\n    investigate these ideas in the context of symbolic differentiation,\n    the representation of sets, and the encoding of information.\n  ","2#p10":"\n    Next we will take up the problem of working with data that may be\n    represented differently by different parts of a program.  This leads\n    to the need to implement \n    generic operations, which must handle many different types of data.\n    Maintaining modularity in the presence of generic operations requires more\n    powerful abstraction barriers than can be erected with simple data\n    abstraction alone. In particular, we introduce data-directed\n    programming as a technique that allows individual data representations\n    to be designed in isolation and then combined \n    additively (i.e., without modification).  To illustrate the power\n    of this approach to system design, we close the chapter by applying what we\n    have learned to the implementation of a package for performing symbolic\n    arithmetic on polynomials, in which the coefficients of the polynomials can\n    be integers, rational numbers, complex numbers, and even other polynomials.\n  ","2#footnote-1":"The ability to directly manipulate\n    procedures\n    provides an analogous increase in the expressive power of a programming\n    language.  For example, in\n    section 1.3.1 we introduced the\n    sumprocedure,\n    which takes a\n    procedureterm as an argument and computes the sum of\n    the values of term over some specified interval.\n    In order to define sum, it is crucial that we\n    be able to speak of a\n    procedure\n    such as term as an entity in its own right,\n    without regard for how term might be expressed\n    with more primitive operations.  Indeed, if we did not have the notion of\n    \"a procedure,\"\n    it is doubtful that we would ever even think of the possibility of defining\n    an operation such as sum.  Moreover, insofar as\n    performing the summation is concerned, the details of how\n    term may be constructed from more primitive\n    operations are irrelevant.","2.1":"2.1  Introduction to Data Abstraction","2.1#p1":"\n    In section 1.1.8, we noted that a\n    procedure\n    used as an element in creating a more complex\n    procedure\n    could be regarded not only as a collection of particular operations but\n    also as a \n    procedural\n    abstraction.  That is, the details of how the\n    procedure\n    was implemented could be suppressed, and the particular\n    procedure\n    itself could be replaced by any other\n    procedure\n    with the same overall behavior.  In other words, we could make an\n    abstraction that would separate the way the\n    procedure\n    would be used from the details of how the\n    procedure\n    would be implemented in terms of more primitive\n    procedures.\n    The analogous notion for compound data is called \n    data abstraction.  Data abstraction is a methodology that enables\n    us to isolate how a compound data object is used from the details of how it\n    is constructed from more primitive data objects.\n  ","2.1#p2":"\n    The basic idea of data abstraction is to structure the programs that are\n    to use compound data objects so that they operate on \n    \"abstract data.\" That is, our programs should use data in such\n    a way as to make no assumptions about the data that are not strictly\n    necessary for performing the task at hand.  At the same time, a \n    \"concrete\" data representation is defined independent of the\n    programs that use the data.  The interface between these two parts of our\n    system will be a set of\n    procedures,\n    called \n    selectors and \n    constructors, that implement the abstract data in terms of the\n    concrete representation.  To illustrate this technique, we will consider\n    how to design a set of\n    procedures\n    for manipulating rational numbers.\n  ","2.1.1":"2.1.1  \n    Example: Arithmetic Operations for Rational Numbers","2.1.1#p1":"\n    Suppose we want to do\n    \n    arithmetic with rational numbers.  We want to be\n    able to add, subtract, multiply, and divide them and to test whether\n    two rational numbers are equal.\n  ","2.1.1#p2":"\n    Let us begin by assuming that we already have a way of constructing a\n    rational number from a numerator and a denominator.  We also assume\n    that, given a rational number, we have a way of extracting (or\n    selecting) its numerator and its denominator.  Let us further assume\n    that the constructor and selectors are available as\n    procedures:(make-rat n d) \n\treturns the\n\trational number whose  numerator is the integer\n\t$n$ and whose denominator is the integer\n\t$d$. \n      (numer x) \n\treturns the numerator of the rational number\n\t$x$.\n      (denom x) \n\treturns the denominator of the rational number\n\t$x$.\n      ","2.1.1#p3":"\n    We are using here a powerful strategy of synthesis: \n    wishful thinking. We haven't yet said how a rational number\n    is represented, or how the\n    proceduresnumer, denom, and \n    make-rat\n    should be implemented.  Even so, if we did have these three\n    procedures,\n    we could then add, subtract, multiply, divide, and test equality by using\n    the following relations:\n\n    \n      \\[\n      \\begin{array}{rll}\n      \\dfrac{n_{1}}{d_{1}}+\\dfrac{n_{2}}{d_{2}}\n      &=&\\dfrac{n_{1}d_{2}+n_{2}d_{1}}{d_{1}d_{2}}\\\\[15pt]\n      \\dfrac{n_{1}}{d_{1}}-\\dfrac{n_{2}}{d_{2}}\n      &=&\\dfrac{n_{1}d_{2}-n_{2}d_{1}}{d_{1}d_{2}}\\\\[15pt]\n      \\dfrac{n_{1}}{d_{1}}\\cdot\\dfrac{n_{2}}{d_{2}}\n      &=&\\dfrac{n_{1}n_{2}}{d_{1}d_{2}}\\\\[15pt]\n      \\dfrac{n_{1}/d_{1}}{n_{2}/d_{2}}\n      &=&\\dfrac{n_{1}d_{2}}{d_{1}n_{2}}\\\\[15pt]\n      \\dfrac{n_{1}}{d_{1}}\n      &=&\\dfrac{n_{2}}{d_{2}}\\ \\quad \\textrm{if and only if}\\ \\ \\  n_{1}d_{2}\\ =\\ n_{2}d_{1}\n      \\end{array}\n      \\]\n    ","2.1.1#p4":"\n    We can express these rules as\n    procedures:(define (add-rat x y)\n  (make-rat (+ (* (numer x) (denom y))\n               (* (numer y) (denom x)))\n            (* (denom x) (denom y))))\n\n(define (sub-rat x y)\n  (make-rat (- (* (numer x) (denom y))\n               (* (numer y) (denom x)))\n            (* (denom x) (denom y))))\n\n(define (mul-rat x y)\n  (make-rat (* (numer x) (numer y))\n            (* (denom x) (denom y))))\n\n(define (div-rat x y)\n  (make-rat (* (numer x) (denom y))\n            (* (denom x) (numer y))))\n\n(define (equal-rat? x y)\n  (= (* (numer x) (denom y))\n     (* (numer y) (denom x))))","2.1.1#p5":"\n    Now we have the operations on rational numbers defined in terms of the\n    selector and constructor\n    proceduresnumer, denom, and\n    make-rat.\n    But we haven't yet defined these. What we need is some way to glue\n    together a numerator and a denominator to form a rational number.\n  ","2.1.1#h1":"Pairs","2.1.1#p6":"\n    To enable us to implement the concrete level of our data abstraction, our\n    \n\tlanguage\n      \n    provides a compound structure called a \n    pair, which can be constructed with the \n      primitive procedurecons.\n      This\n      procedure\n      takes two arguments and returns a compound data object that contains the\n      two arguments as parts.  Given a pair, we can extract the parts using the\n      primitive\n      procedurescar \n      and\n      cdr.\n      Thus, we can use\n      cons,car,\n      and\n      cdr\n      as follows:\n      (define x (cons 1 2)) (car x) (cdr x) \n      Notice that a pair is a data object that can be given a name and\n      manipulated, just like a primitive data object.  Moreover,\n      cons\n      can be used to form pairs whose elements are pairs, and so on:\n      (define x (cons 1 2))\n\n(define y (cons 3 4))\n\n(define z (cons x y)) (car (car z)) (car (cdr z)) \n      In section 2.2 we will see how this\n      ability to combine pairs means that pairs can be used as general-purpose\n      building blocks to create all sorts of complex data structures.  The single\n      compound-data primitive pair, implemented by the\n      procedurescons,car,\n      and\n      cdr,\n      is the only glue we need.  Data objects constructed from pairs are called \n      list-structured data.\n    ","2.1.1#footnote-link-1":"1","2.1.1#h2":"Representing rational numbers","2.1.1#p7":"\n      Pairs offer a natural way to complete the\n      \n      rational-number system.\n      Simply represent a rational number as a pair of two integers: a numerator\n      and a denominator.  Then\n      make-rat,numer, and denom\n      are readily implemented as follows:(define (make-rat n d) (cons n d))\n\n(define (numer x) (car x))\n\n(define (denom x) (cdr x)) \n    Also, in order to display the results of our computations, we can \n    \n    print rational numbers by printing the numerator, a slash, and the \n    \n\tdenominator:(define (print-rat x)\n  (newline)\n  (display (numer x))\n  (display \"/\")\n  (display (denom x))) \n    Now we can try our rational-number\n    procedures:(define one-half (make-rat 1 2))\n\n(print-rat one-half) (define one-third (make-rat 1 3))(print-rat (add-rat one-half one-third)) (print-rat (mul-rat one-half one-third)) (print-rat (add-rat one-third one-third)) ","2.1.1#footnote-link-2":"2","2.1.1#footnote-link-3":"3","2.1.1#p8":"\n    As the final example shows, our rational-number implementation does not\n    \n    reduce rational numbers to lowest terms.  We can remedy this by changing\n    make-rat.\n    If we have a \n    gcdprocedure\n    like the one in section 1.2.5 that produces\n    \n    the greatest common divisor of two integers, we can use\n    gcd to reduce the numerator and the\n    denominator to lowest terms before constructing the pair:\n    (define (make-rat n d)\n  (let ((g (gcd n d)))\n    (cons (/ n g) (/ d g)))) \n    Now we have\n    (print-rat (add-rat one-third one-third)) \n    as desired.  This modification was accomplished by changing the constructor\n    make-rat without changing any of the\n      procedures\n      (such as\n      add-rat\n      and\n      mul-rat)\n      that implement the actual operations.\n  ","2.1.1#ex-2.1":" \n    Define a better version of\n    make-rat\n    that handles both positive and negative arguments.\n    Make-rat\n    should normalize the sign so that if the rational number is positive, both\n    the numerator and denominator are positive, and if the rational number is\n    negative, only the numerator is negative.\n    ","2.1.1#footnote-1":"The name \n      cons stands for \"construct.\"\n      The names \n      car and \n      cdr derive from the original implementation\n      of Lisp on the \n      \n      IBM 704.  That machine had an addressing scheme that allowed one to\n      reference the \"address\" and \"decrement\" parts of\n      a memory location.  Car stands for\n      \"Contents of Address part of Register\" and\n      cdr (pronounced \"could-er\")\n      stands for \"Contents of Decrement part of\n      Register.\"","2.1.1#footnote-2":"\n      Another way to define the selectors and constructor is\n      (define make-rat cons)\n(define numer car)\n(define denom cdr)\n      The first definition associates the name\n      make-rat\n      with the value of the expression\n      cons,\n      which is the primitive\n      procedure\n      that constructs pairs.  Thus\n      make-rat\n      and\n      cons\n      are names for the same primitive constructor.\n      \n        Defining selectors and constructors in this way is efficient: Instead of\n\tmake-ratcallingcons,make-ratiscons,\n\tso there is only one\n        procedure\n        called, not two, when\n\tmake-rat\n\tis called.  On the other hand, doing this defeats debugging aids that\n\ttrace\n        procedure\n        calls or put breakpoints on\n        procedure\n        calls:\n        You may want to watch\n\tmake-rat\n\tbeing called, but you certainly don't want to watch every call to\n\tcons.\n        We have chosen not to use this style of definition in this book.\n\t","2.1.1#footnote-3":"Display is\n\tthe Scheme primitive for\n\t\n\tprinting data.  The Scheme primitive\n\tnewline starts a new line for printing.\n\tNeither of these procedures returns a useful value, so in the uses of\n\tprint-rat below, we show only what\n\tprint-rat prints, not what the interpreter\n\tprints as the value returned by\n\tprint-rat.","2.1.2":"2.1.2  \n    Abstraction Barriers","2.1.2#p1":"\n    Before continuing with more examples of compound data and data\n    abstraction, let us consider some of the issues raised by the\n    rational-number example.  We defined the rational-number operations in\n    terms of a constructor\n    make-rat\n    and selectors numer and\n    denom.  In general, the underlying idea of data\n    abstraction is to identify for each type of data object a basic set of\n    operations in terms of which all manipulations of data objects of that type\n    will be expressed, and then to use only those operations in manipulating the\n    data.\n  ","2.1.2#fig-":"","2.1.2#p2":"\n    We can envision the structure of the rational-number system as\n    shown in\n    \n\tfigure .\n      \n    The horizontal lines represent abstraction barriers that isolate\n    different \"levels\" of the system.  At each level, the barrier\n    separates the programs (above) that use the data abstraction from the\n    programs (below) that implement the data abstraction.  Programs that\n    use rational numbers manipulate them solely in terms of the\n    procedures\n    supplied \"for public use\" by the rational-number package:\n    add-rat,sub-rat,mul-rat,div-rat,\n    and\n    equal-rat?.\n    These, in turn, are implemented solely in terms of the\n    \n    constructor and\n    \n    selectors\n    make-rat,numer, and denom,\n    which themselves are implemented in terms of pairs.  The details of how\n    pairs are implemented are irrelevant to the rest of the rational-number\n    package so long as pairs can be manipulated by the use of\n    cons,car,\n    and\n    cdr.\n    In effect,\n    procedures\n    at each level are the interfaces that define the abstraction barriers and\n    connect the different levels.\n  ","2.1.2#p3":"\n    This simple idea has many advantages.  One advantage is that it makes\n    programs much easier to maintain and to modify.  Any complex data\n    structure can be represented in a variety of ways with the primitive\n    data structures provided by a programming language.  Of course, the\n    choice of representation influences the programs that operate on it;\n    thus, if the representation were to be changed at some later time, all\n    such programs might have to be modified accordingly.  This task could\n    be time-consuming and expensive in the case of large programs unless\n    the dependence on the representation were to be confined by design to\n    a very few program modules.\n  ","2.1.2#p4":"\n    For example, an alternate way to address the problem of\n    \n    reducing rational\n    numbers to lowest terms is to perform the reduction whenever we\n    access the parts of a rational number, rather than when we construct\n    it.  This leads to different constructor and selector\n    procedures:(define (make-rat n d)\n  (cons n d))\n\n(define (numer x)\n  (let ((g (gcd (car x) (cdr x))))\n    (/ (car x) g)))\n\n(define (denom x)\n  (let ((g (gcd (car x) (cdr x))))\n    (/ (cdr x) g))) \n    The difference between this implementation and the previous one lies in when\n    we compute the gcd. If in our typical use of\n    rational numbers we access the numerators and denominators of the same\n    rational numbers many times, it would be preferable to compute the\n    gcd when the rational numbers are constructed.\n    If not, we may be better off waiting until access time to compute the\n    gcd.  In any case, when we change from one\n    representation to the other, the\n    proceduresadd-rat,sub-rat,\n    and so on do not have to be modified at all.\n  ","2.1.2#p5":"\n    Constraining the dependence on the representation to a few interface\n    procedures\n    helps us design programs as well as modify them, because it allows us to\n    maintain the flexibility to consider alternate implementations.  To continue\n    with our simple example, suppose we are designing a rational-number package\n    and we can't decide initially whether to perform the\n    gcd at construction time or at selection time.\n    The data-abstraction methodology gives us a way to defer that decision\n    without losing the ability to make progress on the rest of the system.\n  ","2.1.2#ex-2.2":"\n    Consider the problem of representing \n    \n    line segments in a plane.  Each segment is represented as a pair of points:\n    a starting point and an ending point.\n    Define\n    a constructor \n     make-segment\n    and selectors \n     start-segment\n    and \n     end-segment\n    that define the representation of segments in\n    terms of points.  Furthermore, a point \n    \n    can be represented as a pair\n    of numbers: the $x$ coordinate and the\n    $y$ coordinate.  Accordingly, specify a\n    constructor \n     make-point\n    and selectors\n    x-point\n    and\n    y-point\n    that define this representation. Finally, using your selectors and\n    constructors,\n    define a proceduremidpoint-segment\n    that takes a line segment as argument and returns its midpoint (the point\n    whose coordinates are the average of the coordinates of the endpoints).\n    To try your\n    procedures,\n    you'll need a way to print points:\n    (define (print-point p)\n  (newline)\n  (display \"(\")\n  (display (x-point p))\n  (display \",\")\n  (display (y-point p))\n  (display \")\"))","2.1.2#ex-2.3":"\n    Implement a representation for\n    \n    rectangles in a plane. (Hint: You may want to\n    make use of exercise 2.2.) In terms of your\n    constructors and selectors, create\n    procedures\n    that compute the perimeter and the area of a given rectangle.  Now implement\n    a different representation for rectangles.  Can you design your system with\n    suitable abstraction barriers, so that the same perimeter and area\n    procedures\n    will work using either representation?\n    ","2.1.3":"2.1.3  \n    What Is Meant by Data?","2.1.3#p1":"\n    We began the rational-number implementation in\n    section 2.1.1 by implementing the\n    rational-number operations\n    add-rat,sub-rat,\n    and so on in terms of three unspecified\n    procedures:make-rat,numer, and\n    denom. At that point, we could think of the\n    operations as being defined in terms of data objects—numerators,\n    denominators, and rational numbers—whose behavior was specified\n    by the latter three\n    procedures.","2.1.3#p2":"\n    But exactly what is meant by data?  It is not enough to say\n    \"whatever is implemented by the given selectors and\n    constructors.\" Clearly, not every arbitrary set of three\n    procedures\n    can serve as an appropriate basis for the rational-number\n    implementation.  We need to guarantee that,\n    \n    if we construct a rational number x from a\n    pair of integers n and\n    d, then extracting the\n    numer and the\n    denom of x and\n    dividing them should yield the same result as dividing\n    n by d. In\n    other words,\n    make-rat,numer, and\n    denom must satisfy the condition that, for\n    any integer n and any nonzero\n    integer d, if x is\n    (make-rat n d),\n    then\n    \n          \\[\n\t  \\begin{array}{lll}\n          \\dfrac{(\\texttt{numer}~\\texttt{x})}{(\\texttt{denom}~\\texttt{x})}\n          &=&\n          \\dfrac{\\texttt{n}}{\\texttt{d}}\n\t  \\end{array}\n          \\]\n        \n    In fact, this is the only condition\n    make-rat,numer, and\n    denom must fulfill in order to form a\n    suitable basis for a rational-number representation.  In general, we can\n    think of data as defined by some collection of selectors and\n    constructors, together with specified conditions that these\n    procedures\n    must fulfill in order to be a valid\n    representation.","2.1.3#footnote-link-1":"1","2.1.3#p3":"\n    This point of view can serve to define not only\n    \"high-level\" data objects, such as rational numbers, but\n    lower-level objects as well.\n    Consider the notion of a\n    \n    pair, which we used in order to define our\n    rational numbers.  We never actually said what a pair was, only that\n    the language supplied\n    procedurescons,car,\n    and\n    cdr\n    for operating on pairs.  But the only thing we need to know about these\n    three operations \n    is that if we glue two objects together using\n    cons\n    we can retrieve the objects using\n    car\n    and\n    cdr.\n    That is, the operations satisfy the condition that, for any objects\n    x and y, if\n    z is\n    (cons x y)\n    then\n    (car z)\n    is x and\n    (cdr z)\n    is y.  Indeed, we mentioned that these three\n    procedures\n    are included as primitives in our language. However, any triple of\n    procedures\n    that satisfies the above condition can be used as the basis for\n    implementing pairs.  This point is illustrated strikingly by the fact\n    that we could implement\n    cons,car,\n    and\n    cdr\n    without using any data structures at all but only using\n    procedures.\n    Here are the definitions:(define (cons x y)\n  (define (dispatch m)\n    (cond ((= m 0) x)\n          ((= m 1) y)\n          (else (error \"Argument not 0 or 1 -- CONS\" m))))\n  dispatch)\n\n(define (car z) (z 0))\n\n(define (cdr z) (z 1)) \n    This use of\n    procedures\n    corresponds to nothing like our intuitive notion of what data should be.\n    Nevertheless, all we need to do to show that this is a valid way to\n    represent pairs is to verify that these\n    procedures\n    satisfy the condition given above.\n  ","2.1.3#p4":"\n    The subtle point to notice is that the value returned by\n    (cons x y)\n    is a\n    procedure—namely\n    the internally defined\n    proceduredispatch, which takes one argument and returns\n    either x or y\n    depending on whether the argument is 0 or 1.  Correspondingly,\n    (car z)\n    is defined to apply z to 0.  Hence, if\n    z is the\n    procedure\n    formed by\n    (cons x y),\n    then z applied to 0 will yield\n    x. Thus, we have shown that\n    (car (cons x y))\n    yields x, as desired.  Similarly,\n    (cdr (cons x y))\n    applies the\n    procedure\n    returned by\n    (cons x y)\n    to 1, which returns y.\n    Therefore, this \n    \n        procedural \n      \n    implementation of pairs is a valid\n    implementation, and if we access pairs using only\n    cons,car,\n    and\n    cdr\n    we cannot distinguish this implementation from one that uses\n    \"real\" data structures.\n  ","2.1.3#p5":"\n    The point of exhibiting the \n    \n        procedural \n      \n    representation of pairs is not that our language works this way \n    (Scheme, and Lisp systems in general,\n      implement pairs directly, for efficiency reasons)\n      \n    but that it could work this way. The \n    \n        procedural \n      \n    representation, although obscure, is a perfectly adequate way to represent\n    pairs, since it fulfills the only conditions that pairs need to fulfill.\n    This example also demonstrates that the ability to manipulate\n    procedures\n    as objects automatically provides the ability to represent compound data.\n    This may seem a curiosity now, but \n    \n        procedural \n      \n    representations of data will play a central role in our programming\n    repertoire.  This style of programming is often called \n    message passing, and we will be using it as a basic tool in\n    chapter 3 when we address the issues of modeling and simulation.\n  ","2.1.3#ex-2.4":" \n    Here is an alternative \n    \n        procedural \n      \n    representation of pairs.  For this\n    representation, verify that\n    (car (cons x y))\n    yields x for any objects\n    x and y.\n    (define (cons x y)\n  (lambda (m) (m x y)))\n\n(define (car z)\n  (z (lambda (p q) p))) \n    What is the corresponding definition of\n    cdr?\n    (Hint: To verify that this works, make use of the substitution model of\n    section 1.1.5.)\n    ","2.1.3#ex-2.5":" \n    Show that we can represent pairs of nonnegative integers using only\n    numbers and arithmetic operations if we represent the pair\n    $a$ and $b$ as the\n    integer that is the product $2^a 3^b$.  Give the\n    corresponding definitions of the\n    procedurescons,car,\n    and\n    cdr.","2.1.3#ex-2.6":" \n    In case representing pairs as\n    procedures\n    (exercise 2.4)\n    wasn't mind-boggling enough, consider that, in a language that can\n    manipulate\n    procedures,\n    we can get by without numbers (at least insofar as nonnegative integers\n    are concerned) by implementing 0 and the operation of adding 1 as\n    (define zero (lambda (f) (lambda (x) x)))\n\n(define (add-1 n)\n  (lambda (f) (lambda (x) (f ((n f) x))))) \n    This representation is known as \n    Church numerals, after its inventor, \n    \n    Alonzo Church, the logician who invented the\n    $\\lambda$ calculus.\n    ","2.1.3#p6":"\n    Define one and two\n    directly (not in terms of zero and\n    add-1).\n    (Hint: Use substitution to evaluate\n    (add-1 zero)).\n    Give a direct definition of the addition\n    procedure +\n    (not in terms of repeated application of\n    add-1).","2.1.3#footnote-1":"Surprisingly, this idea is very difficult to\n    formulate rigorously. There are two approaches to giving such a\n    formulation.  One, pioneered by\n    \n    C. A. R. Hoare (1972), is known as the method of \n    abstract models.  It formalizes the\n    \"procedures plus conditions\"\n    specification as outlined in the rational-number example above.  Note\n    that the condition on the rational-number representation was stated in\n    terms of facts about integers (equality and division).  In general,\n    abstract models define new kinds of data objects in terms of previously\n    defined types of data objects.  Assertions about data objects can\n    therefore be checked by reducing them to assertions about previously\n    defined data objects. Another approach, introduced by \n    \n    Zilles at MIT, by \n    \n    Goguen, \n    \n    Thatcher,\n    \n    Wagner, and \n    \n    Wright at IBM (see Thatcher, Wagner, and Wright\n    1978), and by \n    \n    Guttag at Toronto (see Guttag 1977),\n    is called \n    algebraic specification.  It regards the\n    \"procedures\"\n    as elements of an abstract algebraic system whose behavior is\n    specified by axioms that correspond to our \"conditions,\"\n    and uses the techniques of abstract algebra to check assertions about\n    data objects.  Both methods are surveyed in the paper by \n    \n    Liskov and Zilles\n    (1975).","2.1.4":"2.1.4  \n    Extended Exercise: Interval Arithmetic","2.1.4#p1":"\n    Alyssa P. Hacker is designing a system to help people solve\n    engineering problems.  One feature she wants to provide in her system\n    is the ability to manipulate inexact quantities (such as measured\n    parameters of physical devices) with known precision, so that when\n    computations are done with such approximate quantities the results\n    will be numbers of known precision.\n  ","2.1.4#p2":"\n    Electrical engineers will be using Alyssa's system to compute\n    electrical quantities.  It is sometimes necessary for them to compute\n    the value of a parallel equivalent resistance\n    $R_{p}$ of two resistors\n    $R_{1}$ and $R_{2}$\n    using the formula\n    \n      \\[\n      \\begin{array}{lll}\n      R_{p} & = & \\dfrac{1}{1/R_{1}+1/R_{2}}\n      \\end{array}\n      \\]\n    \n    Resistance values are usually known only up to some \n    \n    tolerance guaranteed by the manufacturer of the resistor.  For example, if\n    you buy a resistor labeled \"6.8 ohms with 10% tolerance\" you can\n    only be sure that the resistor has a resistance between\n    $6.8-0.68=6.12$ and\n    $6.8+0.68=7.48$ ohms.  Thus, if you have a\n    6.8-ohm 10% resistor in parallel with a 4.7-ohm\n    5% resistor, the resistance of the combination can range from about\n    2.58 ohms (if the two resistors are at the lower bounds) to about 2.97 ohms\n    (if the two resistors are at the upper bounds).\n  ","2.1.4#p3":"\n    Alyssa's idea is to implement \"interval arithmetic\" as a\n    set of arithmetic operations for combining \"intervals\" (objects\n    that represent the range of possible values of an inexact quantity). The\n    result of adding, subtracting, multiplying, or dividing two intervals is\n    itself an interval, representing the range of the result.\n  ","2.1.4#p4":"\n    Alyssa postulates the existence of an abstract object called an\n    \"interval\" that has two endpoints: a lower bound and an upper bound.\n    She also presumes that, given the endpoints of an interval, she can\n    construct the interval using the data constructor \n     make-interval.\n    Alyssa first writes a\n    procedure\n    for adding two intervals.  She reasons that the minimum value the sum could\n    be is the sum of the two lower bounds and the maximum value it could be is\n    the sum of the two upper bounds:\n    (define (add-interval x y)\n  (make-interval (+ (lower-bound x) (lower-bound y))\n                 (+ (upper-bound x) (upper-bound y)))) \n    Alyssa also works out the product of two intervals by finding the\n    minimum and the maximum of the products of the bounds and using them\n    as the bounds of the resulting interval.\n    (Min\n    and\n    max\n    are\n    primitives that find the minimum or maximum of any number of arguments.)\n    (define (mul-interval x y)\n  (let ((p1 (* (lower-bound x) (lower-bound y)))\n        (p2 (* (lower-bound x) (upper-bound y)))\n        (p3 (* (upper-bound x) (lower-bound y)))\n        (p4 (* (upper-bound x) (upper-bound y))))\n    (make-interval (min p1 p2 p3 p4)\n                   (max p1 p2 p3 p4)))) \n    To divide two intervals, Alyssa multiplies the first by the reciprocal of\n    the second.  Note that the bounds of the reciprocal interval are\n    the reciprocal of the upper bound and the reciprocal of the lower bound, in\n    that order.\n    (define (div-interval x y)\n  (mul-interval x \n                (make-interval (/ 1.0 (upper-bound y))\n                               (/ 1.0 (lower-bound y))))) ","2.1.4#ex-2.7":"\n    Alyssa's program is incomplete because she has not specified the\n    implementation of the interval abstraction.  Here is a definition of\n    the interval constructor:\n    (define (make-interval a b) (cons a b))\n    Define selectors \n    upper-bound\n    and \n     lower-bound\n    to complete the implementation.\n    ","2.1.4#ex-2.8":" \n    Using reasoning analogous to Alyssa's, describe how the difference\n    of two intervals may be computed.  Define a corresponding subtraction\n    procedure,\n    called \n     sub-interval.","2.1.4#ex-2.9":" \n    The\n    width of an interval is half of the difference between its\n    upper and lower bounds.  The width is a measure of the uncertainty of\n    the number specified by the interval.  For some arithmetic operations\n    the width of the result of combining two intervals is a function only\n    of the widths of the argument intervals, whereas for others the width\n    of the combination is not a function of the widths of the argument\n    intervals.  Show that the width of the sum (or difference) of two\n    intervals is a function only of the widths of the intervals being\n    added (or subtracted).  Give examples to show that this is not true\n    for multiplication or division.\n    ","2.1.4#ex-2.10":"\n    Ben Bitdiddle, an expert systems programmer, looks over Alyssa's\n    shoulder and comments that it is not clear what it means to\n    \n    divide by an interval that spans zero.  Modify Alyssa's program to\n    check for this condition and to signal an error if it occurs.\n    ","2.1.4#ex-2.11":"\n    In passing, Ben also cryptically comments: \"By testing the signs of\n    the endpoints of the intervals, it is possible to break\n    mul-interval\n    into nine cases, only one of which requires more than two\n    multiplications.\"  Rewrite this\n    procedure\n    using Ben's suggestion.\n    ","2.1.4#p5":"\n    After debugging her program, Alyssa shows it to a potential user, who\n    complains that her program solves the wrong problem.  He wants a program\n    that can deal with numbers represented as a center value and an additive\n    tolerance; for example, he wants to work with intervals such as\n    $3.5\\pm 0.15$ rather than\n    $[3.35, 3.65]$. Alyssa returns to her desk and\n    fixes this problem by supplying an alternate constructor and alternate\n    selectors:\n    (define (make-center-width c w)\n  (make-interval (- c w) (+ c w)))\n\n(define (center i)\n  (/ (+ (lower-bound i) (upper-bound i)) 2))\n\n(define (width i)\n  (/ (- (upper-bound i) (lower-bound i)) 2)) ","2.1.4#p6":"\n    Unfortunately, most of Alyssa's users are engineers.  Real engineering\n    situations usually involve measurements with only a small uncertainty,\n    measured as the ratio of the width of the interval to the midpoint of the\n    interval.  Engineers usually specify percentage tolerances on the parameters\n    of devices, as in the resistor specifications given earlier.\n  ","2.1.4#ex-2.12":"\n    Define a constructor \n     make-center-percent\n    that takes a center and a percentage tolerance and produces the desired\n    interval.  You must also define a selector\n    percent that produces the percentage tolerance\n    for a given interval.  The center selector is\n    the same as the one shown above.\n    ","2.1.4#ex-2.13":" \n    Show that under the assumption of small percentage tolerances there is\n    a simple formula for the approximate percentage tolerance of the\n    product of two intervals in terms of the tolerances of the factors.\n    You may simplify the problem by assuming that all numbers are\n    positive.\n    ","2.1.4#p7":"\n    After considerable work, Alyssa P. Hacker delivers her finished\n    system.  Several years later, after she has forgotten all about it, she\n    gets a frenzied call from an irate user,  Lem E. Tweakit.\n    It seems that Lem has\n    noticed that the\n    \n    formula for parallel resistors can be written in two\n    algebraically equivalent ways:\n    \n      \\[\n      \\dfrac{R_{1}R_{2}}{R_{1}+R_{2}}\n      \\]\n    \n    and\n    \n      \\[\n      \\dfrac{1}{1/R_{1}+1/R_{2}}\n      \\]\n    \n    He has written the following two programs, each of which computes the\n    parallel-resistors formula differently:\n    (define (par1 r1 r2)\n  (div-interval (mul-interval r1 r2)\n                (add-interval r1 r2)))\n\n(define (par2 r1 r2)\n  (let ((one (make-interval 1 1))) \n    (div-interval one\n                  (add-interval (div-interval one r1)\n                                (div-interval one r2))))) \n    Lem complains that Alyssa's program gives different answers for\n    the two ways of computing. This is a serious complaint.\n  ","2.1.4#ex-2.14":" \n    Demonstrate that Lem is right. Investigate the behavior of the\n    system on a variety of arithmetic expressions. Make some intervals\n    $A$ and $B$,\n    and use them in computing the expressions $A/A$\n    and $A/B$.  You will get the most insight by\n    using intervals whose width is a small percentage of the center value.\n    Examine the results of the computation in center-percent form (see\n    exercise 2.12).\n    ","2.1.4#ex-2.15":"\n    Eva Lu Ator, another user, has also noticed the different intervals\n    computed by different but algebraically equivalent expressions. She\n    says that a formula to compute with intervals using Alyssa's system\n    will produce tighter error bounds if it can be written in such a form\n    that no\n    \n\tvariable\n      \n    that represents an uncertain number is repeated. Thus, she says,\n    par2 is a \"better\" program for\n    parallel resistances than par1.  Is she right?\n    Why?\n    ","2.1.4#ex-2.16":"\n    Explain, in general, why equivalent algebraic expressions may lead to\n    different answers.  Can you devise an interval-arithmetic package that\n    does not have this shortcoming, or is this task impossible?  (Warning:\n    This problem is very difficult.)\n\n    ","2.2":"2.2  Hierarchical Data and the Closure Property","2.2#p1":"\n    As we have seen, pairs provide a primitive \"glue\" that we can\n    use to construct compound data objects.\n    \n\tFigure \n    shows a standard way to visualize a \n    \n    pair—in this case, the pair formed by\n    (cons 1 2).\n\tIn this representation, which is called \n\tbox-and-pointer\n        notation, each object is shown as a \n\tpointer to a box.  The box\n\tfor a primitive object contains a representation of the object.  For\n\texample, the box for a number contains a numeral.  The box for a pair\n\tis actually a double box, the left part containing (a pointer to) the\n\tcar of the pair and the right part\n\tcontaining the cdr.\n      ","2.2#fig-":"","2.2#p2":"\n    We have already seen that\n    cons\n    can be used to combine not only numbers but pairs as well.  (You made use\n    of this fact, or should have, in doing\n    exercises 2.2\n    and 2.3.)  As a consequence, pairs provide\n    a universal building block from which we can construct all sorts of data\n    structures.\n    \n\tFigure \n    shows two ways to use pairs to combine the numbers 1, 2, 3, and 4.\n    ","2.2#p3":"\n    The ability to create pairs whose elements are pairs is the essence of\n    list structure's importance as a representational tool.  We refer to\n    this ability as the \n    closure property of\n    cons.\n    In general, an operation for combining data objects satisfies the closure\n    property if the results of combining things with that operation can\n    themselves be combined using the same  operation.\n    Closure is the key to power in any means of combination because it permits\n    us to create \n    hierarchical structures—structures made up of parts, which\n      themselves are made up of parts, and so on.\n  ","2.2#footnote-link-1":"1","2.2#p4":"\n    From the outset of chapter 1, we've made essential use of\n    closure in dealing with\n    procedures,\n    because all but the very simplest programs rely on the fact that the\n    elements of a combination can themselves be combinations.  In this section,\n    we take up the consequences of closure for compound data.  We describe some\n    conventional techniques for using pairs to represent sequences and trees,\n    and we exhibit a graphics language that illustrates closure in a vivid \n    way.","2.2#footnote-link-2":"2","2.2#footnote-1":"The use of the\n    word \n    \"closure\" here comes from abstract algebra, where a set of\n    elements is said to be\n    closed under an operation if applying the operation\n    to elements in the set produces an element that is again an element of the\n    set.  The\n    Lisp\n    community also (unfortunately) uses the word \"closure\" to\n    describe a totally unrelated concept: A closure\n    is an implementation technique for representing\n    procedures with free variables.\n      \n    We do not use the word \"closure\" in this second sense in this\n    book.","2.2#footnote-2":"The notion that a means of\n    \n    combination should satisfy closure is a straightforward idea. Unfortunately,\n    the data combiners provided in many popular programming languages do not\n    satisfy closure, or make closure cumbersome to exploit.  In \n    \n    Fortran or \n    \n    Basic, one typically combines data elements by assembling them into\n    arrays—but one cannot form arrays whose elements are themselves\n    arrays.  \n    \n    Pascal and \n    \n    C admit structures whose elements are structures.  However, this requires\n    that the programmer manipulate pointers explicitly, and adhere to the\n    restriction that each field of a structure can contain only elements of a\n    prespecified form. Unlike Lisp with its pairs, these languages have no\n    built-in general-purpose glue that makes it easy to manipulate compound\n    data in a uniform way. This limitation lies behind Alan \n    \n    Perlis's comment in his foreword to this book: \"In Pascal the\n    plethora of declarable data structures induces a specialization within\n    functions that inhibits and penalizes casual cooperation.  It is better to\n    have 100 functions operate on one data structure than to have 10 functions\n    operate on 10 data structures.\"","2.2.1":"2.2.1  \n    Representing Sequences","2.2.1#p1":"\n\n    One of the useful structures we can build with pairs is a \n    sequence—an ordered collection of data objects.  There\n    are, of course, many ways to represent sequences in terms of pairs.  One\n    particularly straightforward representation is illustrated in\n    figure ,\n    where the sequence 1, 2, 3, 4 is represented as a chain of pairs.  The\n    car\n    of each pair is the\n    corresponding item in the chain, and the\n    cdr\n    of the pair is the next pair in the chain.  The\n    cdr\n    of the final pair signals the end of the\n    \n\tsequence by pointing to a\n\tdistinguished value that is not a pair,\n      \n    represented in box-and-pointer\n    diagrams as a diagonal line\n    \n    and in programs as\n    the value of the variable nil.\n    The entire sequence is constructed by nested\n    cons\n    operations:\n    (cons 1\n      (cons 2\n            (cons 3\n                  (cons 4 nil)))) ","2.2.1#fig-":"","2.2.1#p2":"\n    Such a sequence of pairs, formed by nested\n    conses,\n      \n    is called a\n    list, and\n    Scheme\n    provides a primitive called \n    list to help in constructing \n    lists.\n    The above sequence could be produced by\n    (list 1 2 3 4).\n    In general, \n    \n(list a$_{1}$ a$_{2}$ $\\ldots$ a$_{n}$)\n      \n    is equivalent to\n    \n(cons a$_{1}$ (cons a$_{2}$ (cons $\\ldots$ (cons a$_{n}$ nil) $\\ldots$)))\n      \n        Lisp systems conventionally print lists by printing the sequence of\n        elements, enclosed in parentheses.  Thus, the data object in\n        figure \n\tis printed as\n\t(1 2 3 4):\n      (define one-through-four (list 1 2 3 4)) one-through-four \n        Be careful not to confuse the expression\n\t(list 1 2 3 4) with the list\n\t(1 2 3 4), which is the result obtained\n\twhen the expression is evaluated.  Attempting to evaluate the\n\texpression (1 2 3 4) will signal an error\n\twhen the interpreter tries to apply the procedure\n        1 to arguments \n        2, 3, \n        and 4.\n    ","2.2.1#footnote-link-1":"1","2.2.1#p3":"\n    We can think of \n    car\n    as selecting the first item in the list, and of \n    cdr\n    as selecting the sublist consisting of all but the first item.  Nested\n    applications of\n    car\n    and\n    cdr\n    can be used to extract the second, third, and subsequent items in the\n    list.\n    The constructor \n    cons\n    makes a list like the original one, but with an additional item at the\n    beginning.\n    (car one-through-four) (cdr one-through-four) (car (cdr one-through-four)) (cons 10 one-through-four) (cons 5 one-through-four) \n        The value of nil, used to terminate the\n\tchain of pairs, can be thought of as a sequence of no elements, the \n        empty list.  The word nil is a contraction of the\n\tLatin word nihil, which means\n\t\"nothing.\"","2.2.1#footnote-link-2":"2","2.2.1#footnote-link-3":"3","2.2.1#h1":"List operations","2.2.1#p4":"\n    The use of pairs to represent sequences of elements as lists is accompanied\n    by conventional programming techniques for manipulating lists by\n    successively \n    \"cdring down\"\n\tthe lists.\n      \n    For example, the\n    procedurelist-ref\n    takes as arguments a list and a number $n$ and\n    returns the $n$th item of the list.  It is\n    customary to number the elements of the list beginning with 0.  The method\n    for computing\n    list-ref\n    is the following:\n    \n\tFor $n=0$,\n\tlist-ref\n\tshould return the\n\tcar\n\tof the list.\n      \n\tOtherwise,\n\tlist-ref\n\tshould return  the $(n-1)$st item of the\n\tcdr\n\tof the list.\n      (define (list-ref items n)\n  (if (= n 0)\n      (car items)\n      (list-ref (cdr items) (- n 1)))) (define squares (list 1 4 9 16 25))\n(list-ref squares 3) ","2.2.1#p5":"\n    Often we\n    cdr down the whole list.\n      \n    To aid in this,\n    Scheme\n    includes a primitive\n    predicate\n    null?,\n    which tests whether its argument is the empty list.  The\n    procedurelength, which returns the number of items in\n    a list, illustrates this typical pattern of use:\n    (define (length items)\n  (if (null? items)\n      0\n      (+ 1 (length (cdr items))))) (define odds (list 1 3 5 7))\n(length odds) \n    The lengthprocedure\n    implements a simple recursive plan. The reduction step is:\n    \n\tThe length of any list is 1 plus the\n\tlength of the\n\tcdr\n\tof the list.\n      \n    This is applied successively until we reach the base case:\n    \n\tThe length of the empty list is 0.\n      \n    We could also compute length in an iterative\n    style:\n    (define (length items)\n  (define (length-iter a count)\n    (if (null? a)\n        count\n        (length-iter (cdr a) (+ 1 count))))\n  (length-iter items 0)) ","2.2.1#p6":"\n    Another conventional programming technique is to \n    \"cons up\"\n\tthe heads and tails of an answer list while\n      cdring down a list,\n      \n    as in the\n    procedureappend, which takes two lists as arguments and\n    combines their elements to make a new list:\n    (append squares odds) (append odds squares) Append\n    is also implemented using a recursive plan. To\n    append lists\n    list1 and list2,\n    do the following:\n    \n\tIf list1 is the empty list, then the\n      result is just list2.\n      \n\tOtherwise, append the\n\tcdr\n\tof list1 and \n\tlist2, and\n\tcons\n\tthe\n\tcar\n\tof list1\n\t    onto the result:\n\t  (define (append list1 list2)\n  (if (null? list1)\n      list2\n      (cons (car list1) (append (cdr list1) list2)))) ","2.2.1#ex-2.17":" \n    Define a\n    procedurelast-pair\n    that returns the list that contains only the last element of a given\n    (nonempty) list:\n    (last-pair (list 23 72 149 34)) ","2.2.1#ex-2.18":"\n    Define a\n    procedurereverse that takes a list as argument and\n    returns a list of the same elements in reverse order:\n    (reverse (list 1 4 9 16 25)) ","2.2.1#ex-2.19":"\n      Consider the \n      \n      change-counting program of\n      section 1.2.2.  It would be nice to be\n      able to easily change the currency used by the program, so that we could\n      compute the number of ways to change a British pound, for example.  As\n      the program is written, the knowledge of the currency is distributed\n      partly into the\n      procedurefirst-denomination\n      and partly into the\n      procedurecount-change\n      (which knows\n      that there are five kinds of U.S. coins).\n      It would be nicer\n      to be able to supply a list of coins to be used for making change.\n    ","2.2.1#p7":"\n      We want to rewrite the\n      procedurecc so that its second argument is a list of\n      the values of the coins to use rather than an integer specifying which\n      coins to use.  We could then have lists that defined each kind of\n      currency:\n      (define us-coins (list 50 25 10 5 1))\n\n(define uk-coins (list 100 50 20 10 5 2 1)) \n      We could then call cc as follows:\n      (cc 100 us-coins) \n      To do this will require changing the program\n      cc somewhat.  It will still have the same\n      form, but it will access its second argument differently, as follows:\n      (define (cc amount coin-values)\n  (cond ((= amount 0) 1)\n        ((or (< amount 0) (no-more? coin-values)) 0)\n        (else\n         (+ (cc amount\n                (except-first-denomination coin-values))\n            (cc (- amount\n                   (first-denomination coin-values))\n                coin-values))))) \n      Define the\n      proceduresfirst-denomination,except-first-denomination,\n      and\n      no-more?\n      in terms of primitive operations on list structures.  Does the order of\n      the list\n      coin-values\n      affect the answer produced by cc?\n      Why or why not?\n    ","2.2.1#ex-2.20":"\n        The procedures\n        +, *, and\n\tlist take arbitrary numbers of arguments.\n\tOne way to define such procedures is to use\n\tdefine with dotted-tail notation.\n\tIn a procedure definition, a parameter list that has a dot before the\n\tlast parameter name indicates that, when the procedure is called, the\n\tinitial parameters (if any) will have as values the initial arguments, \n        as usual, but the final parameter's value will be a list\n\tof any remaining arguments. For instance, given the definition\n        (define (f x y . z) $\\langle \\textit{body} \\rangle$) \n        the procedure f can be called with two or\n\tmore arguments. If we evaluate\n        (f 1 2 3 4 5 6) \n        then in the body of f,\n\tx will be 1,\n\ty will be 2, and\n\tz will be the list\n\t(3 4 5 6). Given the definition\n        (define (g . w) $\\langle \\textit{body} \\rangle$) \n        the procedure g can be called with zero or\n\tmore arguments. If we evaluate\n        (g 1 2 3 4 5 6) \n        then in the body of g,\n\tw will be the list\n\t(1 2 3 4 5 6).\n        Use this notation to write a procedure\n\tsame-parity that takes one or more integers\n        and returns a list of all the arguments that have the same even-odd\n        parity as the first argument.  For example, \n        ;; same-parity to be given by student (same-parity 1 2 3 4 5 6 7) (same-parity 2 3 4 5 6 7) ","2.2.1#footnote-link-4":"4","2.2.1#h2":"Mapping over lists","2.2.1#p8":"\n    One extremely useful operation is to apply some transformation to each\n    element in a list and generate the list of results. For instance, the\n    following\n    procedure\n    scales each number in a list by a given factor:\n    (define (scale-list items factor)\n  (if (null? items)\n      nil\n      (cons (* (car items) factor)\n            (scale-list (cdr items) factor)))) (scale-list (list 1 2 3 4 5) 10) ","2.2.1#p9":"\n    We can abstract this general idea and capture it as a common pattern\n    expressed as a higher-order\n    procedure,\n    just as in section 1.3.  The\n    higher-order\n    procedure\n    here is called map.\n    Map\n    takes as arguments a\n    procedure\n    of one argument and a list, and returns a list of the results produced by\n    applying the\n    procedure\n    to each element in the list:(define (map proc items)\n  (if (null? items)\n      nil\n      (cons (proc (car items))\n            (map proc (cdr items))))) (map abs (list -10 2.5 -11.6 17)) (map (lambda (x) (* x x))\n     (list 1 2 3 4)) \n    Now we can give a new definition of\n    scale-list\n    in terms of map:\n    (define (scale-list items factor)\n  (map (lambda (x) (* x factor))\n       items)) ","2.2.1#footnote-link-5":"5","2.2.1#p10":"Map\n    is an important construct, not only because it captures a common pattern,\n    but because it establishes a higher level of abstraction in dealing with\n    lists.  In the original definition of\n    scale-list,\n    the recursive structure of the program draws attention to the\n    element-by-element processing of the list.  Defining\n    scale-list\n    in terms of map suppresses that level of\n    detail and emphasizes that scaling transforms a list of elements to a list\n    of results.  The difference between the two definitions is not that the\n    computer is performing a different process (it isn't) but that we\n    think about the process differently.  In effect,\n    map helps establish an abstraction barrier\n    that isolates the implementation of\n    procedures\n    that transform lists from the details of how the elements of the list are\n    extracted and combined.  Like the barriers shown in\n    \n\tfigure ,\n      \n    this abstraction gives us the flexibility to change the low-level details\n    of how sequences are implemented, while preserving the conceptual framework\n    of operations that transform sequences to sequences.\n    Section 2.2.3 expands\n    on this use of sequences as a framework for organizing programs.\n  ","2.2.1#ex-2.21":"\n    The\n    proceduresquare-list\n    takes a list of numbers as argument and returns a list of the squares of\n    those numbers.\n    ;; square-list to be given by student\n(square-list (list 1 2 3 4)) \n    Here are two different definitions of\n    square-list.\n    Complete both of them by filling in the missing expressions:\n    \n(define (square-list items)\n  (if (null? items)\n      nil\n      (cons ?? ??)))\n      \n(define (square-list items)\n  (map ?? ??))\n      ","2.2.1#ex-2.22":" \n    Louis Reasoner tries to rewrite the first\n    square-listprocedure\n    of exercise 2.21 so that it evolves an\n    iterative process:\n    (define (square-list items)\n  (define (iter things answer)\n    (if (null? things)\n        answer\n        (iter (cdr things) \n              (cons (square (car things))\n                    answer))))\n  (iter items nil)) \n    Unfortunately, defining\n    square-list\n    this way produces the answer list in the reverse order of the one desired.\n    Why?\n    ","2.2.1#p11":"\n      Louis then tries to fix his bug by interchanging the arguments to\n    cons:(define (square-list items)\n  (define (iter things answer)\n    (if (null? things)\n        answer\n        (iter (cdr things)\n              (cons answer\n                    (square (car things))))))\n  (iter items nil)) \n    This doesn't work either.  Explain.\n    ","2.2.1#ex-2.23":"\n    The\n    procedurefor-each\n    is similar to map.  It takes as arguments a\n    procedure\n    and a list of elements.  However, rather than forming a list of the\n    results,\n    for-each\n    just applies the\n    procedure\n    to each of the elements in turn, from left to right. The values returned by\n    applying the\n    procedure\n    to the elements are not used\n    at all—for-each\n    is used with\n    procedures\n    that perform an action, such as printing.  For example, \n    (for-each \n   (lambda (x) (newline) (display x))\n   (list 57 321 88)) \n    The value returned by the call to\n    for-each\n    (not illustrated above) can be something arbitrary, such as true.  Give an\n    implementation of\n    for-each.","2.2.1#footnote-1":"In this book, we use list to mean a chain of\n    pairs terminated by the end-of-list marker.  In contrast, the term\n    list structure refers to any data structure made out of pairs, \n    not just to lists.","2.2.1#footnote-2":"Since nested applications of\n    car and cdr are\n    cumbersome to write, Lisp dialects provide abbreviations for\n    them—for instance, \n    \n(cadr $\\langle arg \\rangle$) = (car (cdr $\\langle arg \\rangle$))\n       \n    The names of all such procedures start with c\n    and end with r.  Each\n    a between them stands for a \n    car operation and each\n    d for a cdr\n    operation, to be applied in the same order in which they appear in the\n    name.  The names car and \n    cdr persist because simple combinations like\n    cadr are\n    pronounceable.","2.2.1#footnote-3":"It's remarkable how much energy\n\tin the standardization of Lisp dialects has been dissipated in\n\targuments that are literally over nothing: Should\n\tnil be an ordinary name? Should the value\n\tof nil be a symbol?  Should it be a list?\n        Should it be a pair?  \n         \n        In Scheme, nil is an ordinary name, which\n\twe use in this section as a variable whose value is the end-of-list\n\tmarker (just as true is an ordinary\n\tvariable that has a true value).  Other dialects of Lisp, including\n\tCommon Lisp, treat nil as a special\n\tsymbol.  The\n        \n        authors of this book, who have endured too many language\n\tstandardization brawls, would like to avoid the entire issue.  Once we\n        have introduced quotation in\n\tsection 2.3, we will denote the\n\tempty list as '() and dispense with the\n        variable nil entirely.","2.2.1#footnote-4":"To define\n\tf and g using\n        lambda we would write\n        (define f (lambda (x y . z) body))\n(define g (lambda w body))","2.2.1#footnote-5":"\t  \n    Scheme standardly provides a \n     map\n    procedure that is more general than the one described here. This more\n    general map takes a procedure of\n    $n$ arguments, together with\n    $n$ lists, and applies the procedure to all the\n    first elements of the lists, all the second elements of the lists, and so\n    on, returning a list of the results.  For example:\n    (map + (list 1 2 3) (list 40 50 60) (list 700 800 900)) (map (lambda (x y) (+ x (* 2 y)))\n     (list 1 2 3)\n     (list 4 5 6)) ","2.2.2":"2.2.2  \n    Hierarchical Structures","2.2.2#p1":"\n    The representation of sequences in terms of lists generalizes naturally to\n    represent sequences whose elements may themselves be sequences.  For\n    example, we can regard the object\n    ((1 2) 3 4)\n    constructed by\n    (cons (list 1 2) (list 3 4)) \n    as a list of three items, the first of which is itself a list,\n    (1 2). Indeed, this is\n\tsuggested by the form in which the result is printed by the\n\tinterpreter.\n      \n\tFigure \n    shows the representation of this structure in terms of pairs.\n    ","2.2.2#fig-":"","2.2.2#p2":"\n    Another way to think of sequences whose elements are sequences is as\n    trees.  The elements of the sequence are the branches of the\n    tree, and elements that are themselves sequences are subtrees.\n    Figure \n    shows the structure in\n    \n\tfigure \n    viewed as a tree.\n    ","2.2.2#p3":"\n    Recursion\n    \n    is a natural tool for dealing with tree structures, since we can\n    often reduce operations on trees to operations on their branches, which\n    reduce in turn to operations on the branches of the branches, and so on,\n    until we reach the leaves of the tree. As an example, compare the\n    lengthprocedure\n    of section 2.2.1 with the\n     count-leavesprocedure,\n    which returns the total number of leaves of a tree:\n    (define x (cons (list 1 2) (list 3 4))) (length x) (count-leaves x) (list x x) (length (list x x)) (count-leaves (list x x)) ","2.2.2#p4":"\n    To implement\n    count-leaves,\n    recall the recursive plan for computing\n    length:Length\n\tof a list x is 1 plus\n\tlength\n\tof the\n        cdr\n\tof x.\n      Length\n\tof the empty list is 0.\n      Count-leaves\n    is similar.  The value for the empty list is the same:\n    Count-leaves\n\tof the empty list is 0.\n      \n    But in the reduction step, where we strip off the\n    car\n    of the list, we must take into account that the\n    car\n    may itself be a tree whose leaves we need to count.  Thus, the appropriate\n    reduction step is\n    Count-leaves\n\tof a tree x is\n\tcount-leaves\n\tof the\n\tcar\n\tof x plus\n\tcount-leaves\n\tof the\n        cdr\n\tof x.\n      \n    Finally, by taking\n    cars\n    we reach actual leaves, so we need another base case:\n    Count-leaves\n\tof a leaf is 1.\n      \n    To aid in writing recursive\n    procedures\n    on trees,\n    Scheme\n    provides the primitive predicate\n    pair?,\n    which tests whether its argument is a pair. Here is the complete\n    procedure:(define (count-leaves x)\n  (cond ((null? x) 0)\n    ((not (pair? x)) 1)\n    (else (+ (count-leaves (car x))\n             (count-leaves (cdr x)))))) ","2.2.2#footnote-link-1":"1","2.2.2#ex-2.24":"\n    Suppose we evaluate the expression\n    (list 1 (list 2 (list 3 4))).\n      \n    Give the result printed by the interpreter, the corresponding\n    box-and-pointer structure, and the interpretation of this as a tree (as in\n    figure ).","2.2.2#ex-2.25":"\n    Give combinations of\n    cars\n    and\n    cdrs\n    that will pick 7 from each of the following\n    lists:(1 3 (5 7) 9)\n\n((7))\n\n(1 (2 (3 (4 (5 (6 7))))))","2.2.2#ex-2.26":"\n    Suppose we define x and\n    y to be two lists:\n    (define x (list 1 2 3))\n\n(define y (list 4 5 6)) \n\tWhat result is printed by the interpreter in response to evaluating\n\teach of the following\n\texpressions:\n      (append x y) (cons x y) (list x y) ","2.2.2#ex-2.27":"\n    Modify your\n    reverseprocedure\n    of exercise 2.18 to produce a\n     deep-reverseprocedure\n    that takes a list as argument and returns as its value the list with its\n    elements reversed and with all sublists deep-reversed as well.  For example,\n    (define x (list (list 1 2) (list 3 4))) x (reverse x) (deep-reverse x) ","2.2.2#ex-2.28":"\n    Write a\n    procedurefringe\n    that takes as argument a tree (represented as a list) and returns a list\n    whose elements are all the leaves of the tree arranged in left-to-right\n    order.  For example,\n    (define x (list (list 1 2) (list 3 4))) (fringe x) (fringe (list x x)) ","2.2.2#ex-2.29":"\n    A binary\n    \n    mobile consists of two branches, a left branch and a right\n    branch.  Each branch is a rod of a certain length, from which hangs\n    either a weight or another binary mobile.  We can represent a binary\n    mobile using compound data by constructing it from two branches (for\n    example, using list):\n    (define (make-mobile left right)\n  (list left right)) \n    A branch is constructed from a length (which\n    must be a number) together with a structure,\n    which may be either a number (representing a simple weight) or another\n    mobile:\n    (define (make-branch length structure)\n  (list length structure)) \n        Write the corresponding selectors\n\tleft-branch\n\tand\n        right-branch,\n\twhich return the branches of a mobile, and\n        branch-length\n\tand\n\tbranch-structure,\n\twhich return the components of a branch.\n      \n        Using your selectors, define a\n        proceduretotal-weight\n        that returns the total weight of a mobile.\n      \n        A mobile is said to be\n        balanced if the torque applied by its top-left branch is equal\n\tto that applied by its top-right branch (that is, if the length of the\n\tleft rod multiplied by the weight hanging from that rod is equal to the\n\tcorresponding product for the right side) and if each of the submobiles\n\thanging off its branches is balanced. Design a predicate that tests\n\twhether a binary mobile is balanced.\n      \n        Suppose we change the representation of mobiles so that the\n\tconstructors are\n        (define (make-mobile left right)\n  (cons left right))\n\n(define (make-branch length structure)\n  (cons length structure)) \n        How much do you need to change your programs to convert to the new\n        representation?\n      ","2.2.2#h1":"Mapping over trees","2.2.2#p5":"\n    Just as map is a powerful abstraction for\n    dealing with sequences, map together with\n    recursion is a powerful abstraction for dealing with trees.  For instance,\n    the\n    scale-treeprocedure,\n    analogous to\n    scale-list\n    of section 2.2.1, takes as arguments a numeric\n    factor and a tree whose leaves are numbers.  It returns a tree of the same\n    shape, where each number is multiplied by the factor. The recursive plan\n    for\n    scale-tree\n    is similar to the one for\n    count-leaves:(define (scale-tree tree factor)\n  (cond ((null? tree) nil)\n     ((not (pair? tree)) (* tree factor))\n     (else (cons (scale-tree (car tree) factor)\n                 (scale-tree (cdr tree) factor))))) (scale-tree (list 1 (list 2 (list 3 4) 5) (list 6 7))\n            10) ","2.2.2#p6":"\n    Another way to implement\n    scale-tree\n    is to regard the tree as a sequence of sub-trees and use\n    map.\n    We map over the sequence, scaling each sub-tree in turn, and return the\n    list of results.  In the base case, where the tree is a leaf, we simply\n    multiply by the factor:\n    (define (scale-tree tree factor)\n  (map (lambda (sub-tree)\n         (if (pair? sub-tree)\n           (scale-tree sub-tree factor)\n           (* sub-tree factor)))\n       tree)) \n    Many tree operations can be implemented by similar combinations of\n    sequence operations and recursion.\n  ","2.2.2#ex-2.30":"Define a proceduresquare-tree\n    analogous to the\n    square-listprocedure\n    of exercise 2.21.  That is,\n    square-tree\n    should behave as follows:\n    (square-tree\n (list 1\n       (list 2 (list 3 4) 5)\n       (list 6 7))) Define square-tree\n    both directly (i.e., without using any higher-order\n    procedures)\n    and also by using\n    map and recursion.\n    ","2.2.2#ex-2.31":"\n    Abstract your answer to exercise 2.30 to\n    produce a\n    proceduretree-map\n    with the property that\n    square-tree\n\tcould be defined as\n      (define (square-tree tree) (tree-map square tree)) ","2.2.2#ex-2.32":"\n    We can represent a\n    \n    set as a list of distinct elements, and we can\n    represent the set of all subsets of the set as a list of lists.  For\n    example, if the set is\n    (1 2 3),\n    then the set of all subsets is\n    (() (3) (2) (2 3) (1) (1 3) (1 2) (1 2 3)).\n      \n    Complete the\n    following \n    definition of a procedure\n    that generates the set of subsets of a set and give a clear explanation of\n    why it works:\n    \n(define (subsets s)\n  (if (null? s)\n    (list nil)\n    (let ((rest (subsets (cdr s))))\n      (append rest (map ?? rest)))))\n      ","2.2.2#footnote-1":"The order of the first two clauses\n    in the cond matters, since the empty list\n    satisfies null? and also is not a\n    pair.","2.2.3":"2.2.3  \n    Sequences as Conventional Interfaces","2.2.3#p1":"\n    In working with compound data, we've stressed how data abstraction\n    permits us to design programs without becoming enmeshed in the details\n    of data representations, and how abstraction preserves for us the\n    flexibility to experiment with alternative representations.  In this\n    section, we introduce another powerful design principle for working\n    with data structures—the use of conventional interfaces.\n  ","2.2.3#p2":"\n    In section 1.3 we saw how\n    program abstractions, implemented as higher-order\n    procedures,\n    can capture common patterns in programs that deal with numerical data. Our\n    ability to formulate analogous operations for working with compound data\n    depends crucially on the style in which we manipulate our data structures.\n    Consider, for example, the following\n    procedure,\n    analogous to the\n    count-leavesprocedure\n    of section 2.2.2, which takes a tree as argument\n    and computes the sum of the squares of the leaves that are odd:\n    (define (sum-odd-squares tree)\n  (cond ((null? tree) 0)  \n        ((not (pair? tree))\n          (if (odd? tree) (square tree) 0))\n        (else (+ (sum-odd-squares (car tree))\n          (sum-odd-squares (cdr tree)))))) \n    On the surface, this\n    procedure\n    is very different from the following one, which constructs a list of all\n    the even Fibonacci numbers\n    ${\\textrm{Fib}}(k)$, where\n    $k$ is less than or equal to a given integer\n    $n$:\n    (define (even-fibs n)\n  (define (next k)\n    (if (> k n)\n      nil\n      (let ((f (fib k)))\n        (if (even? f)\n          (cons f (next (+ k 1)))\n          (next (+ k 1))))))\n  (next 0)) ","2.2.3#p3":"\n    Despite the fact that these two\n    procedures\n    are structurally very different, a more abstract description of the two\n    computations reveals a great deal of similarity.  The first program\n    \n\tenumerates the leaves of a tree;\n      \n\tfilters them, selecting the odd ones;\n      \n\tsquares each of the selected ones; and\n      \n\taccumulates the results using\n\t+,\n\tstarting with 0.\n      \n    The second program\n    \n\tenumerates the integers from 0 to $n$;\n      \n\tcomputes the Fibonacci number for each integer;\n      \n\tfilters them, selecting the even ones; and\n      \n\taccumulates the results using\n\tcons,\n\tstarting with the empty list.\n      ","2.2.3#p4":"\n    A signal-processing engineer would find it natural to conceptualize these\n    processes in terms of\n    \n    signals flowing through a cascade of stages, each of\n    which implements part of the program plan, as shown in\n    figure .\n    In\n    sum-odd-squares,\n    we begin with an \n    enumerator, which generates a \"signal\" consisting of\n    the leaves of a given tree.  This signal is passed through a \n    filter, which eliminates all but the odd elements. The resulting\n    signal is in turn passed through a \n    map, which is a \"transducer\" that applies the\n    squareprocedure\n    to each element.  The output of the map is then fed to an \n    accumulator, which combines the elements using\n    +,\n    starting from an initial 0. The plan for\n    even-fibs\n    is analogous.\n    ","2.2.3#fig-":"","2.2.3#p5":"\n    Unfortunately, the two\n    procedure definitions\n    above fail to exhibit this signal-flow structure.  For instance, if we\n    examine the\n    sum-odd-squaresprocedure,\n    we find that the enumeration is implemented partly by the\n    null?\n    and\n    pair?\n    tests and partly by the tree-recursive structure of the\n    procedure.\n    Similarly, the accumulation is found partly in the tests and partly in the\n    addition used in the recursion.  In general, there are no distinct parts of\n    either\n    procedure\n    that correspond to the elements in the signal-flow description. Our two\n    procedures\n    decompose the computations in a different way, spreading the enumeration\n    over the program and mingling it with the map, the filter, and the\n    accumulation.  If we could organize our programs to make the signal-flow\n    structure manifest in the\n    procedures\n    we write, this would increase the conceptual clarity of the resulting\n    code.","2.2.3#h1":"Sequence Operations","2.2.3#p6":"\n    The key to organizing programs so as to more clearly reflect the\n    signal-flow structure is to concentrate on the \"signals\" that\n    flow from one stage in the process to the next.  If we represent these\n    signals as lists, then we can use list operations to implement the\n    processing at each of the stages.  For instance, we can implement the\n    mapping stages of the signal-flow diagrams using the\n    mapprocedure\n    from section 2.2.1:\n    (map square (list 1 2 3 4 5)) ","2.2.3#p7":"\n    Filtering a sequence to select only those elements that satisfy a given\n    predicate is accomplished by\n    (define (filter predicate sequence)\n  (cond ((null? sequence) nil)\n        ((predicate (car sequence))\n          (cons (car sequence)\n                (filter predicate (cdr sequence))))\n        (else (filter predicate (cdr sequence))))) \n    For example,\n    (filter odd? (list 1 2 3 4 5)) ","2.2.3#p8":"\n  Accumulations can be implemented by\n    (define (accumulate op initial sequence)\n  (if (null? sequence)\n    initial\n    (op (car sequence)\n        (accumulate op initial (cdr sequence))))) (accumulate + 0 (list 1 2 3 4 5)) (accumulate * 1 (list 1 2 3 4 5)) (accumulate cons nil (list 1 2 3 4 5)) ","2.2.3#p9":"\n    All that remains to implement signal-flow diagrams is to enumerate the\n    sequence of elements to be processed.  For\n    even-fibs,\n    we need to generate the sequence of integers in a given range, which we\n    can do as follows:\n    (define (enumerate-interval low high)\n   (if (> low high)\n      nil\n      (cons low (enumerate-interval (+ low 1) high)))) (enumerate-interval 2 7) \n    To enumerate the leaves of a tree, we can use(define (enumerate-tree tree)\n   (cond ((null? tree) nil)\n         ((not (pair? tree)) (list tree))\n         (else (append (enumerate-tree (car tree))\n                       (enumerate-tree (cdr tree)))))) (enumerate-tree (list 1 (list 2 (list 3 4)) 5)) ","2.2.3#footnote-link-1":"1","2.2.3#p10":"\n    Now we can reformulate\n    sum-odd-squares\n    and\n    even-fibs\n    as in the signal-flow diagrams.  For\n    sum-odd-squares,\n    we enumerate the sequence of leaves of the tree, filter this to keep only\n    the odd numbers in the sequence, square each element, and sum the results:\n    (define (sum-odd-squares tree)\n   (accumulate +\n               0\n               (map square\n                 (filter odd?\n                  (enumerate-tree tree))))) \n    For\n    even-fibs,\n    we enumerate the integers from 0 to $n$, generate\n    the Fibonacci number for each of these integers, filter the resulting\n    sequence to keep only the even elements, and accumulate the results\n    into a list:\n    (define (even-fibs n)\n   (accumulate cons\n               nil\n               (filter even?\n                 (map fib\n                      (enumerate-interval 0 n))))) ","2.2.3#p11":"\n    The value of expressing programs as sequence operations is that this\n    helps us make program designs that are modular, that is, designs that\n    are constructed by combining relatively independent pieces.  We can\n    encourage modular design by providing a library of standard components\n    together with a conventional interface for connecting the components\n    in flexible ways.\n  ","2.2.3#p12":"\n    Modular construction\n    \n    is a powerful strategy for controlling complexity in\n    engineering design.  In real signal-processing applications, for example,\n    designers regularly build systems by cascading elements selected from\n    standardized families of filters and transducers.  Similarly, sequence\n    operations provide a library of standard program elements that we can mix\n    and match.  For instance, we can reuse pieces from the\n    sum-odd-squares\n    and\n    even-fibsprocedures\n    in a program that constructs a list of the squares of the first\n    $n+1$ Fibonacci numbers:\n    (define (list-fib-squares n)\n  (accumulate cons\n              nil\n              (map square\n                   (map fib\n                        (enumerate-interval 0 n))))) (list-fib-squares 10) \n    We can rearrange the pieces and use them in computing the product of the\n    squares of the odd integers in a sequence:\n    (define (product-of-squares-of-odd-elements sequence)\n  (accumulate *\n              1\n              (map square\n                (filter odd? sequence)))) (product-of-squares-of-odd-elements (list 1 2 3 4 5)) ","2.2.3#p13":"\n    We can also formulate conventional data-processing applications in terms of\n    sequence operations.  Suppose we have a sequence of personnel records and\n    we want to find the salary of the highest-paid programmer. Assume that we\n    have a selector salary that returns the salary\n    of a record, and a predicate\n    programmer?\n    that tests if a record is for a programmer.  Then we can write\n    (define (salary-of-highest-paid-programmer records)\n  (accumulate max\n              0\n              (map salary\n                   (filter programmer? records)))) \n    These examples give just a hint of the vast range of operations that\n    can be expressed as sequence operations.","2.2.3#footnote-link-2":"2","2.2.3#p14":"\n    Sequences, implemented here as lists, serve as a conventional interface\n    that permits us to combine processing modules.  Additionally, when we\n    uniformly represent structures as sequences, we have localized the\n    data-structure dependencies in our programs to a small number of sequence\n    operations.  By changing these, we can experiment with alternative\n    representations of sequences, while leaving the overall design of our\n    programs intact.  We will exploit this capability in\n    section 3.5, when we generalize the\n    sequence-processing paradigm to admit infinite sequences.\n  ","2.2.3#ex-2.33":"\n    Fill in the missing expressions to complete the following definitions of\n    some basic list-manipulation operations as accumulations:\n    \n(define (map p sequence)\n  (accumulate (lambda (x y) ??) nil sequence))\n\n(define (append seq1 seq2)\n  (accumulate cons ?? ??))\n\n(define (length sequence)\n  (accumulate ?? 0 sequence))\n      ","2.2.3#ex-2.34":"\n    Evaluating a\n    polynomial in $x$ at a given value\n    of $x$ can be formulated as an accumulation.\n    We evaluate the polynomial\n    \n      \\[ a_{n} x^n +a_{n-1}x^{n-1}+\\cdots + a_{1} x+a_{0} \\]\n    \n    using a well-known algorithm called \n    Horner's rule, which structures the computation as\n    \n      \\[ \\left(\\cdots (a_{n}  x+a_{n-1})x+\\cdots +a_{1}\\right) x+a_{0} \\]\n    \n    In other words, we start with $a_{n}$, multiply\n    by $x$, add $a_{n-1}$,\n    multiply by $x$, and so on, until we reach\n    $a_{0}$.\n    Fill in the following template to produce a\n    procedure\n    that evaluates a polynomial using Horner's rule. Assume that the\n    coefficients of the polynomial are arranged in a sequence, from\n    $a_{0}$ through\n    $a_{n}$.\n    \n(define (horner-eval x coefficient-sequence)\n   (accumulate (lambda (this-coeff higher-terms) ??)\n               0\n               coefficient-sequence))\n      \n    For example, to compute $1+3x+5x^3+x^5$ at\n    $x=2$ you would evaluate\n    (horner-eval 2 (list 1 3 0 5 0 1)) ","2.2.3#footnote-link-3":"3","2.2.3#ex-2.35":"\n    Redefine\n    count-leaves\n    from section 2.2.2 as an accumulation:\n    \n(define (count-leaves t)\n   (accumulate ?? ?? (map ?? ??)))\n          ","2.2.3#ex-2.36":"\n    The\n    procedureaccumulate-n\n    is similar to\n    accumulate\n    except that it takes as its third argument a sequence of sequences, which\n    are all assumed to have the same number of elements.  It applies the\n    designated accumulation\n    procedure\n    to combine all the first elements of the sequences, all the second elements\n    of the sequences, and so on, and returns a sequence of the results.  For\n    instance, if s is a sequence containing four\n    sequences\n    ((1 2 3) (4 5 6) (7 8 9) (10 11 12))\n    then the value of\n    (accumulate-n + 0 s)\n    should be the sequence\n    (22 26 30).\n    Fill in the missing expressions in the following definition of\n    accumulate-n:\n(define (accumulate-n op init seqs)\n  (if (null? (car seqs))\n     nil\n     (cons (accumulate op init ??)\n           (accumulate-n op init ??))))\n      ","2.2.3#ex-2.37":"\n    Suppose we represent vectors $v=(v_{i})$ as\n    \n    sequences of numbers, and matrices $m=(m_{ij})$\n    as sequences of vectors (the rows of the matrix). For example, the matrix\n    \n      \\[ \\left[ \n      \\begin{array}{llll}\n      1 & 2 & 3 & 4\\\\\n      4 & 5 & 6 & 6\\\\\n      6 & 7 & 8 & 9\\\\\n      \\end{array}\n      \\right] \\]\n    \n        is represented as the sequence\n\t((1 2 3 4) (4 5 6 6) (6 7 8 9)).\n      \n    With this representation, we can use sequence operations to concisely\n    express the basic matrix and vector operations.  These operations\n    (which are described in any book on matrix algebra) are the following:\n    (dot-product $v$ $w$)\n\treturns the sum $\\sum_{i}v_{i} w_{i}$;\n\t(matrix-*-vector \t  $m$$v$)\n\treturns the vector $t$, where\n\t$t_{i} =\\sum_{j}m_{ij}v_{j}$;\n\t(matrix-*-matrix $m\\ n$)\n\treturns the matrix $p$, where\n\t$p_{ij}=\\sum_{k} m_{ik}n_{kj}$;\n\t(transpose \t    )\n\treturns the matrix $n$, where\n\t$n_{ij}=m_{ji}$.\n\t\n    We can define the dot product as(define (dot-product v w)\n  (accumulate + 0 (map * v w))) \n    Fill in the missing expressions in the following\n    procedures\n    for computing the other matrix operations.  (The\n    procedureaccumulate-n\n\tis defined in \t\n      \n    exercise 2.36.)\n    \n(define (matrix-*-vector m v)\n  (map ?? m))\n\n(define (transpose mat)\n  (accumulate-n ?? ?? mat))\n\n(define (matrix-*-matrix m n)\n  (let ((cols (transpose n)))\n    (map ?? m)))\n      ","2.2.3#footnote-link-4":"4","2.2.3#ex-2.38":"\n    The\n     accumulateprocedure\n    is also known as\n    fold-right,\n    because it combines the first element of the sequence with the result\n    of combining all the elements to the right.  There is also a\n    fold-left,\n    which is similar to\n    fold-right,\n    except that it combines elements working in the opposite direction:\n    (define (fold-left op initial sequence)\n  (define (iter result rest)\n    (if (null? rest)\n      result\n      (iter (op result (car rest))\n            (cdr rest))))\n  (iter initial sequence)) \n    What are the values of\n    (fold-right / 1 (list 1 2 3)) (fold-left / 1 (list 1 2 3)) (fold-right list nil (list 1 2 3)) (fold-left list nil (list 1 2 3)) \n    Give a property that\n    op\n    should satisfy to guarantee that\n    fold-right\n    and\n    fold-left\n    will produce the same values for any sequence.\n    ","2.2.3#ex-2.39":"\n    Complete the following definitions of reverse\n    (exercise 2.18) in terms of\n    fold-right\n    and\n    fold-left\n    from exercise 2.38:\n    \n(define (reverse sequence)\n  (fold-right (lambda (x y) ??) nil sequence))\n      \n(define (reverse sequence)\n  (fold-left (lambda (x y) ??) nil sequence))\n      ","2.2.3#h2":"Nested Mappings","2.2.3#p15":"\n    We can extend the sequence paradigm to include many computations that are\n    commonly expressed using nested loops.\n    Consider this problem: Given a positive integer\n    $n$, find all ordered pairs of distinct positive\n    integers $i$ and $j$,\n    where $1\\leq j < i\\leq n$, such that\n    $i +j$ is prime.  For example, if\n    $n$ is 6, then the pairs are the following:\n    \n      \\[\n      \\begin{array}{c|ccccccc}\n      i     & 2 & 3 & 4 & 4 & 5 & 6 & 6 \\\\\n      j     & 1 & 2 & 1 & 3 & 2 & 1 & 5 \\\\\n      \\hline\n      i+j   & 3 & 5 & 5 & 7 & 7 & 7 & 11\n      \\end{array}\n      \\]\n    \n    A natural way to organize this computation is to generate the sequence\n    of all ordered pairs of positive integers less than or equal to\n    $n$, filter to select those pairs whose sum is\n    prime, and then, for each pair $(i, j)$ that\n    passes through the filter, produce the triple\n    $(i, j, i+j)$.\n  ","2.2.3#footnote-link-5":"5","2.2.3#p16":"\n    Here is a way to generate the sequence of pairs: For each integer\n    $i\\leq n$, enumerate the integers\n    $j < i$, and for each such\n    $i$ and $j$\n    generate the pair $(i, j)$.  In terms of\n    sequence operations, we map along the sequence\n    (enumerate-interval 1 n).\n    For each $i$ in this sequence, we map along the\n    sequence\n    (enumerate-interval 1 (- i 1)).\n      \n    For each $j$ in this latter sequence, we\n    generate the pair\n    (list i j).\n    This gives us a sequence of pairs for each $i$.\n    Combining all the sequences for all the $i$ (by\n    accumulating with append) produces the\n    required sequence of pairs:;; replace n below by the desired number\n(accumulate append\n  nil\n  (map (lambda (i)\n         (map (lambda (j) (list i j))\n              (enumerate-interval 1 (- i 1))))\n       (enumerate-interval 1 n))) \n    The combination of mapping and accumulating with\n    append is so common in this sort of program\n    that we will isolate it as a separate\n    procedure:(define (flatmap proc seq)\n  (accumulate append nil (map proc seq))) \n    Now filter this sequence of pairs to find those whose sum is prime. The\n    filter predicate is called for each element of the sequence; its argument\n    is a pair and it must extract the integers from the pair. Thus, the\n    predicate to apply to each element in the sequence is\n    (define (prime-sum? pair)\n  (prime? (+ (car pair) (cadr pair)))) \n    Finally, generate the sequence of results by mapping over the filtered\n    pairs using the following\n    procedure,\n    which constructs a triple consisting of the two elements of the pair along\n    with their sum:\n    (define (make-pair-sum pair)\n  (list (car pair) (cadr pair) (+ (car pair) (cadr pair)))) \n    Combining all these steps yields the complete\n    procedure:(define (prime-sum-pairs n)\n  (map make-pair-sum\n       (filter prime-sum?\n               (flatmap\n                 (lambda (i)\n                   (map (lambda (j) (list i j))\n                        (enumerate-interval 1 (- i 1))))\n                 (enumerate-interval 1 n))))) ","2.2.3#footnote-link-6":"6","2.2.3#p17":"\n    Nested mappings are also useful for sequences other than those that\n    enumerate intervals.  Suppose we wish to generate all the \n    \n    permutations\n    of a set $S$; that is, all the ways of ordering\n    the items in the set.  For instance, the permutations of\n    $\\{1, 2, 3\\}$ are\n    $\\{1, 2, 3\\}$,\n    $\\{ 1, 3, 2\\}$,\n    $\\{2, 1, 3\\}$,\n    $\\{ 2, 3, 1\\}$,\n    $\\{ 3, 1, 2\\}$, and\n    $\\{ 3, 2, 1\\}$.  Here is a plan for generating\n    the permutations of $S$: For each item\n    $x$ in $S$,\n    recursively generate the sequence of permutations of\n    $S-x$, and adjoin\n    $x$ to the front of each one.  This yields, for\n    each $x$ in $S$, the\n    sequence of permutations of $S$ that begin\n    with $x$.  Combining these sequences for\n    all $x$ gives all the permutations\n    of $S$:(define (permutations s)\n  (if (null? s)             ; empty set?\n    (list nil)              ; sequence containing empty set\n    (flatmap (lambda (x)\n               (map (lambda (p) (cons x p))\n                    (permutations (remove x s))))\n             s))) \n    Notice how this strategy reduces the problem of generating permutations of\n    $S$ to the problem of generating the\n    permutations of sets with fewer elements than\n    $S$.  In the terminal case, we work our way down\n    to the empty list, which represents a set of no elements. For this, we\n    generate\n    (list nil),\n    which is a sequence with one item, namely the set with no elements.  The\n    removeprocedure\n    used in permutations returns all the items in\n    a given sequence except for a given item.  This can be expressed as a\n    simple filter:\n    (define (remove item sequence)\n  (filter (lambda (x) (not (= x item)))\n          sequence)) ","2.2.3#footnote-link-7":"7","2.2.3#footnote-link-8":"8","2.2.3#ex-2.40":"\n    Write a\n    procedureunique-pairs\n    that, given an integer $n$, generates the\n    sequence of pairs $(i, j)$ with\n    $1\\leq j < i\\leq n$.  Use\n    unique-pairs\n    to simplify the definition of\n    prime-sum-pairs\n    given above.\n    ","2.2.3#ex-2.41":"\n    Write a\n    procedure\n    to find all ordered triples of distinct positive integers\n    $i$, $j$,\n    and $k$ less than or equal to a given\n    integer $n$ that sum to a given integer\n    $s$.\n    ","2.2.3#fig-2.8":"","2.2.3#ex-2.42":"\n      The \n      \"eight-queens puzzle\" asks how to place eight queens on a\n      chessboard so that no queen is in check from any other (i.e., no two\n      queens are in the same row, column, or diagonal).  One possible solution\n      is shown in figure 2.8.  One way to solve the\n      puzzle is to work across the board, placing a queen in each column.\n      Once we have placed $k-1$ queens, we must place\n      the $k$th queen in a position where it does not\n      check any of the queens already on the board.  We can formulate this\n      approach recursively: Assume that we have already generated the sequence\n      of all possible ways to place $k-1$ queens in\n      the first $k-1$ columns of the board.  For\n      each of these ways, generate an extended set of positions by placing a\n      queen in each row of the $k$th column.  Now\n      filter these, keeping only the positions for which the queen in the\n      $k$th column is safe with respect to the other\n      queens.  This produces the sequence of all ways to place\n      $k$ queens in the first\n      $k$ columns.  By continuing this process, we\n      will produce not only one solution, but all solutions to the puzzle.\n      ","2.2.3#p18":"\n      We implement this solution as a\n      procedurequeens,\n      which returns a sequence of all solutions to the problem of placing\n      $n$ queens on an\n      $n\\times n$ chessboard.\n      Queens\n      has an internal\n      procedurequeen-cols\n      that returns the sequence of all ways to place queens in the first\n      $k$ columns of the board.\n    (define (queens board-size)\n  (define (queen-cols k)\n    (if (= k 0)\n      (list empty-board)\n      (filter\n        (lambda (positions) (safe? k positions))\n        (flatmap\n          (lambda (rest-of-queens)\n            (map (lambda (new-row)\n                   (adjoin-position new-row k rest-of-queens))\n                 (enumerate-interval 1 board-size)))\n          (queen-cols (- k 1))))))\n  (queen-cols board-size)) \n    In this\n    procedurerest-of-queens\n    is a way to place $k-1$ queens in the first\n    $k-1$ columns, and\n    new-row\n    is a proposed row in which to place the queen for the\n    $k$th column.  Complete the program by\n    implementing the representation for sets of board positions, including the\n    procedureadjoin-position,\n    which adjoins a new row-column position to a set of positions, and\n    empty-board,\n    which represents an empty set of positions.  You must also write the\n    proceduresafe?,\n    which determines for a set of positions whether the queen in the\n    $k$th column is safe with respect to the others.\n    (Note that we need only check whether the new queen is safe—the\n    other queens are already guaranteed safe with respect to each other.)\n      ","2.2.3#ex-2.43":"\n    Louis Reasoner is having a terrible time doing\n    exercise 2.42.  His\n    queensprocedure\n    seems to work, but it runs extremely slowly. (Louis never does manage to\n    wait long enough for it to solve even the\n    $6\\times 6$ case.)  When Louis asks Eva Lu Ator\n    for help, she points out that he has interchanged the order of the nested\n    mappings in the\n    flatmap,\n    writing it as\n    (flatmap\n  (lambda (new-row)\n    (map (lambda (rest-of-queens)\n           (adjoin-position new-row k rest-of-queens))\n         (queen-cols (- k 1))))\n  (enumerate-interval 1 board-size))\n    Explain why this interchange makes the program run slowly.  Estimate\n    how long it will take Louis's program to solve the eight-queens\n    puzzle, assuming that the program in\n    exercise 2.42 solves the puzzle in time\n    $T$.\n    ","2.2.3#footnote-1":"This is, in fact,\n    precisely the \n     fringeprocedure\n    from exercise 2.28.  Here we've renamed it\n    to emphasize that it is part of a family of general sequence-manipulation\n    procedures.","2.2.3#footnote-2":"\n    Richard Waters (1979) developed a program that automatically analyzes\n    traditional \n    \n    Fortran programs, viewing them in terms of maps, filters, and accumulations.\n    He found that fully 90 percent of the code in the Fortran Scientific\n    Subroutine Package fits neatly into this paradigm.  One of the reasons\n    for the success of Lisp as a programming language is that lists provide a\n    standard medium for expressing ordered collections so that they can be\n    manipulated using higher-order operations. Many modern languages, such as\n    Python, have learned this lesson.","2.2.3#footnote-3":"According to \n    \n    Knuth (1997b), this rule was formulated by\n    \n    W. G. Horner early in the nineteenth century, but the method was actually\n    used by Newton over a hundred years earlier.  Horner's rule evaluates\n    the polynomial using fewer additions and multiplications than does the\n    straightforward method of first computing\n    $a_{n} x^n$, then adding\n    $a_{n-1}x^{n-1}$, and so on.  In fact, it is\n    possible to prove that any algorithm for evaluating arbitrary polynomials\n    must use at least as many additions and multiplications as does\n    Horner's rule, and thus Horner's rule is an \n    \n    optimal algorithm for polynomial evaluation.  This was proved (for the\n    number of additions) by\n    \n    A. M. Ostrowski in a 1954 paper that essentially founded the modern study\n    of optimal algorithms.  The analogous statement for multiplications was\n    proved by \n    \n    V. Y. Pan in 1966.  The book by \n    \n      Borodin and Munro (1975)\n     \n    provides an overview of these and other results about optimal\n    algorithms.","2.2.3#footnote-4":"This definition uses\n    \n\tthe extended version of map\n\tdescribed in footnote .\n      ","2.2.3#footnote-5":"This approach to nested\n    mappings was shown to us by \n    \n    David Turner, whose languages \n    \n    KRC and \n    \n    Miranda provide elegant formalisms for dealing with these constructs.  The\n    examples in this section (see also\n    exercise 2.42) are adapted from Turner 1981.\n    In section 3.5.3, we'll see\n    how this approach generalizes to infinite sequences.","2.2.3#footnote-6":"We're representing a pair here\n    as a list of two elements rather than as\n    a Lisp pair.\n    Thus, the \"pair\"$(i, j)$ is\n    represented as\n    (list i j),\n    not\n    (cons i j).","2.2.3#footnote-7":"The set\n    $S-x$ is the set of all elements of\n    $S$, excluding\n    $x$.","2.2.3#footnote-8":"Semicolons in Scheme code are\n    used to introduce comments. Everything from\n    the semicolon\n    to the end of the line is ignored by the interpreter.  In this book we\n    don't use many comments; we try to make our programs self-documenting\n    by using descriptive names.","2.2.4":"2.2.4  \n    Example: A Picture Language","2.2.4#p1":"\n    This section presents a simple language for drawing pictures that\n    illustrates the power of data abstraction and closure, and also exploits\n    higher-order\n    procedures\n    in an essential way.  The language is designed to make it easy to\n    experiment with patterns such as the ones in\n    figure 2.9, which are composed of\n    repeated elements that are shifted and scaled. In this language, the data objects being\n    combined are represented as\n    procedures\n    rather than as list structure. Just as\n    cons,\n    which satisfies the\n    \n    closure property, allowed us to easily build arbitrarily complicated list\n    structure, the operations in this language, which also satisfy the closure\n    property, allow us to easily build arbitrarily complicated patterns.\n    ","2.2.4#footnote-link-1":"1","2.2.4#fig-2.9":"","2.2.4#h1":"The picture language","2.2.4#p2":"\n    When we began our study of programming in\n    section 1.1, we emphasized the\n    importance of describing a language by focusing on the language's\n    primitives, its means of combination, and its means of abstraction.\n    We'll follow that framework here.\n  ","2.2.4#p3":"\n    Part of the elegance of this picture language is that there is only one\n    kind of element, called a\n    painter.  A painter draws an image that is shifted and scaled to\n    fit within a designated\n    \n    parallelogram-shaped frame.  For example, there's a primitive painter\n    we'll call wave\n    that makes a crude line drawing,\n    as shown in figure 2.10.\n    \n    The actual shape of the drawing depends on the frame—all four\n    images in figure 2.10 are produced by the same\n    wave painter, but with respect to four\n    different frames. Painters can be more elaborate than this: The primitive\n    painter called rogers paints a picture of\n    MIT's founder, William Barton Rogers, as shown in\n    figure 2.11.\n    The four images in figure 2.11\n    are drawn with respect to the same four frames\n    as the wave images in\n    figure 2.10.\n    ","2.2.4#fig-2.10":"","2.2.4#footnote-link-2":"2","2.2.4#fig-2.11":"","2.2.4#p4":"\n    To combine images, we use various\n    \n    operations that construct new painters\n    from given painters. For example, the\n     beside operation takes two painters and\n    produces a new, compound painter that draws the first painter's image\n    in the left half of the frame and the second painter's image in the\n    right half of the frame. Similarly,\n     below takes two painters and produces a\n    compound painter that draws the first painter's image below the\n    second painter's image. Some operations transform a single painter\n    to produce a new painter.  For example,\n     flip-vert\n    takes a painter and produces a painter that draws its image upside-down, and\n     flip-horiz\n    produces a painter that draws the original painter's image\n    left-to-right reversed.\n  ","2.2.4#p5":"\n    Figure 2.12 shows the drawing of a\n    painter called wave4\n    that is built up in two stages starting from\n    wave:\n    (define wave2\n  (beside wave (flip-vert wave)))\n(define wave4\n  (below wave2 wave2)) \n    In building up a complex image in this manner we are exploiting the fact\n    that painters are\n    \n    closed under the language's means of combination.\n    The beside or\n    below of two painters is itself a painter;\n    therefore, we can use it as an element in making more complex painters.\n    As with building up list structure using\n    cons,\n    the closure of our data under the means of combination is crucial to the\n    ability to create complex structures while using only a few operations.\n  ","2.2.4#fig-2.12":"","2.2.4#p6":"\n    Once we can combine painters, we would like to be able to abstract typical\n    patterns of combining painters. We will implement the painter operations as \n    Scheme procedures.\n    This means that we don't need a special abstraction mechanism in the\n    picture language: Since the means of combination are ordinary \n    Scheme procedures,\n    we automatically have the capability to do anything with painter operations\n    that we can do with\n    procedures.\n    For example, we can abstract the pattern in\n    wave4 as\n    (define (flipped-pairs painter)\n  (let ((painter2 (beside painter (flip-vert painter))))\n    (below painter2 painter2))) \n    and\n    definewave4 as an instance of this\n    pattern:\n    (define wave4 (flipped-pairs wave)) ","2.2.4#p7":"\n    We can also define recursive operations. Here's one that makes\n    painters split and branch towards the right as shown in\n    \n\tfigures \n    and\n    :\n      (define (right-split painter n)\n  (if (= n 0)\n    painter\n    (let ((smaller (right-split painter (- n 1))))\n      (beside painter (below smaller smaller))))) \n    We can produce balanced patterns by branching upwards as well as towards\n    the right (see exercise 2.44 and\n    figures 2.13\n    and 2.14):\n    (define (corner-split painter n)\n  (if (= n 0)\n    painter\n    (let ((up (up-split painter (- n 1)))\n          (right (right-split painter (- n 1))))\n      (let ((top-left (beside up up))\n            (bottom-right (below right right))\n            (corner (corner-split painter (- n 1))))\n        (beside (below painter top-left)\n                (below bottom-right corner)))))) \n    By placing four copies of a\n    corner-split\n    appropriately, we obtain a pattern called\n    square-limit,\n    whose application to wave and\n    rogers is shown in\n    figure 2.9:\n    (define (square-limit painter n)\n  (let ((quarter (corner-split painter n)))\n    (let ((half (beside (flip-horiz quarter) quarter)))\n      (below (flip-vert half) half)))) ","2.2.4#fig-":"","2.2.4#ex-2.44":"Define the procedureup-split\n    used by\n    corner-split.\n    It is similar to\n    right-split,\n    except that it switches the roles of below\n    and beside.\n    ","2.2.4#h2":"Higher-order operations","2.2.4#p8":"\n    In addition to abstracting patterns of combining painters, we can work at a\n    higher level, abstracting patterns of combining painter operations. That\n    is, we can view the painter operations as elements to manipulate and can\n    write means of combination for these\n    elements—procedures\n    that take painter operations as arguments and create new painter operations.\n  ","2.2.4#p9":"\n    For example,\n    flipped-pairs\n    and\n    square-limit\n    each arrange four copies of a painter's image in a square pattern;\n    they differ only in how they orient the copies. One way to abstract this\n    pattern of painter combination is with the following\n    procedure,\n    which takes four one-argument painter operations and produces a painter\n    operation that transforms a given painter with those four operations and\n    arranges the results in a square.Tl,tr, bl, and\n    br are the transformations to apply to the\n    top left copy, the top right copy, the bottom left copy, and the bottom\n    right copy, respectively.\n    (define (square-of-four tl tr bl br)\n  (lambda (painter)\n    (let ((top (beside (tl painter) (tr painter)))\n          (bottom (beside (bl painter) (br painter))))\n      (below bottom top)))) \n    Then\n    flipped-pairs\n      can be defined in terms of\n      square-of-four as follows:(define (flipped-pairs painter)\n  (let ((combine4 (square-of-four identity flip-vert\n                                  identity flip-vert)))\n    (combine4 painter))) \n    and\n    square-limit\n    can be expressed as(define (square-limit painter n)\n  (let ((combine4 (square-of-four flip-horiz identity\n                                  rotate180 flip-vert)))\n    (combine4 (corner-split painter n)))) ","2.2.4#footnote-link-3":"3","2.2.4#footnote-link-4":"4","2.2.4#ex-2.45":"Right-split\n    and\n    up-split\n    can be expressed as instances of a general splitting operation. \n    Define a proceduresplit with the property that evaluating\n    (define right-split (split beside below))\n(define up-split (split below beside)) \n    produces\n    proceduresright-split\n    and\n    up-split\n      with the same behaviors as the ones already defined.\n      ","2.2.4#h3":"Frames","2.2.4#p10":"\n    Before we can show how to implement painters and their means of\n    combination, we must first consider\n    \n    frames.  A frame can be described by three vectors—an origin vector\n    and two edge vectors.  The origin vector specifies the offset of the\n    frame's origin from some absolute origin in the plane, and the edge\n    vectors specify the offsets of the frame's corners from its origin.\n    If the edges are perpendicular, the frame will be rectangular.\n    Otherwise the frame will be a more general parallelogram.\n  ","2.2.4#p11":"\n    Figure 2.15 shows a frame and its associated\n    vectors.  In accordance with data abstraction, we need not be specific yet\n    about how frames are represented, other than to say that there is a\n    constructor\n      make-frame,\n    which takes three vectors and produces a frame, and three corresponding\n    selectors\n      origin-frame,edge1-frame,\n    and\n      edge2-frame\n    (see exercise 2.47).\n\n    ","2.2.4#fig-2.15":"","2.2.4#p12":"\n    We will use coordinates in the\n    \n    unit square\n    ($0\\leq x, y\\leq 1$) to specify images. With\n    each frame, we associate a\n    frame coordinate map, which will be used to shift and scale images\n    to fit the frame.  The map transforms the unit square into the frame by\n    mapping the vector $\\mathbf{v}=(x, y)$ to the\n    vector sum\n    \n      \\[\n      \\text{Origin(Frame)} + x\\cdot \\text{ Edge}_1\\text{ (Frame)}\n      + y\\cdot \\text{ Edge}_2\\text{ (Frame)}\n      \\]\n    \n    For example, $(0, 0)$ is mapped to the origin of\n    the frame, $(1, 1)$ to the vertex diagonally\n    opposite the origin, and $(0.5, 0.5)$ to the\n    center of the frame.  We can create a frame's coordinate map with\n    the following\n    procedure:(define (frame-coord-map frame)\n  (lambda (v)\n    (add-vect\n      (origin-frame frame)\n      (add-vect (scale-vect (xcor-vect v)\n                            (edge1-frame frame))\n                (scale-vect (ycor-vect v)\n                            (edge2-frame frame)))))) \n    Observe that applying\n    frame-coord-map\n    to a frame returns a\n    procedure\n    that, given a vector, returns a vector. If the argument vector is in the\n    unit square, the result vector will be in the frame.  For example, \n    ((frame-coord-map a-frame) (make-vect 0 0)) \n    returns the same vector as\n    (origin-frame a-frame) ","2.2.4#footnote-link-5":"5","2.2.4#ex-2.46":"\n    A two-dimensional\n    \n    vector $v$ running from the\n    origin to a point can be represented as a pair consisting of an\n    $x$-coordinate and a\n    $y$-coordinate.  Implement a data abstraction\n    for vectors by giving a constructor\n    make-vect\n    and corresponding selectors\n    xcor-vect\n    and\n    ycor-vect.\n    In terms of your selectors and constructor, implement\n    proceduresadd-vect,sub-vect,\n    and\n    scale-vect\n    that perform the operations vector addition, vector subtraction, and\n    multiplying a vector by a scalar:\n    \n      \\[\\begin{array}{lll}\n      (x_1, y_1)+(x_2, y_2) &=& (x_1+x_2, y_1+y_2)\\\\\n      (x_1, y_1)-(x_2, y_2) &=& (x_1-x_2, y_1-y_2)\\\\\n      s\\cdot(x, y)&= &(sx, sy)\n      \\end{array}\\]\n    ","2.2.4#ex-2.47":"\n    Here are two possible constructors for frames:\n    (define (make-frame origin edge1 edge2)\n  (list origin edge1 edge2))\n\n(define (make-frame origin edge1 edge2)\n  (cons origin (cons edge1 edge2)))\n    For each constructor supply the appropriate selectors to produce an\n    implementation for frames.","2.2.4#h4":"Painters","2.2.4#p13":"\n    A painter is represented as a\n    procedure\n    that, given a frame as argument, draws a particular image shifted and\n    scaled to fit the frame. That is to say, if\n    p is a painter and\n    f is a frame, then we produce\n    p's image in\n    f by calling p\n    with f as argument.\n  ","2.2.4#p14":"\n    The details of how primitive painters are implemented depend on the\n    particular characteristics of the graphics system and the type of image to\n    be drawn.  For instance, suppose we have a\n    proceduredraw-line\n    that draws a line on the screen between two specified points.  Then we can\n    create painters for line drawings, such as the\n    wave\n    painter in figure 2.10, from lists of line\n    segments as follows:(define (segments->painter segment-list)\n  (lambda (frame)\n    (for-each\n      (lambda (segment)\n        (draw-line\n          ((frame-coord-map frame) (start-segment segment))\n          ((frame-coord-map frame) (end-segment segment))))\n      segment-list))) \n    The segments are given using coordinates with respect to the unit square.\n    For each segment in the list, the painter transforms the segment endpoints\n    with the frame coordinate map and draws a line between the transformed\n    points.\n  ","2.2.4#footnote-link-6":"6","2.2.4#p15":"\n    Representing painters as\n    procedures\n    erects a powerful abstraction barrier in the picture language.  We can\n    create and intermix all sorts of primitive painters, based on a variety of\n    graphics capabilities. The details of their implementation do not matter.\n    Any\n    procedure\n    can serve as a painter, provided that it takes a frame as argument and\n    draws something scaled to fit the frame.","2.2.4#footnote-link-7":"7","2.2.4#ex-2.48":"\n    A directed line segment in the plane can be represented as a pair of\n    \n    vectors—the vector running from the origin to the start-point of\n    the segment, and the vector running from the origin to the end-point of\n    the segment. Use your vector representation from\n    exercise 2.46 to define a representation for\n    segments with a constructor\n    make-segment\n    and selectors\n    start-segment\n    and\n    end-segment.","2.2.4#ex-2.49":"\n    Use\n    segments->painter\n    to define the following primitive painters:\n    \n        The painter that draws the outline of the designated frame.\n      \n        The painter that draws an \"X\" by connecting opposite corners of\n        the frame.\n      \n        The painter that draws a diamond shape by connecting the midpoints of\n        the sides of the frame.\n      \n        The wave painter.\n      ","2.2.4#h5":"Transforming and combining painters","2.2.4#p16":"\n    An operation on painters (such as\n    flip-vert\n    or beside)\n    works by creating a painter that invokes the original painters with respect\n    to frames derived from the argument frame. Thus, for example,\n    flip-vert\n    doesn't have to know how a painter works in order to flip\n    it—it just has to know how to turn a frame upside down: The flipped\n    painter just uses the original painter, but in the inverted frame.\n  ","2.2.4#p17":"\n    Painter operations are based on the\n    proceduretransform-painter,\n    which takes as arguments a painter and information on how to transform a\n    frame and produces a new painter.  The transformed painter, when called on\n    a frame, transforms the frame and calls the original painter on the\n    transformed frame. The arguments to\n    transform-painter\n    are points (represented as vectors) that specify the corners of the new\n    frame: When mapped into the frame, the first point specifies the new\n    frame's origin and the other two specify the ends of its edge vectors.\n    Thus, arguments within the unit square specify a frame contained within the\n    original frame.\n    (define (transform-painter painter origin corner1 corner2)\n  (lambda (frame)\n    (let ((m (frame-coord-map frame)))\n      (let ((new-origin (m origin)))\n        (painter\n          (make-frame new-origin\n            (sub-vect (m corner1) new-origin)\n            (sub-vect (m corner2) new-origin))))))) ","2.2.4#p18":"\n    Here's how to flip painter images vertically:\n    (define (flip-vert painter)\n  (transform-painter painter\n    (make-vect 0.0 1.0)   ; new origin\n    (make-vect 1.0 1.0)   ; new end of edge1\n    (make-vect 0.0 0.0))) ; new end of edge2 \n    Using\n    transform-painter,\n      we can easily define new transformations.\n      For example, we can define a\n      painter that shrinks its image to the upper-right quarter of the frame it\n      is given:\n      (define (shrink-to-upper-right painter)\n  (transform-painter painter\n    (make-vect 0.5 0.5)\n    (make-vect 1.0 0.5)\n    (make-vect 0.5 1.0))) \n    Other transformations rotate images counterclockwise by 90\n    degrees(define (rotate90 painter)\n  (transform-painter painter\n    (make-vect 1.0 0.0)\n    (make-vect 1.0 1.0)\n    (make-vect 0.0 0.0))) \n    or squash images towards the center of the frame:(define (squash-inwards painter)\n  (transform-painter painter\n    (make-vect 0.0 0.0)\n    (make-vect 0.65 0.35)\n    (make-vect 0.35 0.65))) ","2.2.4#footnote-link-8":"8","2.2.4#footnote-link-9":"9","2.2.4#p19":"\n    Frame transformation is also the key to\n    defining means of combining two or more painters.\n    The besideprocedure,\n    for example, takes two painters, transforms them to paint in the left and\n    right halves of an argument frame respectively, and produces a new,\n    compound painter. When the compound painter is given a frame, it calls the\n    first transformed painter to paint in the left half of the frame and calls\n    the second transformed painter to paint in the right half of the frame:\n    (define (beside painter1 painter2)\n  (let ((split-point (make-vect 0.5 0.0)))\n    (let ((paint-left\n            (transform-painter painter1\n              (make-vect 0.0 0.0)\n              split-point\n              (make-vect 0.0 1.0)))\n          (paint-right\n            (transform-painter painter2\n              split-point\n              (make-vect 1.0 0.0)\n              (make-vect 0.5 1.0))))\n      (lambda (frame)\n        (paint-left frame)\n        (paint-right frame))))) ","2.2.4#p20":"\n    Observe how the painter data abstraction, and in particular the\n    representation of painters as\n    procedures,\n    makes\n    beside easy to implement.  The\n    besideprocedure\n    need not know anything about the details of the component painters other\n    than that each painter will draw something in its designated frame.\n  ","2.2.4#ex-2.50":"Define\n    the transformation\n    flip-horiz,\n    which flips painters horizontally, and transformations that rotate painters\n    counterclockwise by 180 degrees and 270 degrees.\n    ","2.2.4#ex-2.51":"Define\n    the\n     below operation for painters.\n    Below\n    takes two painters as arguments. The resulting painter, given a frame, \n\tdraws with the first painter in the bottom of the frame and with the\n\tsecond painter in the top.\n    Define below in two different\n    ways—first by writing a\n    procedure\n    that is analogous to the\n    besideprocedure\n    given above, and again in terms of beside and\n    suitable rotation operations (from exercise 2.50).\n    ","2.2.4#h6":"Levels of language for robust design","2.2.4#p21":"\n    The picture language exploits some of the critical ideas we've\n    introduced about abstraction with\n    procedures\n    and data.  The fundamental data abstractions, painters, are implemented\n    using\n    \n        procedural\n      \n    representations, which enables the language to handle different basic\n    drawing capabilities in a uniform way.  The means of combination satisfy\n    the closure property, which permits us to easily build up complex designs.\n    Finally, all the tools for abstracting\n    procedures\n    are available to us for abstracting means of combination for painters.\n  ","2.2.4#p22":"\n    We have also obtained a glimpse of another crucial idea about languages and\n    program design.  This is the approach of\n    stratified design, the notion that a complex system should be\n    structured as a sequence of levels that are described using a sequence of\n    languages. Each level is constructed by combining parts that are regarded\n    as primitive at that level, and the parts constructed at each level are\n    used as primitives at the next level.  The language used at each level\n    of a stratified design has primitives, means of combination, and means\n    of abstraction appropriate to that level of detail.\n  ","2.2.4#p23":"\n    Stratified design pervades the engineering of complex systems.  For\n    example, in computer engineering, resistors and transistors are\n    combined (and described using a language of analog circuits) to\n    produce parts such as and-gates and or-gates, which form the\n    primitives of a language for digital-circuit design. These parts are combined to build\n    processors, bus structures, and memory systems, which are in turn\n    combined to form computers, using languages appropriate to computer\n    architecture.  Computers are combined to form distributed systems,\n    using languages appropriate for describing network interconnections,\n    and so on.\n  ","2.2.4#footnote-link-10":"10","2.2.4#p24":"\n    As a tiny example of stratification, our picture language uses primitive\n    elements (primitive painters) that specify points and lines to provide the\n    shapes of a painter like rogers. The bulk of\n    our description of the picture language focused on combining these\n    primitives, using geometric combiners such as\n    beside and below.\n    We also worked at a higher level, regarding\n    beside and below\n    as primitives to be manipulated in a language whose operations, such as\n    square-of-four,\n    capture common patterns of combining geometric combiners.\n  ","2.2.4#p25":"\n    Stratified design helps make programs\n    robust, that is, it makes\n    it likely that small changes in a specification will require\n    correspondingly small changes in the program.  For instance, suppose we\n    wanted to change the image based on wave\n    shown in figure 2.9.  We could work\n    at the lowest level to change the detailed appearance of the\n    wave element; we could work at the middle\n    level to change the way\n    corner-split\n    replicates the wave; we could work at the\n    highest level to change how\n    square-limit\n    arranges the four copies of the corner. In general, each level of a\n    stratified design provides a different vocabulary for expressing the\n    characteristics of the system, and a different kind of ability to change it.\n  ","2.2.4#ex-2.52":"\n    Make changes to the square limit of wave\n    shown in figure 2.9 by working at\n    each of the levels described above.  In particular:\n    \n        Add some segments to the primitive\n\twave painter\n        of exercise 2.49 (to add a smile, for\n\texample).\n      \n        Change the pattern constructed by\n\tcorner-split\n        (for example, by using only one copy of the\n        up-split\n\tand\n\tright-split\n\timages instead of two).\n      \n        Modify the version of\n\tsquare-limit\n\tthat uses\n\tsquare-of-four\n        so as to assemble the corners in a different pattern.\n\t(For example, you might make the big Mr. Rogers look outward \n\tfrom each corner of the square.)\n      ","2.2.4#footnote-1":"The picture\n    language is based on the language\n    \n    Peter Henderson created to construct images like\n    \n    M.C. Escher's \"Square Limit\" woodcut (see\n    Henderson 1982). The woodcut incorporates a repeated\n    scaled pattern, similar to the arrangements drawn using the\n    square-limitprocedure\n    in this section.","2.2.4#footnote-2":"\n      William Barton Rogers (1804–1882) was the founder and first\n      president of MIT.  A geologist and talented teacher, he taught at\n      William and Mary College and at the University of Virginia.  In 1859\n      he moved to Boston, where he had more time for research, worked on a\n      plan for establishing a \"polytechnic institute,\" and\n      served as Massachusetts's first State Inspector of Gas Meters.\n      \n      When MIT was established in 1861, Rogers was elected its first\n      president. Rogers espoused an ideal of \"useful learning\"\n      that was different from the university education of the time, with its\n      overemphasis on the classics, which, as he wrote, \"stand in the\n      way of the broader, higher and more practical instruction and\n      discipline of the natural and social sciences.\"  This\n      education was likewise to be different from narrow trade-school\n      education.  In Rogers's words:\n      \n        The world-enforced distinction between the practical and the\n        scientific worker is utterly futile, and the whole experience of\n        modern times has demonstrated its utter worthlessness.\n      \n      Rogers served as president of MIT until 1870, when he resigned due to\n      ill health.  In 1878 the second president of MIT,\n      \n      John Runkle, resigned under the pressure of a financial crisis\n      brought on by the Panic of 1873 and strain of fighting off attempts\n      by Harvard to take over MIT.  Rogers returned to hold the office of\n      president until 1881.\n      \n      Rogers collapsed and died while addressing MIT's graduating\n      class at the commencement exercises of 1882.  Runkle quoted\n      Rogers's last words in a memorial address delivered that same\n      year:\n      \"As I stand here today and see what the Institute is, … I call\n        to mind the beginnings of science.  I remember one hundred and fifty\n        years ago Stephen Hales published a pamphlet on the subject of\n        illuminating gas, in which he stated that his researches had\n        demonstrated that 128 grains of bituminous coal—\"\"Bituminous coal,\" these were his last words on\n\t  earth.  Here he bent forward, as if consulting some notes on the\n\t  table before him, then slowly regaining an erect position, threw\n\t  up his hands, and was translated from the scene of his earthly\n\t  labors and triumphs to \"the tomorrow of death,\"\n\t  where the mysteries of life are solved, and the disembodied\n\t  spirit finds unending satisfaction in contemplating the new and\n\t  still unfathomable mysteries of the infinite future.\n      \n      In the words of  Francis A. Walker\n      \n      (MIT's third president):\n      \n        All his life he had borne himself most faithfully and heroically,\n\tand he died as so good a knight would surely have wished, in\n\tharness, at his post, and in the very part and act of public duty.\n      ","2.2.4#footnote-3":"Equivalently, we could\n    write\n    (define flipped-pairs\n  (square-of-four identity flip-vert identity flip-vert)) ","2.2.4#footnote-4":"Rotate180\n    rotates a painter by 180 degrees. Instead of\n    rotate180\n    we could say\n    (compose flip-vert flip-horiz),\n      \n    using the\n    composeprocedure\n    from exercise 1.42.","2.2.4#footnote-5":"Frame-coord-map\n    uses the vector operations described in\n    exercise 2.46 below, which we assume have been\n    implemented using some representation for vectors. Because of data\n    abstraction, it doesn't matter what this vector representation is,\n    so long as the vector operations behave correctly.","2.2.4#footnote-6":"Segments->painter\n    uses the representation for line segments described in\n    exercise 2.48 below. It also uses the\n    for-eachprocedure\n    described in exercise 2.23.","2.2.4#footnote-7":"\n    For example, the rogers painter of\n    figure 2.11 was constructed from a gray-level\n    image. For each point in a given frame, the\n    rogers painter determines the point in\n    the image that is mapped to it under the frame coordinate map, and\n    shades it accordingly.\n    \n    By allowing different types of painters, we are capitalizing on the\n    abstract data idea discussed in section 2.1.3,\n    where we argued that a rational-number representation could be anything at\n    all that satisfies an appropriate condition.  Here we're using the\n    fact that a painter can be implemented in any way at all, so long as it\n    draws something in the designated frame.\n    \n    Section 2.1.3 also showed how pairs could be\n    implemented as\n    procedures.\n    Painters are our second example of a\n    \n        procedural\n      \n    representation for data.","2.2.4#footnote-8":"Rotate90\n    is a pure rotation only for square frames, because it also stretches and\n    shrinks the image to fit into the rotated frame.","2.2.4#footnote-9":"\n    The diamond-shaped images in\n    figures 2.10\n    and 2.11 were created with\n    squash-inwards\n    applied to wave and\n    rogers.\n    ","2.2.4#footnote-10":"\n    Section 3.3.4 describes one such\n    language.","2.3":"2.3  Symbolic Data\n\n  All the compound data objects we have used so far were constructed\n  ultimately from numbers.  In this section we extend the representational\n  capability of our language by introducing the ability to work with\n  arbitrary symbols\n  as data.\n\n  ","2.3.1":"2.3.1","2.3.1#p1":"\n        If we can form compound data using symbols, we can have lists such as\n        (a b c d)\n(23 45 17)\n((Norah 12) (Molly 9) (Anna 7) (Lauren 6) (Charlotte 4))\n        Lists containing symbols can look just like the expressions of our\n        language:\n        (* (+ 23 45) (+ x 9))\n\n(define (fact n) (if (= n 1) 1 (* n (fact (- n 1)))))","2.3.1#p2":"\n        In order to manipulate symbols we need a new element in our language:\n        the ability to quote a data object.  Suppose we want to\n        construct the list (a b).  We can't \n        accomplish this with (list a b), because \n        this expression constructs\n        a list of the values of a and\n\tb rather than the symbols themselves.\n\tThis issue is well known in the context of\n        \n        natural languages, where words and sentences may be regarded either as\n        semantic entities or as character strings (syntactic entities).  The\n        common practice in natural languages is to use quotation marks to\n        indicate that a word or a sentence is to be treated literally as a\n        string of characters.  For instance, the first letter of\n\t\"John\" is clearly \"J.\"  If we tell somebody\n\t\"say your name aloud,\" we expect to hear that\n\tperson's name.  However, if we tell somebody\n\t\"say 'your name' aloud,\" we expect to hear\n\tthe words \"your name.\"  Note that we are forced to nest\n\tquotation marks to describe what somebody else might\n\tsay.","2.3.1#footnote-link-1":"1","2.3.1#p3":"\n        We can follow this same practice to identify lists and symbols that are\n        to be treated as data objects rather than as expressions to be\n        evaluated.  However, our format for quoting differs from that of\n        natural languages in that we place a quotation mark (traditionally,\n        the single \n        \n        quote symbol ') only at the\n\tbeginning of the object to be quoted.  We can get away with this in\n\tScheme syntax because we rely on blanks and parentheses to delimit\n\tobjects.  Thus, the meaning of the single quote character is to quote\n\tthe next object.","2.3.1#footnote-link-2":"2","2.3.1#p4":"\n        Now we can distinguish between symbols and their values:\n        (define a 1)\n\n(define b 2) (list a b) (list 'a 'b) (list 'a b) ","2.3.1#p5":"\n        Quotation also allows us to type in compound objects, using the\n        conventional printed representation for lists:(car '(a b c)) (cdr '(a b c)) \n        In keeping with this, we can obtain the empty list by evaluating '(), \n        and thus dispense with the variable nil.\n      ","2.3.1#footnote-link-3":"3","2.3.1#p6":"\n        One additional primitive used in manipulating symbols is \n        eq?,\n        which takes two symbols as arguments and tests whether they are the\n        same.\n\tUsing eq?, we can implement a useful\n\tprocedure called memq.  This takes two\n        arguments, a symbol and a list.  If the symbol is not contained in the\n        list (i.e., is not eq? to any item in the\n\tlist), then memq returns false.\n\tOtherwise, it returns the sublist of the list beginning with the\n\tfirst occurrence of the symbol:\n        (define (memq item x)\n  (cond ((null? x) false)\n        ((eq? item (car x)) x)\n        (else (memq item (cdr x))))) \n        For example, the value of\n        (memq 'apple '(pear banana prune)) \n        is false, whereas the value of\n        (memq 'apple '(x (apple sauce) y apple pear)) \n        is (apple pear).\n      ","2.3.1#footnote-link-4":"4","2.3.1#ex-2.53":"\n        What would the interpreter print in response to evaluating each of the\n        following expressions?\n        (list 'a 'b 'c) (list (list 'george)) (cdr '((x1 x2) (y1 y2))) (cadr '((x1 x2) (y1 y2))) (pair? (car '(a short list))) (memq 'red '((red shoes) (blue socks))) (memq 'red '(red shoes blue socks)) ","2.3.1#ex-2.54":" \n        Two lists are said to be \n        equal? if they contain equal elements\n        arranged in the same order.  For example,\n        (equal? '(this is a list) '(this is a list)) \n        is true, but\n        (equal? '(this is a list) '(this (is a) list)) \n        is false.  To be more precise, we can define\n\tequal? recursively in terms of the basic\n\teq? equality of symbols by saying that\n\ta and b are\n\tequal? if they are both symbols and the\n\tsymbols are eq?, or if they are both\n\tlists such that (car a) is\n\tequal? to\n\t(car b) and \n        (cdr a) is\n\tequal? to\n\t(cdr b).  \n        Using this idea, implement equal? as a\n        procedure.","2.3.1#footnote-link-5":"5","2.3.1#ex-2.55":"\n        Eva Lu Ator types to the interpreter the expression\n        (car ''abracadabra) \n        To her surprise, the interpreter prints back\n\tquote.  Explain.\n        ","2.3.1#footnote-1":"Allowing quotation in a language wreaks havoc\n        with the ability to reason about the language in simple terms, because\n        it destroys the notion that equals can be substituted for equals.  For\n        example, three is one plus two, but the word \"three\" is\n\tnot the phrase \"one plus two.\"  Quotation is powerful\n\tbecause it gives us a way to build expressions that manipulate other\n\texpressions (as we will see when we write an interpreter in\n\tchapter 4). But allowing statements in a language that talk\n\tabout other statements in that language makes it very difficult to\n\tmaintain any coherent principle of what \"equals can be\n\tsubstituted for equals\" should mean.  For example, if we know\n\tthat\n        \n        the evening star is the morning star, then from the statement \"the\n        evening star is Venus\" we can deduce \"the morning star is\n\tVenus.\" However, given that \"John knows that the evening\n\tstar is Venus\" we cannot infer that \"John knows that the\n\tmorning star is Venus.\"","2.3.1#footnote-2":"The single quote is different \n        \n        from the double quote we have been using to enclose character strings\n\tto be printed.  Whereas the single quote can be used to denote lists\n\tor symbols, the double quote is used only with character strings.\n\tIn this book, the only use for character strings is as items to be\n\tprinted.","2.3.1#footnote-3":"Strictly, our\n        use of the quotation mark violates the general rule that all compound\n        expressions in our language should be delimited by parentheses\n        and look like lists.  We\n        \n        can recover this consistency by introducing a special form\n\tquote, which serves the same purpose as\n\tthe quotation mark.  Thus, we would type\n\t(quote a) instead of\n\t'a, and we would type\n\t(quote (a b c)) instead of\n\t'(a b c).  This is precisely how the\n        interpreter works.  The quotation mark is just a single-character\n        abbreviation for wrapping the next complete expression with \n        quote to form\n\t(quote expression).  This is important\n        because it maintains the principle that any expression seen by the\n        interpreter can be manipulated as a data object.  For instance, we\n        could construct the expression\n        (car '(a b c)), which is the same as\n\t(car (quote (a b c))), by evaluating\n\t(list 'car (list 'quote '(a b c))).","2.3.1#footnote-4":"We can consider two symbols to be\n\t\"the same\" if they consist of the same characters in the\n\tsame order.  Such a definition skirts a deep issue that we are not yet\n\tready to address: the meaning of \"sameness\" in a\n\tprogramming language.  We will return to this in\n        chapter 3\n\t(section 3.1.3).","2.3.1#footnote-5":"In practice, programmers use\n\tequal? to compare lists that contain\n\tnumbers as well as symbols.  Numbers are not considered to be symbols.\n\tThe question\n        \n        of whether two numerically equal numbers (as tested by\n\t=) are also\n\teq? is highly implementation-dependent.\n\tA better definition of equal? (such as\n\tthe one that comes as a primitive in Scheme) would also stipulate that\n\tif a and b are\n        both numbers, then a and\n\tb are equal?\n\tif they are numerically equal.","2.3.2":"2.3.2  \n    Example: Symbolic Differentiation","2.3.2#p1":"\n    As an illustration of symbol manipulation and a further illustration\n    of data abstraction, consider the design of a\n    procedure\n    that performs symbolic differentiation of algebraic expressions.  We would\n    like the\n    procedure\n    to take as arguments an algebraic expression and a variable and to return\n    the derivative of the expression with respect to the variable. For example,\n    if the arguments to the\n    procedure\n    are $ax^2 + bx +c$ \n    and $x$, the\n    procedure\n    should return $2ax+b$.  Symbolic differentiation\n    is of special historical significance in\n    \n\tLisp.\n      \n    It was one of the\n    motivating examples behind the development of a computer language for\n    symbol manipulation.  Furthermore, it marked the beginning of the line of\n    research that led to the development of powerful systems for symbolic\n    mathematical work, which are\n    \n    currently being used by a growing number of\n    applied mathematicians and physicists.\n      ","2.3.2#p2":"\n    In developing the symbolic-differentiation program, we will follow the same\n    strategy of data abstraction that we followed in developing the\n    rational-number system of section 2.1.1.  That\n    is, we will first define a differentiation algorithm that operates on\n    abstract objects such as \"sums,\"\"products,\" and\n    \"variables\" without worrying about how these are to be\n    represented.  Only afterward will we address the representation problem.\n  ","2.3.2#h1":"The differentiation program with abstract data","2.3.2#p3":"\n\tIn order to\n      \n    keep things simple, we will consider a very simple\n    symbolic-differentiation program that handles expressions that are built up\n    using only the operations of addition and multiplication with two\n    arguments.  Differentiation of any such expression can be carried out by\n    applying the following\n    \n    reduction rules:\n    \n      \\[\n      \\begin{array}{rll}\n      \\dfrac{dc}{dx}     & = & \n                         0\\quad \\text{for $c$ a constant or a variable different from $x$} \\\\[12pt]\n      \\dfrac{dx}{dx}     & = &  1  \\\\[12pt]\n      \\dfrac{d(u+v)}{dx} & = & \\dfrac{du}{dx}+\\dfrac{dv}{dx} \\\\[12pt]\n      \\dfrac{d(uv)}{dx}  & = &  u\\left( \\dfrac{dv}{dx}\\right)+v\\left(\n      \\dfrac{du}{dx}\\right)\n      \\end{array}\n      \\]\n    ","2.3.2#p4":"\n  Observe that the latter two rules are recursive in nature.  That is, to\n    obtain the derivative of a sum we first find the derivatives of the terms\n    and add them.  Each of the terms may in turn be an expression that needs to\n    be decomposed.  Decomposing into smaller and smaller pieces will eventually\n    produce pieces that are either constants or variables, whose derivatives\n    will be either $0$ or\n    $1$.\n  ","2.3.2#p5":"\n    To embody these rules in a\n    procedure\n    we indulge in a little\n    \n    wishful thinking, as we did in designing the rational-number implementation.\n    If we had a means for representing algebraic expressions, we should be able\n    to tell whether an expression is a sum, a product, a constant, or a\n    variable.  We should be able to extract the parts of an expression. For a\n    sum, for example, we want to be able to extract the addend (first term) and\n    the augend (second term).  We should also be able to construct expressions\n    from parts.  Let us assume that we already have\n    procedures\n    to implement the following selectors, constructors, and predicates:\n    (variable? e)\n          Is e a variable?\n        (same-variable? v1 v2)\n          Are v1 and\n\t  v2 the same variable?\n        (sum? e)\n          Is e a sum?\n        (addend e)\n          Addend of the sum e.\n        (augend e)\n          Augend of the sum e.\n        (make-sum a1 a2)\n          Construct the sum of a1 and\n\t  a2.\n        (product? e)\n          Is e a product?\n        (multiplier e)\n          Multiplier of the product e.\n        (multiplicand e)\n          Multiplicand of the product e.\n        (make-product m1 m2)\n          Construct the product of m1 and\n\t  m2.\n        \n    Using these, and the primitive predicate\n    number?,\n    which identifies numbers, we can express the differentiation rules as the\n    following\n    procedure:(define (deriv exp var)\n  (cond ((number? exp) 0)\n        ((variable? exp)\n          (if (same-variable? exp var) 1 0))\n        ((sum? exp)\n          (make-sum (deriv (addend exp) var)\n                    (deriv (augend exp) var)))\n        ((product? exp)\n          (make-sum\n            (make-product (multiplier exp)\n                          (deriv (multiplicand exp) var))\n            (make-product (deriv (multiplier exp) var)\n                          (multiplicand exp))))\n        (else\n          (error \"unknown expression type -- DERIV\" exp)))) \n    This derivprocedure\n    incorporates the complete differentiation algorithm. Since it is expressed\n    in terms of abstract data, it will work no matter how we choose to\n    represent algebraic expressions, as long as we design a proper set of\n    selectors and constructors.  This is the issue we must address next.\n  ","2.3.2#h2":"Representing algebraic expressions","2.3.2#p6":"\n    We can imagine many ways to use list structure to represent algebraic\n    expressions.  For example, we could use lists of symbols that mirror the\n    usual algebraic notation, representing $ax+b$ as\n    the list (a * x + b).\n\tHowever, one especially straightforward choice is to use the same\n\tparenthesized prefix notation that Lisp uses for combinations; that\n\tis, to represent $ax+b$ as\n\t(+ (* a x) b). Then our \n      \n    data representation for the differentiation problem is as follows:\n    \n\tThe variables are\n\tsymbols.\n\tThey are identified by the primitive predicate\n\tsymbol?:(define (variable? x) (symbol? x)) \n\tTwo variables are the same if the\n\tsymbols representing them are\n\t  eq?:\n          (define (same-variable? v1 v2)\n  (and (variable? v1) (variable? v2) (eq? v1 v2))) \n\tSums and products are constructed as lists:\n\t(define (make-sum a1 a2) (list '+ a1 a2))\n(define (make-product m1 m2) (list '* m1 m2)) \n\tA sum is a list whose first element is the\n        symbol +:(define (sum? x)\n  (and (pair? x) (eq? (car x) '+))) \n\tThe addend is the second item of the sum list:\n        (define (addend s) (cadr s)) \n\tThe augend is the third item of the sum list:\n        (define (augend s) (caddr s)) \n\tA product is a list whose first element is the\n        symbol *:(define (product? x)\n  (and (pair? x) (eq? (car x) '*))) \n\tThe multiplier is the second item of the product list:\n        (define (multiplier p) (cadr p)) \n\tThe multiplicand is the third item of the product list:\n        (define (multiplicand p) (caddr p)) \n    Thus, we need only combine these with the algorithm as embodied by\n    deriv in order to have a working\n    symbolic-differentiation program.  Let us look at some examples of its\n    behavior:\n    (deriv '(+ x 3) 'x) (deriv '(* x y) 'x) (deriv '(* (* x y) (+ x 3)) 'x) \n    The program produces answers that are correct; however, they are\n    unsimplified.  It is true that\n    \n      \\[\n      \\begin{array}{lll}\n      \\dfrac{d(xy)}{dx} & = & x\\cdot 0+1\\cdot y\n      \\end{array}\n      \\]\n    \n    but we would like the program to know that\n    $x\\cdot 0 = 0$,\n    $1\\cdot y = y$, and\n    $0+y = y$.  The answer for the second example\n    should have been simply y.  As the\n    third example shows, this becomes a serious issue when the expressions are\n    complex.\n  ","2.3.2#p7":"\n    Our difficulty is much like the one we encountered with the rational-number\n    implementation:\n    \n    we haven't reduced answers to simplest form.  To\n    accomplish the rational-number reduction, we needed to change only the\n    constructors and the selectors of the implementation. We can adopt a similar strategy here.  We won't change deriv at\n    all.  Instead, we will change\n    make-sum\n    so that if both summands are numbers,\n    make-sum\n    will add them and return their sum.  Also, if one of the summands is 0,\n    then\n    make-sum\n    will return the other summand.\n    (define (make-sum a1 a2)\n  (cond ((=number? a1 0) a2)\n        ((=number? a2 0) a1)\n        ((and (number? a1) (number? a2)) (+ a1 a2))\n        (else (list '+ a1 a2)))) \n    This uses the\n    procedure=number?,\n    which checks whether an expression is equal to a given number:\n    (define (=number? exp num)\n  (and (number? exp) (= exp num))) \n    Similarly, we will change\n    make-product\n    to build in the rules that 0 times anything is 0 and 1 times anything is\n    the thing itself:\n    (define (make-product m1 m2)\n  (cond ((or (=number? m1 0) (=number? m2 0)) 0)\n        ((=number? m1 1) m2)\n        ((=number? m2 1) m1)\n        ((and (number? m1) (number? m2)) (* m1 m2))\n        (else (list '* m1 m2)))) \n    Here is how this version works on our three examples:\n    (deriv '(+ x 3) 'x) (deriv '(* x y) 'x) (deriv '(* (* x y) (+ x 3)) 'x) \n    Although this is quite an improvement, the third example shows that there\n    is still a long way to go before we get a program that puts expressions\n    into a form that we might agree is \"simplest.\"  The problem\n    of algebraic simplification is complex because, among other reasons, a\n    form that may be simplest for one purpose may not be for another.\n    ","2.3.2#ex-2.56":"\n    Show how to extend the basic differentiator to handle more kinds of\n    expressions.\n    \n    For instance, implement the differentiation rule\n    \n      \\[\n      \\begin{array}{lll}\n      \\dfrac {d(u^{n})}{dx} & = & nu^{n-1}\\left( \\dfrac{du}{dx}\\right)\n      \\end{array}\n      \\]\n    \n    by adding a new clause to the deriv program\n    and defining appropriate\n    proceduresexponentiation?,base, exponent,\n    and\n    make-exponentiation.\n    (You may use\n    the symbol **\n    to denote exponentiation.) Build in the rules that anything raised to the\n    power 0 is 1 and anything raised to the power 1 is the thing itself.\n    ","2.3.2#ex-2.57":"\n    Extend the differentiation program to handle sums and products of arbitrary\n    numbers of (two or more) terms. Then the last example above could be\n    expressed as\n    (deriv '(* x y (+ x 3)) 'x) \n    Try to do this by changing only the\n    representation for sums and products, without changing the\n    derivprocedure\n    at all.  For example, the addend of a sum would\n    be the first term, and the augend would be the\n    sum of the rest of the terms.\n    ","2.3.2#ex-2.58":"\n    Suppose we want to modify the differentiation program so that it works\n    with ordinary mathematical notation, in which\n    +\n    and\n    *\n    are\n    \n    infix rather than prefix operators.  Since the differentiation program\n    is defined in terms of abstract data, we can modify it to work with\n    different representations of expressions solely by changing the predicates,\n    selectors, and constructors that define the representation of the algebraic\n    expressions on which the differentiator is to operate.\n    \n        Show how to do this in order to differentiate algebraic expressions\n\tpresented in infix form,\n\t\n\t    such as\n\t    (x + (3 * (x + (y + 2)))).\n\t  \n        To simplify the task, assume that\n\t+\n\tand\n\t*\n\talways take two arguments and that expressions are fully parenthesized.\n      \n        The problem becomes substantially harder if we allow\n        \n            standard algebraic notation, such as\n            (x + 3 * (x + y + 2))\n            which drops unnecessary parentheses and assumes that multiplication is done before addition.\n          \n        Can you design appropriate predicates, selectors, and constructors for\n\tthis notation such that our derivative program still works?\n      ","2.3.3":"2.3.3  \n    Example: Representing Sets","2.3.3#p1":"\n    In the previous examples we built representations for two kinds of\n    compound data objects: rational numbers and algebraic expressions.  In\n    one of these examples we had the choice of simplifying (reducing) the\n    expressions at either construction time or selection time, but other\n    than that the choice of a representation for these structures in terms\n    of lists was straightforward. When we turn to the representation of\n    sets, the choice of a representation is not so obvious.  Indeed, there\n    are a number of possible representations, and they differ\n    significantly from one another in several ways.\n  ","2.3.3#p2":"\n    Informally, a set is simply a collection of distinct objects.  To give\n    a more precise definition we can employ the method of data\n    abstraction.  That is, we define \"set\" by specifying the\n    \n    operations that are to be used on sets.  These are\n    union-set,intersection-set,element-of-set?,\n    and\n    adjoin-set.Element-of-set?\n    is a predicate that determines whether a given element is a member of a set.\n    Adjoin-set\n    takes an object and a set as arguments and returns a set that contains the\n    elements of the original set and also the adjoined element.\n    Union-set\n    computes the union of two sets, which is the set containing each element\n    that appears in either argument.\n    Intersection-set\n    computes the intersection of two sets, which is the set containing only\n    elements that appear in both arguments.  From the viewpoint of data\n    abstraction, we are free to design any representation that implements these\n    operations in a way consistent with the interpretations given\n    above.","2.3.3#footnote-link-1":"1","2.3.3#h1":"Sets as unordered lists","2.3.3#p3":"\n    One way to represent a set is as a list of its elements in which no\n    element appears more than once.  The empty set is represented by the\n    empty list.  In this representation,\n    element-of-set?\n    is similar to the\n    procedurememq\n      of section .\t\n      \n    It uses\n    equal?\n    instead of\n    eq?\n    so that the set elements need not be\n    symbols:(define (element-of-set? x set)\n  (cond ((null? set) false)\n        ((equal? x (car set)) true)\n        (else (element-of-set? x (cdr set))))) \n    Using this, we can write\n    adjoin-set.\n    If the object to be adjoined is already in the set, we just return the set.\n    Otherwise, we use\n    cons\n    to add the object to the list that represents the set:\n    (define (adjoin-set x set)\n  (if (element-of-set? x set)\n    set\n    (cons x set))) \n    For\n    intersection-set\n    we can use a recursive strategy.  If we know how to form the intersection\n    of set2 and the\n    cdr\n    of set1, we only need to decide whether to\n    include the\n    car\n    of set1 in this.  But this depends on whether\n    (car set1)\n    is also in set2.  Here is the resulting\n    procedure:(define (intersection-set set1 set2)\n  (cond ((or (null? set1) (null? set2)) '())\n        ((element-of-set? (car set1) set2)\n          (cons (car set1)\n            (intersection-set (cdr set1) set2)))\n        (else (intersection-set (cdr set1) set2)))) ","2.3.3#p4":"\n    In designing a representation, one of the issues we should be concerned\n    with is efficiency.  Consider the number of steps required by our set\n    operations.  Since they all use\n    element-of-set?,\n    the speed of this operation has a major impact on the efficiency of the set\n    implementation as a whole.  Now, in order to check whether an object is a\n    member of a set,\n    element-of-set?\n    may have to scan the entire set. (In the worst case, the object turns out\n    not to be in the set.)  Hence, if the set has\n    $n$ elements,\n    element-of-set?\n    might take up to $n$ steps.  Thus, the number of\n    steps required grows as $\\Theta(n)$. The number\n    of steps required by\n    adjoin-set,\n      \n    which uses\n    this operation, also grows as $\\Theta(n)$.  For\n    intersection-set,\n    which does an\n    element-of-set?\n    check for each element of set1, the number of\n    steps required grows as the product of the sizes of the sets involved, or\n    $\\Theta(n^{2})$ for two sets of size\n    $n$.  The same will be true of\n    union-set.","2.3.3#ex-2.59":"\n    Implement the\n    union-set\n    operation for the unordered-list representation of sets.\n    ","2.3.3#ex-2.60":"\n    We specified that a set would be represented as a list with no duplicates.\n    Now suppose we allow duplicates.  For instance, the set\n    $\\{1,2,3\\}$ could be represented as the list\n    (2 3 2 1 3 2 2).\n    Design\n    procedureselement-of-set?,adjoin-set,union-set,\n    and\n    intersection-set\n    that operate on this representation.  How does the efficiency of each\n    compare with the corresponding\n    procedure\n    for the non-duplicate representation?  Are there applications for which\n    you would use this representation in preference to the non-duplicate one?\n    ","2.3.3#h2":"Sets as ordered lists","2.3.3#p5":"\n    One way to speed up our set operations is to change the representation\n    so that the set elements are listed in increasing order.  To do this,\n    we need some way to compare two objects so that we can say which is\n    bigger.  For example, we could compare\n    \n\tsymbols\n      \n    lexicographically, or\n    we could agree on some method for assigning a unique number to an\n    object and then compare the elements by comparing the corresponding\n    numbers.  To keep our discussion simple, we will consider only the\n    case where the set elements are numbers, so that we can compare\n    elements using > and\n    <. We will represent a set of\n    numbers by listing its elements in increasing order.  Whereas our\n    first representation above allowed us to represent the set\n    $\\{1,3,6,10\\}$ by listing the elements in any\n    order, our new representation allows only the list\n    (1 3 6 10).","2.3.3#p6":"\n    One advantage of ordering shows up in\n    element-of-set?:\n    In checking for the presence of an item, we no longer have to scan the\n    entire set.  If we reach a set element that is larger than the item we\n    are looking for, then we know that the item is not in the set:\n    \n(define (element-of-set? x set)\n  (cond ((null? set) false)\n        ((= x (car set)) true)\n        ((< x (car set)) false)\n        (else (element-of-set? x (cdr set)))))\n      \n    How many steps does this save?  In the worst case, the item we are\n    looking for may be the largest one in the set, so the number of steps\n    is the same as for the unordered representation.  On the other hand,\n    if we search for items of many different sizes we can expect that\n    sometimes we will be able to stop searching at a point near the\n    beginning of the list and that other times we will still need to\n    examine most of the list.  On the average we should expect to have to\n    examine about half of the items in the set.  Thus, the average\n    number of steps required will be about $n/2$.\n    This is still $\\Theta(n)$ growth, but\n    it does save us, on the average, a factor of 2 in number of steps over the\n    previous implementation.\n  ","2.3.3#p7":"\n    We obtain a more impressive speedup with\n    intersection-set.\n    In the unordered representation this operation required\n    $\\Theta(n^2)$ steps, because we performed a\n    complete scan of set2 for each element of\n    set1.  But with the ordered representation,\n    we can use a more clever method.  Begin by comparing the initial elements,\n    x1 and\n    x2, of the two sets.  If\n    x1 equals\n    x2, then that gives an element of the\n    intersection, and the rest of the intersection is the intersection of the\n    cdrs\n    of the two sets.  Suppose, however, that x1\n    is less than x2. Since\n    x2 is the smallest element in\n    set2, we can immediately conclude that\n    x1 cannot appear anywhere in\n    set2 and hence is not in the intersection.\n    Hence, the intersection is equal to the intersection of\n    set2 with the\n    cdr\n    of set1.  Similarly, if\n    x2 is less than\n    x1, then the intersection is given by the\n    intersection of set1 with the\n    cdr\n    of set2.  Here is the\n    procedure:\n(define (intersection-set set1 set2)\n  (if (or (null? set1) (null? set2))\n    '()\n    (let ((x1 (car set1)) (x2 (car set2)))\n      (cond ((= x1 x2)\n              (cons x1\n                    (intersection-set (cdr set1)\n                                      (cdr set2))))\n            ((< x1 x2)\n              (intersection-set (cdr set1) set2))\n            ((< x2 x1)\n              (intersection-set set1 (cdr set2)))))))\n      \n    To estimate the number of steps required by this process, observe that at\n    each step we reduce the intersection problem to computing intersections of\n    smaller sets—removing the first element from\n    set1 or set2\n    or both.  Thus, the number of steps required is at most the sum of the sizes\n    of set1 and set2,\n    rather than the product of the sizes as with the unordered representation.\n    This is $\\Theta(n)$ growth rather than\n    $\\Theta(n^2)$—a considerable speedup,\n    even for sets of moderate size.\n  ","2.3.3#ex-2.61":"\n    Give an implementation of\n    adjoin-set\n    using the ordered representation.  By analogy with\n    element-of-set?\n    show how to take advantage of the ordering to produce a\n    procedure\n    that requires on the average about half as many steps as with the unordered\n    representation.\n    ","2.3.3#ex-2.62":"\n    Give a $\\Theta(n)$ implementation of\n     union-set\n    for sets represented as ordered lists.\n    ","2.3.3#h3":"Sets as binary trees","2.3.3#p8":"\n    We can do better than the ordered-list representation by arranging the set\n    elements in the form of a tree.  Each node of the tree holds one element of\n    the set, called the \"entry\" at that node, and a link to each\n    of two other (possibly empty) nodes.  The \"left\" link points to\n    elements smaller than the one at the node, and the \"right\"\n    link to elements greater than the one at the node.\n    Figure 2.16 shows some trees that represent\n    the set $\\{1,3,5,7,9,11\\}$.  The same set may be\n    represented by a tree in a number of different ways.  The only thing we\n    require for a valid representation is that all elements in the left subtree\n    be smaller than the node entry and that all elements in the right subtree be\n    larger.\n    ","2.3.3#fig-2.16":"","2.3.3#p9":"\n    The advantage of the tree representation is this: Suppose we want to check\n    whether a number $x$ is contained in a set.  We\n    begin by comparing $x$ with the entry in the\n    top node.  If $x$ is less than this, we know\n    that we need only search the left subtree; if $x$\n    is greater, we need only search the right subtree.  Now, if the tree is\n    \"balanced,\" each of these subtrees will be about half the size\n    of the original.  Thus, in one step we have reduced the problem of\n    searching a tree of size $n$ to searching a tree\n    of size $n/2$.  Since the size of the tree is\n    halved at each step, we should expect that the number of steps needed to\n    search a tree of size $n$ grows as\n    $\\Theta(\\log n)$. For\n    large sets, this will be a significant speedup over the previous\n    representations.\n  ","2.3.3#footnote-link-2":"2","2.3.3#p10":"\n    We can represent trees by using\n    \n    lists.  Each node will be a list of\n    three items: the entry at the node, the left subtree, and the right\n    subtree.  A left or a right subtree of the empty list will indicate\n    that there is no subtree connected there.  We can describe this\n    representation by the following\n    procedures:(define (entry tree) (car tree))\n\n(define (left-branch tree) (cadr tree))\n\n(define (right-branch tree) (caddr tree))\n\n(define (make-tree entry left right)\n  (list entry left right)) ","2.3.3#footnote-link-3":"3","2.3.3#p11":"\n    Now we can write \n    the element-of-set? procedure\n    using the strategy described above:\n    \n(define (element-of-set? x set)\n  (cond ((null? set) false)\n        ((= x (entry set)) true)\n        ((< x (entry set))\n          (element-of-set? x (left-branch set)))\n        ((> x (entry set))\n          (element-of-set? x (right-branch set)))))\n      ","2.3.3#p12":"\n    Adjoining an item to a set is implemented similarly and also requires\n    $\\Theta(\\log n)$ steps.  To adjoin an item\n    x, we compare\n    x with the node entry to determine whether\n    x should be added to the right or to the left\n    branch, and having adjoined\n    x to the appropriate branch we piece this\n    newly constructed branch together with the original entry and the other\n    branch.  If x is equal to the entry, we just\n    return the node.  If we are asked to adjoin\n    x to an empty tree, we generate a tree that\n    has x as the entry and empty right and left\n    branches.  Here is the\n    procedure:\n(define (adjoin-set x set)\n  (cond ((null? set) (make-tree x '() '()))\n        ((= x (entry set)) set)\n        ((< x (entry set))\n          (make-tree (entry set)\n                     (adjoin-set x (left-branch set))\n                     (right-branch set)))\n        ((> x (entry set))\n          (make-tree (entry set)\n                     (left-branch set)\n                     (adjoin-set x (right-branch set))))))\n      ","2.3.3#p13":"\n    The above claim that searching the tree can be performed in a logarithmic\n    number of steps rests on the assumption that the tree is\n    \"balanced,\" i.e., that the\n    left and the right subtree of every tree have approximately the same\n    number of elements, so that each subtree contains about half the\n    elements of its parent.  But how can we be certain that the trees we\n    construct will be balanced?  Even if we start with a balanced tree,\n    adding elements with\n    adjoin-set\n    may produce an unbalanced result.  Since the position of a newly adjoined\n    element depends on how the element compares with the items already in the\n    set, we can expect that if we add elements \"randomly\" the tree\n    will tend to be balanced on the average.  But this is not a guarantee.  For\n    example, if we start with an empty set and adjoin the numbers 1 through 7\n    in sequence we end up with the highly unbalanced tree shown in\n    figure 2.17.  In this tree all the left\n    subtrees are empty, so it has no advantage over a simple ordered list.  One\n    way to solve this problem is to define an operation that transforms an\n    arbitrary tree into a balanced tree with the same elements.  Then we can perform this transformation after every few\n    adjoin-set\n    operations to keep our set in balance.  There are also other ways to solve\n    this problem, most of which involve designing new data structures for which\n    searching and insertion both can be done in\n    $\\Theta(\\log n)$\n    steps.","2.3.3#footnote-link-4":"4","2.3.3#fig-2.17":"","2.3.3#ex-2.63":"\n    Each of the following two\n    procedures\n    converts a\n    \n    binary tree to a list.\n    (define (tree->list-1 tree)\n  (if (null? tree)\n    '()\n    (append (tree->list-1 (left-branch tree))\n            (cons (entry tree)\n                  (tree->list-1 (right-branch tree)))))) (define (tree->list-2 tree)\n  (define (copy-to-list tree result-list)\n    (if (null? tree)\n      result-list\n      (copy-to-list (left-branch tree)\n                    (cons (entry tree)\n                          (copy-to-list (right-branch tree)\n                                        result-list)))))\n  (copy-to-list tree '())) \n        Do the two\n        procedures\n        produce the same result for every tree? If not, how do the results\n\tdiffer?  What lists do the two\n\tprocedures\n        produce for the trees in figure 2.16?\n        \n        Do the two\n        procedures\n        have the same order of growth in the number of steps required to\n\tconvert a balanced tree with $n$ elements\n\tto a list? If not, which one grows more slowly?\n      ","2.3.3#ex-2.64":"\n    The following\n    procedurelist->tree\n    converts an ordered list to a balanced binary tree.  The helper\n    procedurepartial-tree\n    takes as arguments an integer $n$ and list of\n    at least $n$ elements and constructs a balanced\n    tree containing the first $n$ elements of the\n    list.  The result returned by\n    partial-tree\n    is a pair (formed with\n    cons)\n    whose\n    car\n    is the constructed tree and whose\n    cdr\n    is the list of elements not included in the tree.\n    (define (list->tree elements)\n  (car (partial-tree elements (length elements))))\n\n(define (partial-tree elts n)\n  (if (= n 0)\n    (cons '() elts)\n    (let ((left-size (quotient (- n 1) 2)))\n      (let ((left-result (partial-tree elts left-size)))\n        (let ((left-tree (car left-result))\n              (non-left-elts (cdr left-result))\n              (right-size (- n (+ left-size 1))))\n          (let ((this-entry (car non-left-elts))\n                (right-result (partial-tree (cdr non-left-elts)\n                                            right-size)))\n            (let ((right-tree (car right-result))\n                  (remaining-elts (cdr right-result)))\n              (cons (make-tree this-entry left-tree right-tree)\n                    remaining-elts)))))))) \n        Write a short paragraph explaining as clearly as you can how\n\tpartial-tree\n\tworks.  Draw the tree produced by\n\tlist->tree\n\tfor the list\n\t(1 3 5 7 9 11).\n        What is the order of growth in the number of steps required by\n\tlist->tree\n\tto convert a list of $n$ elements?\n      ","2.3.3#ex-2.65":"\n    Use the results of exercises 2.63\n    and 2.64 to give\n    $\\Theta(n)$ implementations of\n    union-set\n    and\n    intersection-set\n    for sets implemented as (balanced) binary trees.","2.3.3#footnote-link-5":"5","2.3.3#h4":"Sets and information retrieval","2.3.3#p14":"\n    We have examined options for using lists to represent sets and have\n    seen how the choice of representation for a data object can have a\n    large impact on the performance of the programs that use the data.\n    Another reason for concentrating on sets is that the techniques\n    discussed here appear again and again in applications involving\n    information retrieval.\n  ","2.3.3#p15":"\n    Consider a data base containing a large number of individual records,\n    \n    such as the personnel files for a company or the transactions in an\n    accounting system.  A typical data-management system spends a large\n    amount of time accessing or modifying the data in the records and\n    therefore requires an efficient method for accessing records.  This is\n    done by identifying a part of each record to serve as an identifying\n    key.  A key can be anything that uniquely identifies the\n    record.  For a personnel file, it might be an employee's ID number.\n    For an accounting system, it might be a transaction number.  Whatever\n    the key is, when we define the record as a data structure we should\n    include a\n     key selector\n    procedure\n    that retrieves the key associated with a given record.\n  ","2.3.3#p16":"\n    Now we represent the data base as a set of records. To locate the record\n    with a given key we use a\n    procedurelookup, which takes as arguments a key and a\n    data base and which returns the record that has that key, or false if there\n    is no such record.\n    Lookup\n    is implemented in almost the same way as\n    element-of-set?.\n    For example, if the set of records is implemented as an unordered list, we\n    could use\n    (define (lookup given-key set-of-records)\n  (cond ((null? set-of-records) false)\n        ((equal? given-key (key (car set-of-records)))\n          (car set-of-records))\n        (else (lookup given-key (cdr set-of-records))))) ","2.3.3#p17":"\n    Of course, there are better ways to represent large sets than as unordered\n    lists.  Information-retrieval systems in which records have to be\n    \"randomly accessed\" are typically implemented by a tree-based\n    method, such as the binary-tree representation discussed previously.\n    In designing such a system the methodology of data abstraction\n    can be a great help.  The designer can create an initial implementation\n    using a simple, straightforward representation such as unordered lists.\n    This will be unsuitable for the eventual system, but it can be useful in\n    providing a \"quick and dirty\" data base with which to test the\n    rest of the system.  Later on, the data representation can be modified to\n    be more sophisticated.  If the data base is accessed in terms of abstract\n    selectors and constructors, this change in representation will not require\n    any changes to the rest of the system.\n  ","2.3.3#ex-2.66":"\n    Implement the lookupprocedure\n    for the case where the set of records is structured as a binary tree,\n    ordered by the numerical values of the keys.\n    ","2.3.3#footnote-1":"If we want to be more formal, we can specify\n    \"consistent with the interpretations given above\" to mean\n    that the operations satisfy a collection of rules such as these:\n    \n\tFor any set S and any object\n\tx,\n\t(element-of-set? x (adjoin-set x S))\n\tis true (informally: \"Adjoining an object to a set produces a\n\tset that contains the object\").\n      \n        For any sets S and\n\tT and any object\n\tx,\n        (element-of-set? x (union-set S T))\n        is equal to\n        (or (element-of-set? x S) (element-of-set? x T)) \t    \n        (informally: \"The elements of\n\t(union-set S T)\n\tare the elements that are in S or in\n\tT\").\n      \n        For any object x,\n        (element-of-set? x '())\n        is false (informally:\n\t\"No object is an element of the empty set\").\n      ","2.3.3#footnote-2":"Halving the size of\n    the problem at each step is the distinguishing characteristic of\n    \n    logarithmic growth, as we saw with the fast-exponentiation algorithm of\n    section 1.2.4 and the half-interval\n    search method of\n    section 1.3.3.","2.3.3#footnote-3":"We\n    are representing sets in terms of trees, and trees in terms of\n    lists—in effect, a data abstraction built upon a data abstraction.\n    We can regard the\n    proceduresentry,\n    left-branch,right-branch,\n    and\n    make-tree\n    as a way of isolating the abstraction of a \"binary tree\" from\n    the particular way we might wish to represent such a tree in terms of list\n    structure.","2.3.3#footnote-4":"Examples of such structures include\n    B-trees and red-black trees.  There is a large literature\n    on data structures devoted to this problem.  See \n    Cormen, Leiserson, Rivest, and Stein 2022.","2.3.3#footnote-5":"\n    Exercises 2.63–2.65\n    are due to\n    \n    Paul Hilfinger.","2.3.4":"2.3.4  \n    Example: Huffman Encoding Trees","2.3.4#p1":"\n    This section provides practice in the use of list structure and data\n    abstraction to manipulate sets and trees.  The application is to\n    methods for representing data as sequences of ones and zeros (bits).\n    For example, the\n    \n    ASCII standard code used to represent text in\n    computers encodes each\n    \n    character as a sequence of seven bits.  Using\n    seven bits allows us to distinguish $2^7$, or\n    128, possible different characters.  In general, if we want to distinguish\n    $n$ different symbols, we will need to use\n    $\\log_2 n$ bits per symbol.  If all our messages\n    are made up of the eight symbols A, B, C, D, E, F, G, and H, we can choose\n    a code with three bits per character, for example\n    A000           C010           E100           G110B001D011F101H111\n    With this code, the message\n    \nBACADAEAFABBAAAGAH\n    is encoded as the string of 54 bits\n    \n001000010000011000100000101000001001000000000110000111","2.3.4#p2":"\n    Codes such as ASCII and the A-through-H code above are known as\n    fixed-length codes, because they represent each symbol in the\n    message with the same number of bits.  It is sometimes advantageous to use\n    variable-length codes, in which different symbols may be\n    represented by different numbers of bits.  For example,\n    \n    Morse code does not use the same number of dots and dashes for each letter\n    of the alphabet.  In particular, E, the most frequent letter, is represented\n    by a single dot.  In general, if our messages are such that some symbols\n    appear very frequently and some very rarely, we can encode data more\n    efficiently (i.e., using fewer bits per message) if we assign shorter\n    codes to the frequent symbols.  Consider the following alternative code for\n    the letters A through H:\n    A0           C1010           E1100           G1110B100D1011F1101H1111\n    With this code, the same message as above is encoded as the string\n    \n100010100101101100011010100100000111001111\n    \n    This string contains 42 bits, so it saves more than 20% in space in\n    comparison with the fixed-length code shown above.\n  ","2.3.4#p3":"\n    One of the difficulties of using a variable-length code is knowing\n    when you have reached the end of a symbol in reading a sequence of\n    zeros and ones.  Morse code solves this problem by using a special\n    separator code (in this case, a pause) after the sequence of\n    dots and dashes for each letter.  Another solution is to design the\n    code in such a way that no complete code for any symbol is the\n    beginning (or prefix) of the code for another symbol.  Such a\n    code is called a\n    prefix code.  In the example above, A is encoded by 0 and B is\n    encoded by 100, so no other symbol can have a code that begins with 0 or\n    with 100.\n  ","2.3.4#p4":"\n    In general, we can attain significant savings if we use\n    variable-length prefix codes that take advantage of the relative\n    frequencies of the symbols in the messages to be encoded.  One\n    particular scheme for doing this is called the Huffman encoding\n    method, after its discoverer,\n    \n    David Huffman.  A Huffman code can be represented as a\n    \n    binary tree whose leaves are the symbols that are encoded.  At each\n    non-leaf node of the tree there is a set containing all the symbols in the\n    leaves that lie below the node.  In addition, each symbol at a leaf is\n    assigned a weight (which is its relative frequency), and each non-leaf\n    node contains a weight that is the sum of all the weights of the leaves\n    lying below it.  The weights are not used in the encoding or the decoding\n    process.  We will see below how they are used to help construct the tree.\n  ","2.3.4#p5":"\n    Figure 2.18 shows the Huffman tree for the\n    A-through-H code given above.  The weights at the leaves indicate that the\n    tree was designed for messages in which A appears with relative frequency\n    8, B with relative frequency 3, and the other letters each with relative\n    frequency 1.\n  ","2.3.4#fig-2.18":"","2.3.4#p6":"\n    Given a Huffman tree, we can find the encoding of any symbol by\n    starting at the root and moving down until we reach the leaf that\n    holds the symbol.  Each time we move down a left branch we add a 0 to\n    the code, and each time we move down a right branch we add a 1.  (We\n    decide which branch to follow by testing to see which branch either is\n    the leaf node for the symbol or contains the symbol in its set.)  For\n    example, starting from the root of the tree in\n    figure 2.18, we arrive at the leaf for D by\n    following a right branch, then a left branch, then a right branch, then a\n    right branch; hence, the code for D is 1011.\n  ","2.3.4#p7":"\n    To decode a bit sequence using a Huffman tree, we begin at the root\n    and use the successive zeros and ones of the bit sequence to determine\n    whether to move down the left or the right branch.  Each time we come\n    to a leaf, we have generated a new symbol in the message, at which\n    point we start over from the root of the tree to find the next symbol.\n    For example, suppose we are given the tree above and the sequence\n    10001010.  Starting at the root, we move down the right branch (since\n    the first bit of the string is 1), then down the left branch (since\n    the second bit is 0), then down the left branch (since the third bit\n    is also 0).  This brings us to the leaf for B, so the first\n    symbol of the decoded message is B.  Now we start again at the root,\n    and we make a left move because the next bit in the string is 0.\n    This brings us to the leaf for A.  Then we start again at the root\n    with the rest of the string 1010, so we move right, left, right, left and\n    reach C. Thus, the entire message is BAC.\n  ","2.3.4#h1":"Generating Huffman trees","2.3.4#p8":"\n    Given an \"alphabet\" of symbols and their relative frequencies,\n    how do we construct the \"best\" code?  (In other words, which\n    tree will encode messages with the fewest bits?)  Huffman gave an algorithm\n    for doing this and showed that the resulting code is indeed the best\n    variable-length code for messages where the relative frequency of the\n    symbols matches the frequencies with which the code was constructed.\n    \n    We will not prove this optimality of Huffman codes here, but we will\n    show how Huffman trees are constructed.","2.3.4#footnote-link-1":"1","2.3.4#p9":"\n    The algorithm for generating a Huffman tree is very simple. The idea\n    is to arrange the tree so that the symbols with the lowest frequency\n    appear farthest away from the root. Begin with the set of leaf nodes,\n    containing symbols and their frequencies, as determined by the initial data\n    from which the code is to be constructed. Now find two leaves with\n    the lowest weights and merge them to produce a node that has these\n    two nodes as its left and right branches. The weight of the new node\n    is the sum of the two weights. Remove the two leaves from the\n    original set and replace them by this new node. Now continue this\n    process. At each step, merge two nodes with the smallest weights,\n    removing them from the set and replacing them with a node that has\n    these two as its left and right branches. The process stops when\n    there is only one node left, which is the root of the entire tree.\n    Here is how the Huffman tree of figure 2.18 was\n    generated:\n    \n\t      Initial leaves\n\t    $\\{$(A 8) (B 3) (C 1) (D 1) (E 1) (F 1) (G 1) (H 1)$\\}$\n\t      Merge\n\t    $\\{$(A 8) (B 3) ($\\{$C D$\\}$ 2) (E 1) (F 1) (G 1) (H 1)$\\}$\n\t      Merge\n\t    $\\{$(A 8) (B 3) ($\\{$C D$\\}$ 2) ($\\{$E F$\\}$ 2) (G 1) (H 1)$\\}$\n\t      Merge\n\t    $\\{$(A 8) (B 3) ($\\{$C D$\\}$ 2) ($\\{$E F$\\}$ 2) ($\\{$G H$\\}$ 2)$\\}$\n\t      Merge\n\t    $\\{$(A 8) (B 3) ($\\{$C D$\\}$ 2) ($\\{$E F G H$\\}$ 4)$\\}$\n\t      Merge\n\t    $\\{$(A 8) ($\\{$B C D$\\}$ 5) ($\\{$E F G H$\\}$ 4)$\\}$\n\t      Merge\n\t    $\\{$(A 8) ($\\{$B C D E F G H$\\}$ 9)$\\}$\n\t      Final merge\n\t    $\\{$($\\{$A B C D E F G H$\\}$ 17)$\\}$\n    The algorithm does not always specify a unique tree, because there may\n    not be unique smallest-weight nodes at each step.  Also, the choice of\n    the order in which the two nodes are merged (i.e., which will be the\n    right branch and which will be the left branch) is arbitrary.\n  ","2.3.4#h2":"Representing Huffman trees","2.3.4#p10":"\n    In the exercises below we will work with a system that uses\n    Huffman trees to encode and decode messages and generates Huffman\n    trees according to the algorithm outlined above.  We will begin by\n    discussing how trees are represented.\n  ","2.3.4#p11":"\n    Leaves of the tree are represented by a list consisting of the\n    symbol leaf,\n    the symbol at the leaf, and the weight:\n    (define (make-leaf symbol weight)\n  (list 'leaf symbol weight))\n\n(define (leaf? object)\n  (eq? (car object) 'leaf))\n\n(define (symbol-leaf x) (cadr x))\n\n(define (weight-leaf x) (caddr x)) \n    A general tree will be a list of\n    \n    a left branch, a right branch, a set\n    of symbols, and a weight.  The set of symbols will be simply a list of\n    the symbols, rather than some more sophisticated set representation.\n    When we make a tree by merging two nodes, we obtain the weight of the\n    tree as the sum of the weights of the nodes, and the set of symbols as\n    the union of the sets of symbols for the nodes.  Since our symbol sets are\n    represented as lists, we can form the union by using the\n    appendprocedure\n    we defined in section 2.2.1:\n    (define (make-code-tree left right)\n  (list left\n    right\n    (append (symbols left) (symbols right))\n    (+ (weight left) (weight right)))) \n    If we make a tree in this way, we have the following selectors:\n    (define (left-branch tree) (car tree))\n\n(define (right-branch tree) (cadr tree))\n\n(define (symbols tree)\n  (if (leaf? tree)\n    (list (symbol-leaf tree))\n    (caddr tree)))\n\n(define (weight tree)\n  (if (leaf? tree)\n    (weight-leaf tree)\n    (cadddr tree))) \n    The\n    proceduressymbols and\n    weight must do something slightly different\n    depending on whether they are called with a leaf or a general tree.\n    These are simple examples of\n    \n      generic\n    procedures(procedures\n    that can handle more than one kind of data), which we will have much more\n    to say about in sections 2.4\n    and 2.5.\n  ","2.3.4#h3":"\n      The decoding\n      procedure","2.3.4#p12":"\n    The following\n    procedure\n    implements the decoding algorithm. It takes as arguments a list of zeros\n    and ones, together with a Huffman tree.\n    (define (decode bits tree)\n  (define (decode-1 bits current-branch)\n    (if (null? bits)\n      '()\n      (let ((next-branch\n              (choose-branch (car bits) current-branch)))\n        (if (leaf? next-branch)\n          (cons (symbol-leaf next-branch)\n                (decode-1 (cdr bits) tree))\n          (decode-1 (cdr bits) next-branch)))))\n  (decode-1 bits tree))\n\n(define (choose-branch bit branch)\n  (cond ((= bit 0) (left-branch branch))\n        ((= bit 1) (right-branch branch))\n        (else (error \"bad bit -- CHOOSE-BRANCH\" bit)))) \n    The\n    proceduredecode-1\n    takes two arguments: the list of remaining bits and the current position in\n    the tree.  It keeps moving \"down\" the tree, choosing a left or\n    a right branch according to whether the next bit in the list is a zero or a\n    one.  (This is done with the\n    procedurechoose-branch.)\n    When it reaches a leaf, it returns the symbol at that leaf as the next\n    symbol in the message by\n    consing\n\tit onto the result of decoding the rest of the message,\n\tstarting at the root of the tree.\n      \n    Note the error check in the final clause of\n    choose-branch,\n      which complains if the\n      procedure\n    finds something other than a zero or a one in the input data.\n  ","2.3.4#h4":"Sets of weighted elements","2.3.4#p13":"\n    In our representation of trees, each non-leaf node contains a set of\n    symbols, which we have represented as a simple list.  However, the\n    tree-generating algorithm discussed above requires that we also work\n    with sets of leaves and trees, successively merging the two smallest\n    items.  Since we will be required to repeatedly find the smallest item\n    in a set, it is convenient to use an ordered representation for this\n    kind of set.\n  ","2.3.4#p14":"\n    We will represent a set of leaves and trees as a list of elements,\n    arranged in increasing order of weight.\n    The following\n    adjoin-setprocedure\n    for constructing sets is similar to the one\n    described in exercise 2.61; however, items\n    are compared by their weights, and the element being added to the set is\n    never already in it.\n    (define (adjoin-set x set)\n  (cond ((null? set) (list x))\n        ((< (weight x) (weight (car set))) (cons x set))\n        (else (cons (car set)\n                    (adjoin-set x (cdr set)))))) ","2.3.4#p15":"\n    The following\n    procedure\n    takes a list of symbol-frequency pairs such as\n    ((A 4) (B 2) (C 1) (D 1))\n    and constructs an initial ordered set of leaves, ready to be merged\n    according to the Huffman algorithm:\n    (define (make-leaf-set pairs)\n  (if (null? pairs)\n    '()\n    (let ((pair (car pairs)))\n      (adjoin-set (make-leaf (car pair)    ; symbol\n                             (cadr pair))  ; frequency\n                  (make-leaf-set (cdr pairs)))))) ","2.3.4#ex-2.67":"Define\n    an encoding tree and a sample message:\n    (define sample-tree\n  (make-code-tree (make-leaf 'A 4)\n                  (make-code-tree\n                    (make-leaf 'B 2)\n                    (make-code-tree (make-leaf 'D 1)\n                                    (make-leaf 'C 1)))))\n\n(define sample-message '(0 1 1 0 0 1 0 1 0 1 1 1 0)) \n    Use the decodeprocedure\n    to decode the message, and give the result.\n    ","2.3.4#ex-2.68":"\n    The encodeprocedure\n    takes as arguments a message and a tree and produces the list of bits that\n    gives the encoded message.\n    (define (encode message tree)\n  (if (null? message)\n    '()\n    (append (encode-symbol (car message) tree)\n            (encode (cdr message) tree)))) Encode-symbol is a procedure,\n\twhich you must write, that returns the list of bits that encodes a given\n\tsymbol according to a given tree.\n      \n    You should design\n    encode-symbol\n    so that it signals an error if the symbol is not in the tree at all.\n    Test your\n    procedure\n    by encoding the result you obtained in\n    exercise 2.67 with the sample tree and\n    seeing whether it is the same as the original sample message.\n    ","2.3.4#ex-2.69":"\n    The following\n    procedure\n    takes as its argument a list of symbol-frequency pairs (where no symbol\n    appears in more than one pair) and generates a Huffman encoding tree\n    according to the Huffman algorithm.\n    (define (generate-huffman-tree pairs)\n  (successive-merge (make-leaf-set pairs))) Make-leaf-set is the procedure\n        given above that transforms the\n        list of pairs into an ordered set of leaves.\n\tSuccessive-merge\n        is the procedure\n        you must write, using make-code-tree to\n        successively merge the smallest-weight elements of the set until there\n        is only one element left, which is the desired Huffman tree.\n      \n    (This\n    procedure\n    is slightly tricky, but not really complicated.  If you find yourself\n    designing a complex\n    procedure,\n    then you are almost certainly doing something wrong.  You can take\n    significant advantage of the fact that we are using an ordered set\n    representation.)\n    ","2.3.4#ex-2.70":"\n    The following eight-symbol alphabet with associated relative\n    frequencies was designed to efficiently encode the lyrics of 1950s\n    \n    rock songs.  (Note that the \"symbols\" of an\n    \"alphabet\" need not be individual letters.)\n    \n\t      A\n\t    \n\t      2\n\t                            \n\t      NA\n\t    \n\t      16\n\t    \n\t      BOOM\n\t    \n\t      1\n\t     \n\t      SHA\n\t      3\n\t    \n\t      GET\n\t    \n\t      2\n\t     \n\t      YIP\n\t      9\n\t    \n\t      JOB\n\t    \n\t      2\n\t     \n\t      WAH\n\t      1\n\t    \n    Use\n    generate-huffman-tree\n    (exercise 2.69) to generate a\n    corresponding Huffman tree, and use encode\n    (exercise 2.68) to encode the following\n    message:\n    Get a jobSha na na na na na na na naGet a jobSha na na na na na na na naWah yip yip yip yip yip yip yip yip yipSha boom\n    How many bits are required for the encoding?  What is the smallest number\n    of bits that would be needed to encode this song if we used a fixed-length\n    code for the eight-symbol alphabet?\n    ","2.3.4#ex-2.71":"\n    Suppose we have a Huffman tree for an alphabet of\n    $n$ symbols, and that the relative frequencies\n    of the symbols are 1, 2, 4, …,\n    $2^{n-1}$.  Sketch the tree for\n    $n$=5; for $n$=10.\n    In such a tree (for general $n$) how may bits\n    are required to encode the most frequent symbol?  the least frequent symbol?\n    ","2.3.4#ex-2.72":"\n    Consider the encoding\n    procedure\n    that you designed in exercise 2.68.  What\n    is the\n    \n    order of growth in the number of steps needed to encode a symbol?\n    Be sure to include the number of steps needed to search the symbol list at\n    each node encountered.  To answer this question in general is difficult.\n    Consider the special case where the relative frequencies of the\n    $n$ symbols are as described in\n    exercise 2.71, and give the order of\n    growth (as a function of $n$) of the number of\n    steps needed to encode the most frequent and least frequent symbols in the\n    alphabet.\n    ","2.3.4#footnote-1":"See\n    Hamming 1980\n    for a discussion of the mathematical properties of Huffman codes.","2.4":"2.4  Multiple Representations for Abstract Data","2.4#p1":"\n    We have introduced data abstraction, a methodology for structuring systems\n    in such a way that much of a program can be specified independent of the\n    choices involved in implementing the data objects that the program\n    manipulates.  For example, we saw in\n    section 2.1.1 how to separate the task of\n    designing a program that uses rational numbers from the task of implementing\n    rational numbers in terms of the computer language's primitive\n    mechanisms for constructing compound data.  The key idea was to erect an\n    \n    abstraction barrier—in this case, the selectors and constructors for\n    rational numbers\n    (make-rat,numer,\n    denom)—that isolates the way rational\n    numbers are used from their underlying representation in terms of list\n    structure.  A similar abstraction barrier isolates the details of the\n    procedures\n    that perform rational arithmetic\n    (add-rat,sub-rat,mul-rat,\n    and\n    div-rat)\n    from the \"higher-level\"procedures\n    that use rational numbers.  The resulting program has the structure shown\n    in figure 2.1.\n  ","2.4#p2":"\n    These data-abstraction barriers are powerful tools for controlling\n    complexity.  By isolating the underlying representations of data\n    objects, we can divide the task of designing a large program into\n    smaller tasks that can be performed separately.  But this kind of data\n    abstraction is not yet powerful enough, because it may not always make\n    sense to speak of \"the underlying representation\" for a\n    data object.\n  ","2.4#p3":"\n    For one thing, there might be more than one useful representation for\n    a data object, and we might like to design systems that can deal with\n    multiple representations.  To take a simple example, complex numbers\n    may be represented in two almost equivalent ways: in rectangular form\n    (real and imaginary parts) and in polar form (magnitude and angle).\n    Sometimes rectangular form is more appropriate and sometimes polar\n    form is more appropriate.  Indeed, it is perfectly plausible to\n    imagine a system in which complex numbers are represented in both\n    ways, and in which the\n    procedures\n    for manipulating complex numbers work with either representation.\n  ","2.4#p4":"\n    More importantly, programming systems are often designed by many\n    people working over extended periods of time, subject to requirements\n    that change over time.  In such an environment, it is simply not\n    possible for everyone to agree in advance on choices of data\n    representation.  So in addition to the data-abstraction barriers that\n    isolate representation from use, we need abstraction barriers that\n    isolate different design choices from each other and permit different\n    choices to coexist in a single program.  Furthermore, since large\n    programs are often created by combining\n    pre-existing\n    modules that were\n    designed in isolation, we need conventions that permit programmers to\n    incorporate modules into larger systems \n    additively, that is,\n    without having to redesign or reimplement these modules.\n  ","2.4#p5":"\n    In this section, we will learn how to cope with data that may be\n    represented in different ways by different parts of a program.  This\n    requires constructing \n    generic procedures—procedures\n    that can operate on data that may be represented in more than one way.  Our\n    main technique for building generic\n    procedures\n    will be to work in terms of data objects that have \n    type tags, that is, data objects that include explicit information\n    about how they are to be processed. We will also discuss \n    data-directed programming, a powerful and convenient\n    implementation strategy for additively assembling systems with generic\n    operations.\n  ","2.4#p6":"\n    We begin with the simple complex-number example. We will see how\n    type tags and data-directed style enable us to design separate\n    rectangular and polar representations for complex numbers while\n    maintaining the notion of an abstract\n    \"complex-number\"\n    data object.\n    We will accomplish this by defining arithmetic\n    procedures\n    for complex numbers\n    (add-complex,sub-complex,mul-complex,\n    and\n    div-complex)\n    in terms of generic selectors that access parts of a complex number\n    independent of how the number is represented.  The resulting complex-number\n    system, as shown in\n    \n\tfigure ,\n      \n    contains two different kinds of\n    \n    abstraction barriers.  The \"horizontal\" abstraction barriers\n    play the same role as the ones in\n    figure 2.1.  They isolate\n    \"higher-level\" operations from \"lower-level\"\n    representations.  In addition, there is a \"vertical\" barrier\n    that gives us the ability to separately design and install alternative\n    representations.\n    ","2.4#fig-":"","2.4#p7":"\n    In section 2.5 we will show how to\n    use type tags and data-directed style to develop a generic arithmetic\n    package.  This provides\n    procedures\n    (add, mul, and so\n    on) that can be used to manipulate all sorts of \"numbers\" and\n    can be easily extended when a new kind of number is needed. In\n    section 2.5.3, we'll show how to\n    use generic arithmetic in a system that performs symbolic algebra.\n  ","2.4.1":"2.4.1  \n    Representations for Complex Numbers","2.4.1#p1":"\n    We will develop a system that performs arithmetic operations on complex\n    numbers as a simple but unrealistic example of a program that uses generic\n    operations.  We begin by discussing two plausible representations for\n    complex numbers as ordered pairs: rectangular form (real part and imaginary\n    part) and polar form (magnitude and angle). Section 2.4.2\n    will show how both representations can be made to coexist in a single\n    system through the use of type tags and generic operations.\n  ","2.4.1#footnote-link-1":"1","2.4.1#p2":"\n    Like rational numbers, complex numbers are naturally represented as ordered\n    pairs.  The set of complex numbers can be thought of as a two-dimensional\n    space with two orthogonal axes, the \"real\" axis and the\n    \"imaginary\" axis. (See\n    figure 2.20.)  From this point of view,\n    the complex number $z=x+iy$ (where\n    $i^{2} =-1$) can be thought of as the point in\n    the plane whose real coordinate is $x$ and whose\n    imaginary coordinate is $y$. Addition of complex\n    numbers reduces in this representation to addition of coordinates:\n    \n      \\[\n        \\begin{array}{lll}\n      \\text{Real-part}(z_{1}+z_{2}) & = & \n         \\text{Real-part}(z_{1})+\\text{Real-part}(z_{2}) \\\\[1ex]\n      \\text{Imaginary-part}(z_{1} +z_{2}) & = & \n         \\text{Imaginary-part}(z_1)+\\text{Imaginary-part}(z_2)\n        \\end{array}\n      \\]\n    ","2.4.1#p3":"\n    When multiplying complex numbers, it is more natural to think in terms\n    of representing a complex number in polar form, as a magnitude and an\n    angle ($r$ and $A$\n    in figure 2.20). The product of two\n    complex numbers is the vector obtained by stretching one complex number by\n    \n    the length of the other and then rotating it through the angle of the other:\n    \n      \\[\n        \\begin{array}{lll}\n      \\text{Magnitude}(z_{1}\\cdot z_{2}) & = & \n      \\text{Magnitude}(z_{1})\\cdot\\text{Magnitude}(z_{2})\\\\[1ex]\n      \\text{Angle}(z_{1}\\cdot z_{2}) & = & \n      \\text{Angle}(z_{1})+\\text{Angle}(z_{2})\n        \\end{array}\n      \\]\n    ","2.4.1#p4":"\n    Thus, there are two different representations for complex numbers,\n    which are appropriate for different operations.  Yet, from the\n    viewpoint of someone writing a program that uses complex numbers, the\n    principle of data abstraction suggests that all the operations for\n    manipulating complex numbers should be available regardless of which\n    representation is used by the computer.  For example, it is often\n    useful to be able to find the magnitude of a complex number that is\n    specified by rectangular coordinates.  Similarly, it is often useful\n    to be able to determine the real part of a complex number that is\n    specified by polar coordinates.\n    ","2.4.1#fig-2.20":"","2.4.1#p5":"\n    To design such a system, we can follow the same \n    \n    data-abstraction strategy we followed in designing the rational-number\n    package in section 2.1.1.  Assume that the\n    operations on complex numbers are implemented in terms of four selectors:\n    real-part,imag-part,magnitude,\n    and angle.  Also assume that we have two\n    procedures\n    for constructing complex numbers:\n    make-from-real-imag\n    returns a complex number with specified real and imaginary parts, and\n    make-from-mag-ang\n    returns a complex number with specified magnitude and angle. These\n    procedures\n    have the property that, for any complex number\n    z, both\n    (make-from-real-imag (real-part z) (imag-part z))\n    and\n    (make-from-mag-ang (magnitude z) (angle z))\n    produce complex numbers that are equal to z.\n  ","2.4.1#p6":"\n    Using these constructors and selectors, we can implement arithmetic on\n    complex numbers using the \"abstract data\" specified by the\n    constructors and selectors, just as we did for rational numbers in\n    section 2.1.1.  As shown in the formulas\n    above, we can add and subtract complex numbers in terms of real and\n    imaginary parts while multiplying and dividing complex numbers in terms of\n    magnitudes and angles:\n    (define (add-complex z1 z2)\n  (make-from-real-imag (+ (real-part z1) (real-part z2))\n                       (+ (imag-part z1) (imag-part z2))))\n\n(define (sub-complex z1 z2)\n  (make-from-real-imag (- (real-part z1) (real-part z2))\n                       (- (imag-part z1) (imag-part z2))))\n\n(define (mul-complex z1 z2)\n  (make-from-mag-ang (* (magnitude z1) (magnitude z2))\n                     (+ (angle z1) (angle z2))))\n\n(define (div-complex z1 z2)\n  (make-from-mag-ang (/ (magnitude z1) (magnitude z2))\n                     (- (angle z1) (angle z2))))","2.4.1#p7":"\n    To complete the complex-number package, we must choose a representation and\n    we must implement the constructors and selectors in terms of primitive\n    numbers and primitive list structure. There are two obvious ways to do\n    this: We can represent a complex number in \"rectangular form\"\n    as a pair (real part, imaginary part) or in \"polar form\" as a\n    pair (magnitude, angle).  Which shall we choose?\n  ","2.4.1#p8":"\n    In order to make the different choices concrete, imagine that there are two\n    programmers, Ben Bitdiddle and Alyssa P. Hacker, who are independently\n    designing representations for the complex-number system.\n    Ben chooses to represent\n    \n    complex numbers in rectangular form.  With this\n    choice, selecting the real and imaginary parts of a complex number is\n    straightforward, as is constructing a complex number with given real and\n    imaginary parts.  To find the magnitude and the angle, or to construct a\n    complex number with a given magnitude and angle, he uses the trigonometric\n    relations\n    \n      \\[\n      \\begin{array}{lllllll}\n      x & = & r\\ \\cos A & \\quad \\quad \\quad & r & = & \\sqrt{x^2 +y^2} \\\\\n      y & = & r\\ \\sin A &             & A &= & \\arctan (y,x)\n      \\end{array}\n      \\]\n    \n    which relate the real and imaginary parts ($x$,\n    $y$) to the magnitude and the angle\n    $(r, A)$.\n    Ben's representation is therefore given by the following selectors\n    and constructors:\n    (define (real-part z) (car z))\n\n(define (imag-part z) (cdr z))\n\n(define (magnitude z)\n  (sqrt (+ (square (real-part z)) (square (imag-part z)))))\n\n(define (angle z)\n  (atan (imag-part z) (real-part z)))\n\n(define (make-from-real-imag x y) (cons x y))\n\n(define (make-from-mag-ang r a) \n  (cons (* r (cos a)) (* r (sin a)))) ","2.4.1#footnote-link-2":"2","2.4.1#p9":"\n    Alyssa, in contrast, chooses to represent complex numbers in\n    \n    polar form.\n    \n    For her, selecting the magnitude and angle is straightforward, but she has\n    to use the \n    \n    trigonometric relations to obtain the real and imaginary parts.\n    Alyssa's representation is:\n    (define (real-part z)\n  (* (magnitude z) (cos (angle z))))\n\n(define (imag-part z)\n  (* (magnitude z) (sin (angle z))))\n\n(define (magnitude z) (car z))\n\n(define (angle z) (cdr z))\n\n(define (make-from-real-imag x y) \n  (cons (sqrt (+ (square x) (square y)))\n        (atan y x)))\n\n(define (make-from-mag-ang r a) (cons r a)) ","2.4.1#p10":"\n    The discipline of data abstraction ensures that the same implementation of\n    add-complex,\n      sub-complex,\n      mul-complex,\n      \n    and\n    div-complex\n    will work with either Ben's representation or Alyssa's\n    representation. \n  ","2.4.1#footnote-1":"In actual computational\n    systems, rectangular form is preferable to polar form most of the time\n    because of \n    \n    roundoff errors in conversion between rectangular and polar form.  This is\n    why the complex-number example is unrealistic.  Nevertheless, it provides a\n    clear illustration of the design of a system using generic operations and a\n    good introduction to the more substantial systems to be developed later in\n    this chapter.","2.4.1#footnote-2":"The arctangent function\n    referred to\n    here,\n    \n\tcomputed by Scheme's\n\tatan procedure,\n      \n    is defined so as to take two arguments\n    $y$ and $x$\n    and to return the angle whose tangent is $y/x$.\n    The signs of the arguments determine the quadrant of the angle.","2.4.2":"2.4.2  \n    Tagged data","2.4.2#p1":"\n    One way to view data abstraction is as an application of the\n    \"principle of least commitment.\"  In implementing the\n    complex-number system in\n    section 2.4.1, we can\n    use either Ben's rectangular representation or Alyssa's polar\n    representation.  The abstraction barrier formed by the selectors and\n    constructors permits us to defer to the last possible moment the choice of\n    a concrete representation for our data objects and thus retain maximum\n    flexibility in our system design.\n  ","2.4.2#p2":"\n    The principle of least commitment can be carried to even further extremes.\n    If we desire, we can maintain the ambiguity of representation even\n    after we have designed the selectors and constructors, and elect\n    to use both Ben's representation and Alyssa's\n    representation.  If both representations are included in a single system,\n    however, we will need some way to distinguish data in polar form from data\n    in rectangular form.  Otherwise, if we were asked, for instance, to find\n    the magnitude of the pair\n    $(3,4)$, we wouldn't know whether to\n    answer 5 (interpreting the number in rectangular form) or\n    3 (interpreting the number in polar form). A straightforward way to\n    accomplish this distinction is to include a\n    type tag—the\n    symbol rectangular\n    or\n    polar—as\n    part of each complex number.  Then when we need to manipulate a complex\n    number we can use the tag to decide which selector to apply.\n  ","2.4.2#p3":"\n    In order to manipulate tagged data, we will assume that we have\n    procedurestype-tag\n    and contents that extract from a data object\n    the tag and the actual contents (the polar or rectangular coordinates, in\n    the case of a complex number).  We will also postulate a\n    procedureattach-tag\n    that takes a tag and contents and produces a tagged data object.  A\n    straightforward way to implement this is to use ordinary list structure:\n    (define (attach-tag type-tag contents)\n  (cons type-tag contents))\n\n(define (type-tag datum)\n  (if (pair? datum)\n    (car datum)\n    (error \"Bad tagged datum -- TYPE-TAG\" datum)))\n\n(define (contents datum)\n  (if (pair? datum)\n    (cdr datum)\n    (error \"Bad tagged datum -- CONTENTS\" datum))) ","2.4.2#p4":"\n        Using these procedures,\n      \n    we can define predicates\n    rectangular?\n    and\n    polar?,\n    which recognize rectangular and polar numbers, respectively:\n    (define (rectangular? z)\n  (eq? (type-tag z) 'rectangular))\n\n(define (polar? z)\n  (eq? (type-tag z) 'polar))","2.4.2#p5":"\n    With type tags, Ben and Alyssa can now modify their code so that their two\n    different representations can coexist in the same system. Whenever Ben\n    constructs a complex number, he tags it as rectangular. Whenever Alyssa\n    constructs a complex number, she tags it as polar. In addition, Ben and\n    Alyssa must make sure that the names of their\n    procedures\n    do not conflict.  One way to do this is for Ben to append the suffix\n    rectangular to the name of each of his\n    representation\n    procedures\n    and for Alyssa to append polar to the names of\n    hers. Here is Ben's revised rectangular representation from\n    section 2.4.1:\n    (define (real-part-rectangular z) (car z))\n\n(define (imag-part-rectangular z) (cdr z))\n\n(define (magnitude-rectangular z)\n  (sqrt (+ (square (real-part-rectangular z))\n           (square (imag-part-rectangular z)))))\n\n(define (angle-rectangular z)\n  (atan (imag-part-rectangular z)\n        (real-part-rectangular z)))\n\n(define (make-from-real-imag-rectangular x y)\n  (attach-tag 'rectangular (cons x y)))\n\n(define (make-from-mag-ang-rectangular r a) \n  (attach-tag 'rectangular\n              (cons (* r (cos a)) (* r (sin a))))) \n    and here is Alyssa's revised polar representation:\n    (define (real-part-polar z)\n  (* (magnitude-polar z) (cos (angle-polar z))))\n\n(define (imag-part-polar z)\n  (* (magnitude-polar z) (sin (angle-polar z))))\n\n(define (magnitude-polar z) (car z))\n\n(define (angle-polar z) (cdr z))\n\n(define (make-from-real-imag-polar x y) \n  (attach-tag 'polar\n              (cons (sqrt (+ (square x) (square y)))\n              (atan y x))))\n\n(define (make-from-mag-ang-polar r a)\n  (attach-tag 'polar (cons r a))) ","2.4.2#p6":"\n    Each generic selector is implemented as a\n    procedure\n    that checks the tag of its argument and calls the appropriate\n    procedure\n    for handling data of that type.  For example, to obtain the real part of\n    a complex number,\n    real-part\n    examines the tag to determine whether to use Ben's\n    real-part-rectangular\n    or Alyssa's\n    real-part-polar.\n    In either case, we use contents to extract the\n    bare, untagged datum and send this to the rectangular or polar\n    procedure\n    as required:\n    (define (real-part z)\n  (cond ((rectangular? z) \n         (real-part-rectangular (contents z)))\n        ((polar? z)\n         (real-part-polar (contents z)))\n        (else (error \"Unknown type -- REAL-PART\" z))))\n\n(define (imag-part z)\n  (cond ((rectangular? z)\n         (imag-part-rectangular (contents z)))\n        ((polar? z)\n         (imag-part-polar (contents z)))\n        (else (error \"Unknown type -- IMAG-PART\" z))))\n\n(define (magnitude z)\n  (cond ((rectangular? z)\n         (magnitude-rectangular (contents z)))\n        ((polar? z)\n         (magnitude-polar (contents z)))\n        (else (error \"Unknown type -- MAGNITUDE\" z))))\n\n(define (angle z)\n  (cond ((rectangular? z)\n         (angle-rectangular (contents z)))\n        ((polar? z)\n         (angle-polar (contents z)))\n        (else (error \"Unknown type -- ANGLE\" z)))) ","2.4.2#p7":"\n    To implement the complex-number arithmetic operations, we can use the same\n    proceduresadd-complex,sub-complex,mul-complex,\n    and\n    div-complex\n    from section 2.4.1,\n    because the selectors they call are generic, and so will work with either\n    representation.  For example, the\n    procedureadd-complex\n    is still\n    (define (add-complex z1 z2)\n  (make-from-real-imag (+ (real-part z1) (real-part z2))\n                       (+ (imag-part z1) (imag-part z2))))","2.4.2#p8":"\n    Finally, we must choose whether to construct complex numbers using\n    Ben's representation or Alyssa's representation.  One\n    reasonable choice is to construct rectangular numbers whenever we have\n    real and imaginary parts and to construct polar numbers whenever we have\n    magnitudes and angles:\n    (define (make-from-real-imag x y)\n  (make-from-real-imag-rectangular x y))\n\n(define (make-from-mag-ang r a)\n  (make-from-mag-ang-polar r a)) ","2.4.2#p9":"The resulting complex-number system has the structure shown in\n    \n\tfigure .\n      \n    The system has been decomposed into three relatively independent parts: the\n    complex-number-arithmetic operations, Alyssa's polar implementation,\n    and Ben's rectangular implementation.  The polar and rectangular\n    implementations could have been written by Ben and Alyssa working\n    separately, and both of these can be used as underlying representations by\n    a third programmer implementing the complex-arithmetic\n    procedures\n    in terms of the abstract constructor/selector interface.\n  ","2.4.2#fig-":"","2.4.2#p10":"\n    Since each data object is tagged with its type, the selectors operate on\n    the data in a\n    \n    generic manner.  That is, each selector is defined to have a\n    behavior that depends upon the particular type of data it is applied to.\n    Notice the general mechanism for interfacing the separate representations:\n    Within a given representation implementation (say, Alyssa's polar\n    package) a complex number is an untyped pair (magnitude, angle).  When a\n    generic selector operates on a number of polar\n    type, it strips off the tag and passes the contents on to Alyssa's\n    code.  Conversely, when Alyssa constructs a number for general use, she\n    tags it with a type so that it can be appropriately recognized by the\n    higher-level\n    procedures.\n    This discipline of stripping off and attaching tags as data objects are\n    passed from level to level can be an important organizational strategy,\n    as we shall see in section 2.5.\n    ","2.4.3":"2.4.3  \n    Data-Directed Programming and Additivity","2.4.3#p1":"\n    The general strategy of checking the type of a datum and calling an\n    appropriate\n    procedure\n    is called\n    dispatching on type.  This is a powerful strategy for obtaining\n    modularity in system design.  On the other hand, implementing the dispatch\n    as in section 2.4.2 has two significant\n    weaknesses.  One weakness is that the generic interface\n    procedures(real-part,imag-part,magnitude, and\n    angle) must know about all the different\n    representations.  For instance, suppose we wanted to incorporate a new\n    representation for complex numbers into our complex-number system.  We\n    would need to identify this new representation with a type, and then add a\n    clause to each of the generic interface\n    procedures\n    to check for the new type and apply the appropriate selector for that\n    representation.\n  ","2.4.3#p2":"\n    Another weakness of the technique is that even though the individual\n    representations can be designed separately, we must guarantee that no two\n    procedures\n    in the entire system have the same name.  This is why Ben and Alyssa had\n    to change the names of their original\n    procedures\n    from section 2.4.1.\n  ","2.4.3#p3":"\n    The issue underlying both of these weaknesses is that the technique for\n    implementing generic interfaces is not additive.  The person\n    implementing the generic selector\n    procedures\n    must modify those\n    procedures\n    each time a new representation is installed, and the people\n    interfacing the individual representations must modify their\n    code to avoid name conflicts.  In each of these cases, the changes\n    that must be made to the code are straightforward, but they must be\n    made nonetheless, and this is a source of inconvenience and error.\n    This is not much of a problem for the complex-number system as it\n    stands, but suppose there were not two but hundreds of different\n    representations for complex numbers.  And suppose that there were many\n    generic selectors to be maintained in the abstract-data interface.\n    Suppose, in fact, that no one programmer knew all the interface\n    procedures\n    or all the representations.  The problem is real and must\n    be addressed in such programs as\n    large-scale data-base-management systems.\n  ","2.4.3#p4":"\n    What we need is a means for modularizing the system design even\n    further.  This is provided by the programming technique known as \n    data-directed programming.  To understand how data-directed\n    programming works, begin with the observation that whenever we deal\n    with a set of generic operations that are common to a set of\n    different types we are, in effect, dealing with a two-dimensional\n    table that contains the possible operations on one axis and the\n    possible types on the other axis.  The entries in the table are the\n    procedures\n    that implement each operation for each type of argument presented.\n    In the complex-number system developed in the previous section, the\n    correspondence between operation name, data type, and  actual\n    procedure\n    was spread out among the various conditional clauses in the generic\n    interface\n    procedures.\n    But the same information could have been organized in a table, as shown in\n    \n\tfigure .\n      ","2.4.3#p5":"\n    Data-directed programming is the technique of designing programs to work\n    with such a\n    \n    table directly.  Previously, we implemented the mechanism that\n    interfaces the complex-arithmetic code with the two representation packages\n    as a set of\n    procedures\n    that each perform an explicit dispatch on type.  Here we will implement the\n    interface as a single\n    procedure\n    that looks up the combination of the operation name and argument type in\n    the table to find the correct\n    procedure\n    to apply, and then applies it to the contents of the argument.  If we do\n    this, then to add a new representation package to the system we need not\n    change any existing\n    procedures;\n    we need only add new entries to the table.\n  ","2.4.3#p6":"\n    To implement this plan, assume that we have two\n    procedures,put and get, for\n    manipulating the\n    \n    operation-and-type table:\n    (put $\\langle \\textit{op} \\rangle\\ \\langle \\textit{type} \t    \\rangle \\ \\langle \\textit{item} \\rangle$ \t    )\n\tinstalls the\n\t$\\langle \\textit{item} \\rangle$\n\tin the table, indexed by the\n\t$\\langle \\textit{op} \\rangle$ and the\n\t    $\\langle \\textit{type} \\rangle$.\n\t  (get $\\langle \\textit{op} \\rangle\\ \\langle \t    \\textit{type}$)\n\tlooks up the\n\t$\\langle \\textit{op} \\rangle$,\n\t    $\\langle \\textit{type} \\rangle$\n\tentry in the table and returns the item found there.\n\tIf no item is found,\n\tget returns\n\tfalse.\n    For now, we can assume that put and\n    get are included in our language.  In\n    chapter 3 (section 3.3.3) we will see\n    how to implement these and other operations for manipulating tables.\n  ","2.4.3#fig-":"","2.4.3#p7":"\n    Here is how data-directed programming can be used in the complex-number\n    system.  Ben, who developed the rectangular representation, implements his\n    code just as he did originally.  He defines a collection of\n    procedures,\n    or a\n    package, and interfaces these to the rest of the system by adding\n    entries to the table that tell the system how to operate on rectangular\n    numbers. This is accomplished by calling the following\n    procedure:(define (install-rectangular-package)\n  ;; internal procedures\n  (define (real-part z) (car z))\n  (define (imag-part z) (cdr z))\n  (define (make-from-real-imag x y) (cons x y))\n  (define (magnitude z)\n    (sqrt (+ (square (real-part z))\n             (square (imag-part z)))))\n  (define (angle z)\n    (atan (imag-part z) (real-part z)))\n  (define (make-from-mag-ang r a)\n    (cons (* r (cos a)) (* r (sin a))))\n\n  ;; interface to the rest of the system\n  (define (tag x) (attach-tag 'rectangular x))\n  (put 'real-part '(rectangular) real-part)\n  (put 'imag-part '(rectangular) imag-part)\n  (put 'magnitude '(rectangular) magnitude)\n  (put 'angle '(rectangular) angle)\n  (put 'make-from-real-imag 'rectangular\n       (lambda (x y) (tag (make-from-real-imag x y))))\n  (put 'make-from-mag-ang 'rectangular\n       (lambda (r a) (tag (make-from-mag-ang r a))))\n  'done) ","2.4.3#p8":"\n    Notice that the internal\n    procedures\n    here are the same\n    procedures\n    from section 2.4.1 that\n    Ben wrote when he was working in isolation.  No changes are necessary in\n    order to interface them to the rest of the system.  Moreover, since these\n    procedure definitions\n    are internal to the installation\n    procedure,\n    Ben needn't worry about name conflicts with other\n    procedures\n    outside the rectangular package.  To interface these to the rest of the\n    system, Ben installs his\n    real-partprocedure\n    under the operation name\n    real-part\n    and the type\n    (rectangular),\n    and similarly for the other selectors.  The interface also defines the\n    constructors to be used by the external system. These are identical to Ben's internally defined\n    constructors, except that they attach the tag.\n  ","2.4.3#footnote-link-1":"1","2.4.3#footnote-link-2":"2","2.4.3#p9":"\n    Alyssa's\n    \n    polar package is analogous:\n    (define (install-polar-package)\n  ;; internal procedures\n  (define (magnitude z) (car z))\n  (define (angle z) (cdr z))\n  (define (make-from-mag-ang r a) (cons r a))\n  (define (real-part z)\n    (* (magnitude z) (cos (angle z))))\n  (define (imag-part z)\n    (* (magnitude z) (sin (angle z))))\n  (define (make-from-real-imag x y)\n    (cons (sqrt (+ (square x) (square y)))\n          (atan y x)))\n\n  ;; interface to the rest of the system\n  (define (tag x) (attach-tag 'polar x))\n  (put 'real-part '(polar) real-part)\n  (put 'imag-part '(polar) imag-part)\n  (put 'magnitude '(polar) magnitude)\n  (put 'angle '(polar) angle)\n  (put 'make-from-real-imag 'polar\n       (lambda (x y) (tag (make-from-real-imag x y))))\n  (put 'make-from-mag-ang 'polar\n       (lambda (r a) (tag (make-from-mag-ang r a))))\n  'done) ","2.4.3#p10":"\n    Even though Ben and Alyssa both still use their original\n    procedures\n    defined with the same names as each other's (e.g.,\n    real-part),\n    these declarations are now internal to different\n    procedures\n    (see section 1.1.8), so there is no name\n    conflict.\n  ","2.4.3#p11":"\n    The complex-arithmetic selectors access the table by means of a general\n    \"operation\"procedure\n    called\n    apply-generic,\n    which applies a generic operation to some arguments.\n    Apply-generic\n    looks in the table under the name of the operation and the types of the\n    arguments and applies the resulting\n    procedure\n    if one is present:(define (apply-generic op . args)\n  (let ((type-tags (map type-tag args)))\n    (let ((proc (get op type-tags)))\n      (if proc\n        (apply proc (map contents args))\n        (error\n          \"No method for these types -- APPLY-GENERIC\"\n          (list op type-tags))))))\n    Using\n    apply-generic,\n    we can define our generic selectors as follows:\n    (define (real-part z) (apply-generic 'real-part z))\n(define (imag-part z) (apply-generic 'imag-part z))\n(define (magnitude z) (apply-generic 'magnitude z))\n(define (angle z) (apply-generic 'angle z)) \n    Observe that these do not change at all if a new representation is\n    added to the system.\n  ","2.4.3#footnote-link-3":"3","2.4.3#p12":"\n    We can also extract from the table the constructors to be used by the\n    programs external to the packages in making complex numbers from real and\n    imaginary parts and from magnitudes and angles. As in\n    section 2.4.2, we construct rectangular\n    numbers whenever we have real and imaginary parts, and polar numbers\n    whenever we have magnitudes and angles:\n    (define (make-from-real-imag x y)\n  ((get 'make-from-real-imag 'rectangular) x y))\n\n(define (make-from-mag-ang r a)\n  ((get 'make-from-mag-ang 'polar) r a)) ","2.4.3#ex-2.73":"\n    Section 2.3.2 described a\n    program that performs\n    \n    symbolic differentiation:\n    (define (deriv exp var)\n  (cond ((number? exp) 0)\n        ((variable? exp) (if (same-variable? exp var) 1 0))\n        ((sum? exp)\n          (make-sum (deriv (addend exp) var)\n                    (deriv (augend exp) var)))\n        ((product? exp)\n          (make-sum\n            (make-product (multiplier exp)\n                          (deriv (multiplicand exp) var))\n            (make-product (deriv (multiplier exp) var)\n                          (multiplicand exp))))\n        ;; more rules can be added here\n        (else (error \"unknown expression type -- DERIV\" exp)))) (deriv '(* (* x y) (+ x 4)) 'x) \n    We can regard this program as performing a dispatch on the type of the\n    expression to be differentiated.  In this situation the\n    \"type tag\" of the datum is the algebraic operator symbol\n    (such as +)\n    and the operation being performed is\n    deriv.  We can transform this program into\n    data-directed style by rewriting the basic derivative\n    procedure\n    as\n    (define (deriv exp var)\n  (cond ((number? exp) 0)\n        ((variable? exp) (if (same-variable? exp var) 1 0))\n        (else ((get 'deriv (operator exp)) (operands exp)\n                                           var))))\n\n(define (operator exp) (car exp))\n\n(define (operands exp) (cdr exp))\n\tExplain what was done above. Why can't we assimilate the\n\tpredicates\n\tnumber?\n\tand\n\tvariable?\n\tinto the data-directed dispatch?\n      \n\tWrite the\n\tprocedures\n\tfor derivatives of sums and products, and the auxiliary code required\n\tto install them in the table used by the program above.\n      \n\tChoose any additional differentiation rule that you like, such as the\n\tone for exponents\n\t(exercise 2.56), and install it\n\tin this data-directed system.\n      \n\tIn this simple algebraic manipulator the type of an expression is the\n\talgebraic operator that binds it together.  Suppose, however, we\n\tindexed the\n\tprocedures\n\tin the opposite way, so that the dispatch line\n\tin deriv looked like\n\t((get (operator exp) 'deriv) (operands exp) var)\n\tWhat corresponding changes to the derivative system are required?\n      ","2.4.3#ex-2.74":"\n    Insatiable\n    \n    Enterprises, Inc., is a highly decentralized conglomerate company\n    consisting of a large number of independent divisions located all over the\n    world.  The company's computer facilities have just been\n    interconnected by means of a clever network-interfacing scheme that makes\n    the entire network appear to any user to be a single computer.\n    Insatiable's president,  in her first attempt to exploit the ability\n    of the network to extract administrative information from division files,\n    is dismayed to discover that, although all the division files have been\n    implemented as data structures in\n    \n\tScheme,\n      \n    the particular data structure used varies from division to division.  A\n    meeting of division managers is hastily called to search for a strategy to\n    integrate the files that will satisfy headquarters' needs while\n    preserving the existing autonomy of the divisions.\n    \n    Show how such a strategy can be implemented with\n    \n    data-directed programming.\n    As an example, suppose that each division's personnel records consist\n    of a single file, which contains a set of records keyed on\n    employees' names.  The structure of the set varies from division to\n    division.  Furthermore, each employee's record is itself a set\n    (structured differently from division to division) that contains\n    information keyed under identifiers such as\n    address and\n    salary.  In particular:\n    \n        Implement for headquarters a\n\tget-recordprocedure\n        that retrieves a specified employee's record from a specified\n\tpersonnel file.  The\n        procedure\n        should be applicable to any division's file. Explain how the\n\tindividual divisions' files should be structured.  In particular,\n\twhat type information must be supplied?\n      \n        Implement for headquarters a\n\tget-salaryprocedure\n        that returns the salary information from a given employee's\n\trecord from any division's personnel file.  How should the record\n\tbe structured in order to make this operation work?\n      \n        Implement for headquarters a\n\tfind-employee-recordprocedure.\n        This should search all the divisions' files for the record of a\n\tgiven employee and return the record.  Assume that this\n        procedure\n        takes as arguments an employee's name and a list of all the\n\tdivisions' files.\n      \n        When Insatiable takes over a new company, what changes must be made in\n\torder to incorporate the new personnel information into the central\n\tsystem?\n      ","2.4.3#h1":"Message passing","2.4.3#p13":"\n    The key idea of data-directed programming is to handle generic operations\n    in programs by dealing explicitly with operation-and-type tables, such as\n    the table in\n    \n\tfigure .\n      \n    The style of programming we used in\n    section 2.4.2 organized the required\n    dispatching on type by having each operation take care of its own\n    dispatching.  In effect, this decomposes the operation-and-type table into\n    rows, with each generic operation\n    procedure\n    representing a row of the table.\n  ","2.4.3#p14":"\n    An alternative implementation strategy is to decompose the table into\n    columns and, instead of using \"intelligent operations\" that\n    dispatch on data types, to work with \"intelligent data\n    objects\" that dispatch on operation names.  We can do this by\n    arranging things so that a data object, such as a rectangular number, is\n    represented as a\n    procedure\n    that takes as input the required operation name and performs the operation\n    indicated.  In such a discipline,\n    make-from-real-imag\n    could be written as\n    (define (make-from-real-imag x y)\n  (define (dispatch op)\n    (cond ((eq? op 'real-part) x)\n          ((eq? op 'imag-part) y)\n          ((eq? op 'magnitude)\n            (sqrt (+ (square x) (square y))))\n          ((eq? op 'angle) (atan y x))\n          (else\n            (error \"Unknown op -- MAKE-FROM-REAL-IMAG\" op))))\n  dispatch) \n    The corresponding\n    apply-genericprocedure,\n    which applies a generic operation to an argument, now simply feeds the\n    operation's name to the data object and lets the object do the\n    work:(define (apply-generic op arg) (arg op)) \n    Note that the value returned by\n    make-from-real-imag\n    is a\n    procedure—the internaldispatchprocedure.\n    This is the\n    procedure\n    that is invoked when\n    apply-generic\n    requests an operation to be performed.\n  ","2.4.3#footnote-link-4":"4","2.4.3#p15":"\n    This style of programming is called message passing.  The name\n    comes from the image that a data object is an entity that receives the\n    requested operation name as a \"message.\"  We have already seen\n    an example of message passing in section 2.1.3,\n    where we saw how\n    cons,car,\n    and\n    cdr\n    could be defined with no data objects but only\n    procedures.\n    Here we see that message passing is not a mathematical trick but a useful\n    technique for organizing systems with generic operations.  In the remainder\n    of this chapter we will continue to use data-directed programming, rather\n    than message passing, to discuss generic arithmetic operations.  In\n    chapter 3 we will return to message passing, and we will see that\n    it can be a powerful tool for structuring simulation programs.\n  ","2.4.3#ex-2.75":"\n    Implement the constructor\n     make-from-mag-ang\n    in message-passing style. This\n    procedure\n    should be analogous to the\n    make-from-real-imagprocedure\n    given above.\n    ","2.4.3#ex-2.76":"\n    As a large system with generic operations evolves, new types of data\n    objects or new operations may be needed.  For each of the three\n    strategies—generic operations with explicit\n    \n    dispatch, data-directed\n    style, and message-passing-style—describe the changes that must be\n    made to a system in order to add new types or new operations.  Which\n    organization would be most appropriate for a system in which new types must\n    often be added?  Which would be most appropriate for a system in which new\n    operations must often be added?\n    ","2.4.3#footnote-1":"We use the list\n    (rectangular)\n    rather than the\n    symbol rectangular\n    to allow for the possibility of operations with multiple arguments, not\n    all of the same type.","2.4.3#footnote-2":"The type the\n    constructors are installed under needn't be a list because a\n    constructor is always used to make an object of one particular\n    type.","2.4.3#footnote-3":"Apply-generic\n\tuses the\n\t\n        dotted-tail notation described in\n        exercise 2.20,\n\tbecause different generic operations may take\n\tdifferent numbers of arguments.\n\tIn\n\tapply-generic,op has as its value the first argument to\n\tapply-generic\n\tand\n\targs has as its value a list of the remaining\n\targuments.\n\tApply-generic also\n\tuses the \n\tprimitive procedure\n\tapply,\n\twhich takes two arguments, a procedure\n\tand a list.\n\tApply\n\tapplies the procedure,\n\tusing the elements in the list as arguments.\n      \n    For example,\n    (apply + (list 1 2 3 4))\n    returns 10.","2.4.3#footnote-4":"One limitation of this organization is it permits only\n    generic\n    procedures\n    of one argument.","2.5":"2.5  Systems with Generic Operations","2.5#p1":"\n    In the previous section, we saw how to design systems in which data\n    objects can be represented in more than one way.  The key idea is to\n    link the code that specifies the data operations to the several\n    representations by means of generic interface\n    procedures.\n    Now we will see how to use this same idea not only to define operations\n    that are generic over different representations but also to define\n    operations that are\n    \n    generic over different kinds of arguments.  We have\n    already seen several different packages of arithmetic operations: the\n    primitive arithmetic (+,\n    -, *,\n    /) built into our language, the\n    rational-number arithmetic\n    (add-rat,sub-rat,mul-rat,div-rat)\n    of section 2.1.1, and the complex-number\n    arithmetic that we implemented in\n    section 2.4.3.  We will now use\n    data-directed techniques to construct a package of arithmetic operations\n    that incorporates all the arithmetic packages we have already constructed.\n  ","2.5#p2":"\n\tFigure \n    shows the structure of the system we\n    shall build.  Notice the \n    \n    abstraction barriers.  From the perspective\n    of someone using \"numbers,\" there is a single\n    procedureadd that operates on whatever numbers are\n    supplied.\n    Add\n    is part of a generic interface that allows the separate ordinary-arithmetic,\n    rational-arithmetic, and complex-arithmetic packages to be accessed\n    uniformly by programs that use numbers.  Any individual arithmetic package\n    (such as the complex package) may itself be accessed through generic\n    procedures\n    (such as\n    add-complex)\n    that combine packages designed for different representations (such as\n    rectangular and polar).  Moreover, the structure of the system is additive,\n    so that one can design the individual arithmetic packages separately and\n    combine them to produce a generic arithmetic system.\n    ","2.5#fig-":"","2.5.1":"2.5.1  \n    Generic Arithmetic Operations","2.5.1#p1":"\n    The task of designing generic arithmetic operations is analogous to that of\n    designing the generic complex-number operations.  We would like, for\n    instance, to have a generic addition\n    procedureadd that acts like ordinary primitive addition\n    + on ordinary numbers, like\n    add-rat\n    on rational numbers, and like\n    add-complex\n    on complex numbers.  We can implement add, and\n    the other generic arithmetic operations, by following the same strategy we\n    used in section 2.4.3 to implement the\n    generic selectors for complex numbers.  We will attach a type tag to each\n    kind of number and cause the generic\n    procedure\n    to dispatch to an appropriate package according to the data type of its\n    arguments.\n  ","2.5.1#p2":"\n    The generic arithmetic\n    procedures\n    are defined as follows:\n    (define (add x y) (apply-generic 'add x y))\n(define (sub x y) (apply-generic 'sub x y))\n(define (mul x y) (apply-generic 'mul x y))\n(define (div x y) (apply-generic 'div x y))","2.5.1#p3":"\n    We begin\n    by installing a package for handling\n    ordinary numbers,\n    that is, the primitive numbers of our language.  We\n    \n\twill\n      \n    tag these\n    with the \n    symbol scheme-number.\n    The arithmetic operations in this package are the primitive arithmetic\n    procedures\n    (so there is no need to define extra\n    procedures\n    to handle the untagged numbers).  Since these operations each take two\n    arguments, they are installed in the table keyed by the list\n    (scheme-number scheme-number):(define (install-scheme-number-package)\n  (define (tag x)\n    (attach-tag 'scheme-number x))    \n  (put 'add '(scheme-number scheme-number)\n    (lambda (x y) (tag (+ x y))))\n  (put 'sub '(scheme-number scheme-number)\n    (lambda (x y) (tag (- x y))))\n  (put 'mul '(scheme-number scheme-number)\n    (lambda (x y) (tag (* x y))))\n  (put 'div '(scheme-number scheme-number)\n    (lambda (x y) (tag (/ x y))))\n  (put 'make 'scheme-number\n    (lambda (x) (tag x)))\n  'done) ","2.5.1#p4":"\n    Users of the\n    Scheme-number package\n    will create (tagged) ordinary numbers by means of the\n    procedure:(define (make-scheme-number n)\n  ((get 'make 'scheme-number) n)) ","2.5.1#p5":"\n    Now that the framework of the generic arithmetic system is in place,\n    we can readily include new kinds of numbers.  Here is a package that\n    performs rational arithmetic.  Notice that, as a benefit of\n    additivity, we can use without modification the rational-number code\n    from section 2.1.1 as the internal\n    procedures\n    in the package:\n    (define (install-rational-package)\n  ;; internal procedures\n  (define (numer x) (car x))\n  (define (denom x) (cdr x))\n  (define (make-rat n d)\n    (let ((g (gcd n d)))\n      (cons (/ n g) (/ d g))))\n  (define (add-rat x y)\n    (make-rat (+ (* (numer x) (denom y))\n                 (* (numer y) (denom x)))\n              (* (denom x) (denom y))))\n  (define (sub-rat x y)\n    (make-rat (- (* (numer x) (denom y))\n                 (* (numer y) (denom x)))\n              (* (denom x) (denom y))))\n  (define (mul-rat x y)\n    (make-rat (* (numer x) (numer y))\n              (* (denom x) (denom y))))\n  (define (div-rat x y)\n    (make-rat (* (numer x) (denom y))\n              (* (denom x) (numer y))))\n\n  ;; interface to rest of the system\n  (define (tag x) (attach-tag 'rational x))\n  (put 'add '(rational rational)\n    (lambda (x y) (tag (add-rat x y))))\n  (put 'sub '(rational rational)\n    (lambda (x y) (tag (sub-rat x y))))\n  (put 'mul '(rational rational)\n    (lambda (x y) (tag (mul-rat x y))))\n  (put 'div '(rational rational)\n    (lambda (x y) (tag (div-rat x y))))\n\n  (put 'make 'rational\n    (lambda (n d) (tag (make-rat n d))))\n  'done)\n\n(define (make-rational n d)\n  ((get 'make 'rational) n d)) ","2.5.1#p6":"\n    We can install a similar package to handle complex numbers, using the tag\n    complex.\n    In creating the package, we extract from the table the operations\n    make-from-real-imag\n    and\n    make-from-mag-ang\n    that were defined by the rectangular and polar packages.  \n    \n    Additivity permits us to use, as the internal operations, the same\n    add-complex,sub-complex,mul-complex,\n    and\n    div-complexprocedures\n    from section 2.4.1.\n    (define (install-complex-package)\n  ;; imported procedures from rectangular and polar packages\n  (define (make-from-real-imag x y)\n    ((get 'make-from-real-imag 'rectangular) x y))\n  (define (make-from-mag-ang r a)\n    ((get 'make-from-mag-ang 'polar) r a))\n\n  ;; internal procedures\n  (define (add-complex z1 z2)\n    (make-from-real-imag (+ (real-part z1) (real-part z2))\n                         (+ (imag-part z1) (imag-part z2))))\n  (define (sub-complex z1 z2)\n    (make-from-real-imag (- (real-part z1) (real-part z2))\n                         (- (imag-part z1) (imag-part z2))))\n  (define (mul-complex z1 z2)\n    (make-from-mag-ang (* (magnitude z1) (magnitude z2))\n                       (+ (angle z1) (angle z2))))\n  (define (div-complex z1 z2)\n    (make-from-mag-ang (/ (magnitude z1) (magnitude z2))\n                       (- (angle z1) (angle z2))))\n\n  ;; interface to rest of the system\n  (define (tag z) (attach-tag 'complex z))\n  (put 'add '(complex complex)\n    (lambda (z1 z2) (tag (add-complex z1 z2))))\n  (put 'sub '(complex complex)\n    (lambda (z1 z2) (tag (sub-complex z1 z2))))\n  (put 'mul '(complex complex)\n    (lambda (z1 z2) (tag (mul-complex z1 z2))))\n  (put 'div '(complex complex)\n    (lambda (z1 z2) (tag (div-complex z1 z2))))\n  (put 'make-from-real-imag 'complex\n    (lambda (x y) (tag (make-from-real-imag x y))))\n  (put 'make-from-mag-ang 'complex\n    (lambda (r a) (tag (make-from-mag-ang r a))))\n  'done) ","2.5.1#p7":"\n    Programs outside the complex-number package can construct complex\n    numbers either from real and imaginary parts or from magnitudes and\n    angles.  Notice how the underlying\n    procedures,\n    originally defined in the rectangular and polar packages, are exported to\n    the complex package, and exported from there to the outside world.\n    (define (make-complex-from-real-imag x y)\n  ((get 'make-from-real-imag 'complex) x y))\n\n(define (make-complex-from-mag-ang r a)\n  ((get 'make-from-mag-ang 'complex) r a)) ","2.5.1#p8":"\n    What we have here is a\n    \n    two-level tag system.  A typical complex number,\n    such as $3+4i$ in rectangular form, would be\n    represented as shown in\n    \n\tfigure .\n      \n      The outer tag\n    (complex)\n    is used to direct the number to the complex package.  Once within the\n    complex package, the next tag\n    (rectangular)\n    is used to direct the number to the rectangular package. In a large and\n    complicated system there might be many levels, each interfaced with the\n    next by means of generic operations.  As a data object is passed\n    \"downward,\" the outer tag that is used to direct it to the\n    appropriate package is stripped off (by applying\n    contents) and the next level of tag (if any)\n    becomes visible to be used for further dispatching.\n    ","2.5.1#fig-":"","2.5.1#p9":"\n    In the above packages, we used\n    add-rat,add-complex,\n    and the other arithmetic\n    procedures\n    exactly as originally written. Once these declarations are internal to\n    different installation\n    procedures,\n    however, they no longer need names that are distinct from each other:\n    we could simply name them add,\n    sub, mul, and\n    div in both packages.\n  ","2.5.1#ex-2.77":" \n    Louis Reasoner tries to evaluate the expression\n    (magnitude z)\n    where\n    z is the object shown in\n    \n\tfigure .\n      \n    To his surprise, instead of the answer $5$\n    he gets an error message from\n    apply-generic,\n    saying there is no method for the operation\n    magnitude on the types\n    (complex).\n    He shows this interaction to Alyssa P. Hacker, who says \"The problem\n    is that the complex-number selectors were never defined for\n    complex\n    numbers, just for\n    polar\n    and\n    rectangular\n    numbers.  All you have to do to make this work is add the following to the\n    complex package:\"(put 'real-part '(complex) real-part)\n(put 'imag-part '(complex) imag-part)\n(put 'magnitude '(complex) magnitude)\n(put 'angle '(complex) angle) \n    Describe in detail why this works.  As an example, trace through all the\n    procedures\n    called in evaluating the expression\n    (magnitude z)\n    where z is the object shown in\n    \n\tfigure .\n      \n    In particular, how many times is\n    apply-generic\n    invoked?  What\n    procedure\n    is dispatched to in each case?\n    ","2.5.1#ex-2.78":" \n    The internal\n    procedures\n    in the\n    scheme-number\n    package are essentially nothing more than calls to the primitive\n    procedures+, -, etc.  It\n    was not possible to use the primitives of the language directly because our\n    type-tag system requires that each data object have a type attached to it.\n    In fact, however, all\n    Lisp\n    implementations do have a type system, which they use internally. Primitive\n    predicates such as\n    symbol?\n    and\n    number?\n    determine whether data objects have particular types.  Modify the\n    definitions of\n    type-tag,contents, and\n    attach-tag\n    from section 2.4.2 so that our generic\n    system takes advantage of\n    Scheme's\n    internal type system.  That is to say, the system should work as before\n    except that ordinary numbers should be represented simply as\n    Scheme\n    numbers rather than as pairs whose\n    car\n    is the\n    symbol scheme-number.","2.5.1#ex-2.79":" \n    Define a generic equality predicate\n    equ?\n    that tests the equality of two numbers, and install it in the generic\n    arithmetic package.  This operation should work for ordinary numbers,\n    rational numbers, and complex numbers.\n    ","2.5.1#ex-2.80":"\n    Define a generic predicate\n    =zero?\n    that tests if its argument is zero, and install it in the generic\n    arithmetic package.  This operation should work for ordinary numbers,\n    rational numbers, and complex numbers.\n    ","2.5.2":"2.5.2  \n    Combining Data of Different Types","2.5.2#p1":"\n    We have seen how to define a unified arithmetic system that\n    encompasses ordinary numbers, complex numbers, rational numbers, and\n    any other type of number we might decide to invent, but we have\n    ignored an important issue.  The operations we have defined so far\n    treat the different data types as being completely independent.  Thus,\n    there are separate packages for adding, say, two ordinary numbers, or\n    two complex numbers.  What we have not yet considered is the fact that\n    it is meaningful to define operations that cross the type boundaries,\n    such as the addition of a complex number to an ordinary number.  We\n    have gone to great pains to introduce barriers between parts of our\n    programs so that they can be developed and understood separately.  We\n    would like to introduce the cross-type operations in some carefully\n    controlled way, so that we can support them\n    without seriously violating our module boundaries.\n  ","2.5.2#p2":"\n    One way to handle\n    \n    cross-type operations is to design a different\n    procedure\n    for each possible combination of types for which the operation is valid.\n    For example, we could extend the complex-number package so that it\n    provides a\n    procedure\n    for adding complex numbers to ordinary numbers and installs this in the\n    table using the tag\n    (complex scheme-number):;; to be included in the complex package\n(define (add-complex-to-schemenum z x)\n  (make-from-real-imag (+ (real-part z) x)\n                       (imag-part z)))\n\n(put 'add '(complex scheme-number)\n     (lambda (z x) (tag (add-complex-to-schemenum z x))))","2.5.2#footnote-link-1":"1","2.5.2#p3":"\n    This technique works, but it is cumbersome.  With such a system, the\n    cost of introducing a new type is not just the construction of the\n    package of\n    procedures\n    for that type but also the construction and installation of the\n    procedures\n    that implement the cross-type operations.  This can easily be much more\n    code than is needed to define the operations on the type itself.  The\n    method also undermines our ability to combine separate packages additively,\n    or least to limit the extent to which the implementors of the individual\n    packages need to take account of other packages.  For instance, in the\n    example above, it seems reasonable that handling mixed operations on\n    complex numbers and ordinary numbers should be the responsibility of\n    the complex-number package.  Combining rational numbers and complex\n    numbers, however, might be done by the complex package, by the rational\n    package, or by some third package that uses operations extracted from\n    these two packages.  Formulating coherent policies on the division of\n    responsibility among packages can be an overwhelming task in designing\n    systems with many packages and many cross-type operations.\n  ","2.5.2#h1":"Coercion","2.5.2#p4":"\n    In the general situation of completely unrelated operations acting on\n    completely unrelated types, implementing explicit cross-type operations,\n    cumbersome though it may be, is the best that one can hope for.\n    Fortunately, we can usually do better by taking advantage of additional\n    structure that may be latent in our type system.  Often the different\n    data types are not completely independent, and there may be ways by which\n    objects of one type may be viewed as being of another type.  This process\n    is called coercion.  For example, if we are asked to\n    arithmetically combine an ordinary number with a complex number, we can\n    view the ordinary number as a complex number whose imaginary part is zero.\n    This transforms the problem to that of combining two complex numbers, which\n    can be handled in the ordinary way by the complex-arithmetic package.\n  ","2.5.2#p5":"\n    In general, we can implement this idea by designing\n    \n    coercion\n    procedures\n    that transform an object of one type into an equivalent\n    object of another type.  Here is a typical coercion\n    procedure,\n    which transforms a given ordinary number to a complex number with that real\n    part and zero imaginary part:\n    (define (scheme-number->complex n)\n  (make-complex-from-real-imag (contents n) 0))\n    We install these coercion\n    procedures\n    in a special coercion table, indexed under the names of the two types:\n    (put-coercion 'scheme-number 'complex scheme-number->complex) \n    (We assume that there are\n    put-coercion\n    and\n    get-coercionprocedures\n    available for manipulating this table.)  Generally some of the slots in\n    the table will be empty, because it is not generally possible to coerce\n    an arbitrary data object of each type into all other types.  For example,\n    there is no way to coerce an arbitrary complex number to an ordinary\n    number, so there will be no general\n    complex->scheme-numberprocedure\n    included in the table.\n  ","2.5.2#p6":"\n    Once the coercion table has been set up, we can handle coercion in a\n    uniform manner by modifying the\n    apply-genericprocedure\n    of section 2.4.3.  When asked to apply an\n    operation, we first check whether the operation is defined for the\n    arguments' types, just as before.  If so, we dispatch to the\n    procedure\n    found in the operation-and-type table. Otherwise, we try coercion.  For\n    simplicity, we consider only the case where there are two\n    arguments.  We check the coercion table to see if objects\n    of the first type can be coerced to the second type.  If so, we coerce the\n    first argument and try the operation again.  If objects of the first type\n    cannot in general be coerced to the second type, we try the coercion the\n    other way around to see if there is a way to coerce the second argument to\n    the type of the first argument. Finally, if there is no known way to coerce\n    either type to the other type, we give up. Here is the\n    procedure:(define (apply-generic op . args)\n  (let ((type-tags (map type-tag args)))\n    (let ((proc (get op type-tags)))\n      (if proc\n        (apply proc (map contents args))\n        (if (= (length args) 2)\n          (let ((type1 (car type-tags))\n                 (type2 (cadr type-tags))\n                 (a1 (car args))\n                 (a2 (cadr args)))\n            (let ((t1->t2 (get-coercion type1 type2))\n                  (t2->t1 (get-coercion type2 type1)))\n              (cond (t1->t2\n                      (apply-generic op (t1->t2 a1) a2))\n                    (t2->t1\n                      (apply-generic op a1 (t2->t1 a2)))\n                    (else\n                      (error \"No method for these types\"\n                        (list op type-tags))))))\n          (error \"No method for these types\"\n            (list op type-tags))))))) ","2.5.2#footnote-link-2":"2","2.5.2#p7":"\n    This coercion scheme has many advantages over the method of defining\n    explicit cross-type operations, as outlined above.  Although we still\n    need to write coercion\n    procedures\n    to relate the types (possibly $n^2$procedures\n    for a system with $n$ types), we need to write\n    only one\n    procedure\n    for each pair of types rather than a different\n    procedure\n    for each collection of types and each generic operation.  What we are counting on here is the fact that the\n    appropriate transformation between types depends only on the types\n    themselves, not on the operation to be applied.\n  ","2.5.2#footnote-link-3":"3","2.5.2#p8":"\n    On the other hand, there may be applications for which our coercion\n    scheme is not general enough.  Even when neither of the objects to be\n    combined can be converted to the type of the other it may still be\n    possible to perform the operation by converting both objects to a\n    third type.  In order to deal with such complexity and still preserve\n    modularity in our programs, it is usually necessary to build systems\n    that take advantage of still further structure in the relations among\n    types, as we discuss next.\n  ","2.5.2#h2":"Hierarchies of types","2.5.2#p9":"\n    The coercion scheme presented above relied on the existence of natural\n    relations between pairs of types.  Often there is more \"global\"\n    structure in how the different types relate to each other.  For\n    instance, suppose we are building a generic arithmetic system to\n    handle integers, rational numbers, real numbers, and complex numbers.\n    In such a system, it is quite natural to regard an integer as a\n    special kind of rational number, which is in turn a special kind of\n    real number, which is in turn a special kind of complex number.  What\n    we actually have is a so-called hierarchy of types, in which,\n    for example, integers are a\n    subtype of rational numbers (i.e.,\n    any operation that can be applied to a rational number can\n    automatically be applied to an integer).  Conversely, we say that\n    rational numbers form a\n    supertype of integers.  The particular\n    hierarchy we have here is of a very simple kind, in which each type\n    has at most one supertype and at most one subtype.  Such a structure,\n    called a tower, is illustrated in\n    figure 2.25.\n    ","2.5.2#fig-2.25":"","2.5.2#p10":"\n    If we have a tower structure, then we can greatly simplify the problem\n    of adding a new type to the hierarchy, for we need only specify how\n    the new type is embedded in the next supertype above it and how it is\n    the supertype of the type below it.  For example, if we want to add an\n    integer to a complex number, we need not explicitly define a special\n    coercion\n    procedureinteger->complex.\n    Instead, we define how an integer can be transformed into a rational\n    number, how a rational number is transformed into a real number, and how\n    a real number is transformed into a complex number.  We then allow the\n    system to transform the integer into a complex number through these steps\n    and then add the two complex numbers.\n  ","2.5.2#p11":"\n    We can redesign our\n    apply-genericprocedure\n    in the following way: For each type, we need to supply a\n    raiseprocedure,\n    which \"raises\" objects of that type one level in the tower.\n    Then when the system is required to operate on objects of different types\n    it can successively raise the lower types until all the objects are at\n    the same level in the tower.  (Exercises 2.83\n    and  2.84\n    concern the details of implementing such a strategy.)\n  ","2.5.2#p12":"\n    Another advantage of a tower is that we can easily implement the notion\n    that every type \"inherits\" all operations defined on a\n    supertype.  For instance, if we do not supply a special\n    procedure\n    for finding the real part of an integer, we should nevertheless expect\n    that\n    real-part\n    will be defined for integers by virtue of the fact that integers are a\n    subtype of complex numbers.  In a tower, we can arrange for this to happen\n    in a uniform way by modifying\n    apply-generic.\n    If the required operation is not directly defined for the type of the\n    object given, we raise the object to its supertype and try again.  We thus\n    crawl up the tower, transforming our argument as we go, until we either\n    find a level at which the desired operation can be performed or hit the\n    top (in which case we give up).\n  ","2.5.2#p13":"\n    Yet another advantage of a tower over a more general hierarchy is that\n    it gives us a simple way to \"lower\" a data object to the\n    simplest representation.  For example, if we add\n    $2+3i$ to $4-3i$,\n    it would be nice to obtain the answer as the integer 6 rather than as the\n    complex number $6+0i$.\n    Exercise 2.85 discusses a way to implement\n    such a lowering operation.  (The trick is that we need a general way\n    to distinguish those objects that can be lowered, such as\n    $6+0i$, from those that cannot, such as\n    $6+2i$.)\n  ","2.5.2#h3":"Inadequacies of hierarchies","2.5.2#p14":"\n    If the data types in our system can be naturally arranged in a tower,\n    this greatly simplifies the problems of dealing with generic operations\n    on different types, as we have seen.  Unfortunately, this is usually\n    not the case.  Figure 2.26\n    illustrates a more complex arrangement of mixed types, this one showing\n    relations among different types of geometric figures.  We see that, in\n    general,\n    \n    a type may have more than one subtype.  Triangles and quadrilaterals,\n    for instance, are both subtypes of polygons.  In addition, a type may\n    have more than one supertype.  For example, an isosceles right\n    triangle may be regarded either as an isosceles triangle or as a right\n    triangle.  This multiple-supertypes issue is particularly thorny,\n    since it means that there is no unique way to \"raise\" a type\n    in the hierarchy.  Finding the \"correct\" supertype in which\n    to apply an operation to an object may involve considerable searching\n    through the entire type network on the part of a\n    procedure\n    such as\n    apply-generic.\n    Since there generally are multiple subtypes for a type, there is a similar\n    problem in coercing a value \"down\" the type hierarchy.\n    Dealing with large numbers of interrelated types while still preserving\n    modularity in the design of large systems is very difficult, and is an area\n    of much current research.","2.5.2#footnote-link-4":"4","2.5.2#fig-2.26":"","2.5.2#ex-2.81":"\n    Louis Reasoner has noticed that\n    apply-generic\n    may try to coerce the arguments to each other's type even if they\n    already have the same type.  Therefore, he reasons, we need to put\n    procedures\n    in the coercion table to \"coerce\" arguments of each type to\n    their own type.  For example, in addition to the\n    scheme-number->complex\n    coercion shown above, he would do:\n    (define (scheme-number->scheme-number n) n)\n(define (complex->complex z) z)\n(put-coercion 'scheme-number 'scheme-number\n  scheme-number->scheme-number)\n(put-coercion 'complex 'complex complex->complex)\n        With Louis's coercion\n        procedures\n        installed, what happens if\n\tapply-generic\n        is called with two arguments of type\n\tscheme-number\n\tor two arguments of\n        type\n\tcomplex\n\tfor an operation that is not found in the table for those types?\n\tFor example, assume that we've defined a generic exponentiation\n        operation:\n        (define (exp x y) (apply-generic 'exp x y))\n        and have put a\n        procedure\n        for exponentiation in the\n\tScheme-number\n        package but not in any other package:\n        \n;; following added to Scheme-number package\n(put 'exp '(scheme-number scheme-number)\n  (lambda (x y) (tag (expt x y)))) ; using primitive expt\n          \n        What happens if we call exp with two\n\tcomplex numbers as arguments?\n      \n        Is Louis correct that something had to be done about coercion with\n\targuments of the same type, or does\n\tapply-generic\n        work correctly as is?\n      \n        Modify\n\tapply-generic\n\tso that it doesn't try coercion if the two arguments have the\n\tsame type.\n      ","2.5.2#ex-2.82":"\n    Show how to generalize\n    apply-generic\n    to handle coercion in the general case of multiple arguments.  One\n    strategy is to attempt to coerce all the arguments to the type of the\n    first argument, then to the type of the second argument, and so on.\n    Give an example of a situation where this strategy (and likewise the\n    two-argument version given above) is not sufficiently general.\n    (Hint: Consider the case where there are some suitable mixed-type\n    operations present in the table that will not be tried.)\n    ","2.5.2#ex-2.83":"\n    Suppose you are designing a generic arithmetic system for dealing with\n    the tower of types shown in figure 2.25:\n    integer, rational, real, complex.  For\n    each type (except complex), design a\n    procedure\n    that raises objects of that type one level in the tower.  Show how to\n    install a generic raise operation that will\n    work for each type (except complex).\n    ","2.5.2#ex-2.84":"\n    Using the raise operation of\n    exercise 2.83, modify the\n    apply-genericprocedure\n    so that it coerces its arguments to have the same type by the method of\n    successive raising, as discussed in this section.  You will need to devise\n    a way to test which of two types is higher in the tower.  Do this in a\n    manner that is \"compatible\" with the rest of the system and\n    will not lead to problems in adding new levels to the tower.\n    ","2.5.2#ex-2.85":"\n    This section mentioned a method for \"simplifying\" a data object\n    by lowering it in the tower of types as far as possible.  Design a\n    proceduredrop that accomplishes this for the tower\n    described in exercise 2.83.  The key is to decide,\n    in some general way, whether an object can be lowered.  For example, the\n    complex number $1.5+0i$ can be lowered as far as\n    real,\n    the complex number $1+0i$ can be lowered as far\n    as\n    integer,\n    and the complex number $2+3i$ cannot be lowered\n    at all.  Here is a plan for determining whether an object can be lowered:\n    Begin by defining a generic operation project\n    that \"pushes\" an object down in the tower.  For example,\n    projecting a complex number would involve throwing away the imaginary part.\n    Then a number can be dropped if, when we\n    project it and\n    raise the result back to the type we started\n    with, we end up with something equal to what we started with.  Show how to\n    implement this idea in detail, by writing a\n    dropprocedure\n    that drops an object as far as possible.  You will need to design the\n    various projection operations and install\n    project as a generic operation in the system.\n    You will also need to make use of a generic equality predicate, such as\n    described in exercise 2.79.  Finally, use\n    drop\n    to rewrite\n    apply-generic\n    from exercise 2.84 so that it\n    \"simplifies\" its answers.\n    ","2.5.2#footnote-link-5":"5","2.5.2#ex-2.86":"\n    Suppose we want to handle complex numbers whose real\n    parts, imaginary parts, magnitudes, and angles can be either ordinary\n    numbers, rational numbers, or other numbers we might wish to add to\n    the system.  Describe and implement the changes to the system needed\n    to accommodate this.  You will have to define operations such as\n    sine and cosine\n    that are generic over ordinary numbers and rational numbers.\n    ","2.5.2#footnote-1":"We\n    also have to supply an almost identical\n    procedure\n    to handle the types(scheme_number complex).","2.5.2#footnote-2":"See exercise 2.82 for\n    generalizations.","2.5.2#footnote-3":"If we are\n    clever, we can usually get by with fewer than\n    $n^2$ coercion\n    procedures.\n    For instance, if we know how to convert from type 1 to type 2 and from\n    type 2 to type 3, then we can use this knowledge to convert from type 1 to\n    type 3.  This can greatly decrease the number of coercion\n    procedures\n    we need to supply explicitly when we add a new type to the system.  If we\n    are willing to build the required amount of sophistication into our system,\n    we can have it search the \"graph\" of relations among types and\n    automatically generate those coercion\n    procedures\n    that can be inferred from the ones that are supplied\n    explicitly.","2.5.2#footnote-4":"This statement, which also appears in\n    the first edition of this book, is just as true now as it was when we wrote\n    it\n    twelve years ago.\n    Developing a useful, general framework for expressing\n    the relations among different types of entities (what philosophers call\n    \"ontology\") seems intractably difficult.  The main difference\n    between the confusion that existed\n    ten years ago\n    and the confusion that\n    exists now is that now a variety of inadequate ontological theories have\n    been embodied in a plethora of correspondingly inadequate programming\n    languages.  For example, much of the complexity of\n    \n    object-oriented programming languages—and the subtle and confusing\n    differences among contemporary object-oriented\n    languages—centers on the treatment of generic operations on\n    interrelated types.  Our own discussion of computational objects in\n    chapter 3 avoids these issues entirely.  Readers familiar with\n    object-oriented programming will notice that we have much to say in\n    chapter 3 about local state, but we do not even mention\n    \"classes\" or \"inheritance.\"  In fact, we suspect\n    that these problems cannot be adequately addressed in terms of\n    computer-language design alone, without also drawing on work in knowledge\n    representation and automated reasoning.","2.5.2#footnote-5":"A real number can be projected to\n    an integer using the\n    round\n    primitive, which returns the closest integer\n    to its argument.","2.5.3":"2.5.3  \n    Example: Symbolic Algebra","2.5.3#p1":"\n    The manipulation of symbolic algebraic expressions is a complex\n    process that illustrates many of the hardest problems that occur in\n    the design of large-scale systems.  An\n    \n    algebraic expression, in\n    general, can be viewed as a hierarchical structure, a tree of\n    operators applied to operands.  We can construct algebraic expressions\n    by starting with a set of primitive objects, such as constants and\n    variables, and combining these by means of algebraic operators, such\n    as addition and multiplication.  As in other languages, we form\n    abstractions that enable us to refer to compound objects in simple\n    terms.  Typical abstractions in symbolic algebra are ideas such as\n    linear combination, polynomial, rational function, or trigonometric\n    function.  We can regard these as compound \"types,\" which are\n    often useful for directing the processing of expressions.  For example, we\n    could describe the expression\n\n    \n      \\[ x^{2}\\, \\sin (y^2+1)+x\\, \\cos 2y+\\cos (y^3 -2y^2) \\]\n    \n    as a polynomial in $x$ with coefficients that\n    are trigonometric functions of polynomials in\n    $y$ whose coefficients are integers.\n  ","2.5.3#p2":"\n    We will not attempt to develop a complete algebraic-manipulation\n    system here.  Such systems are exceedingly complex programs, embodying\n    deep algebraic knowledge and elegant algorithms.  What we will do is\n    look at a simple but important part of algebraic manipulation: the\n    arithmetic of polynomials.  We will illustrate the kinds of decisions\n    the designer of such a system faces, and how to apply the ideas of\n    abstract data and generic operations to help organize this effort.\n  ","2.5.3#h1":"Arithmetic on polynomials","2.5.3#p3":"\n    Our first task in designing a system for performing arithmetic on\n    polynomials is to decide just what a polynomial is.  Polynomials are\n    normally defined relative to certain variables (the \n    indeterminates of the polynomial).  For simplicity, we will\n      restrict ourselves to polynomials having just one indeterminate \n      (univariate polynomials). We will define a polynomial to\n      be a sum of terms, each of which is either a coefficient, a power of the\n      indeterminate, or a product of a coefficient and a power of the\n      indeterminate.  A coefficient is defined as an algebraic expression\n      that is not dependent upon the indeterminate of the polynomial.  For\n      example,\n      \n        \\[ 5x^2 +3x +7 \\]\n      \n      is a simple polynomial in $x$, and\n      \n        \\[ (y^2 +1)x^3 +(2y)x+1 \\]\n      \n      is a polynomial in $x$ whose coefficients are\n      polynomials in $y$.\n  ","2.5.3#footnote-link-1":"1","2.5.3#p4":"\n    Already we are skirting some thorny issues.  Is the first of these\n    polynomials the same as the polynomial\n    $5y^2 +3y +7$, or not?  A reasonable answer\n    might be \"yes, if we are considering a polynomial purely as a\n    mathematical function, but no, if we are considering a polynomial to be a\n    syntactic form.\"  The second polynomial is algebraically equivalent\n    to a polynomial in $y$ whose coefficients are\n    polynomials in $x$.  Should our system recognize\n    this, or not? Furthermore, there are other ways to represent a\n    polynomial—for example, as a product of factors, or (for a\n    univariate polynomial) as the set of roots, or as a listing of the values\n    of the polynomial at a specified set of points.\n    We can finesse these questions by deciding that in our\n    algebraic-manipulation system a \"polynomial\" will be a\n    particular syntactic form, not its underlying mathematical meaning.\n  ","2.5.3#footnote-link-2":"2","2.5.3#p5":"\n    Now we must consider how to go about doing arithmetic on polynomials.\n    In this simple system, we will consider only addition and\n    multiplication.  Moreover, we will insist that two polynomials to be\n    combined must have the same indeterminate.\n  ","2.5.3#p6":"\n    We will approach the design of our system by following the familiar\n    discipline of data abstraction.  We will represent polynomials using a\n    data structure called a \n    poly, which consists of a variable and a\n    \n    collection of terms.  We assume that we have selectors\n    variable and\n    term-list\n    that extract those parts from a poly and a constructor\n    make-poly\n    that assembles a poly from a given variable and a term list.\n    A variable will be just a\n    \n\tsymbol,\n      \n    so we can use the \n     same-variable?procedure\n    of section 2.3.2 to compare\n    variables.\n    The following\n    procedures\n    define\n    \n    addition and multiplication of polys:\n    (define (add-poly p1 p2)\n  (if (same-variable? (variable p1) (variable p2))\n    (make-poly (variable p1)\n               (add-terms (term-list p1)\n                          (term-list p2)))\n  (error \"Polys not in same var - - ADD-POLY\"\n         (list p1 p2))))\n\n(define (mul-poly p1 p2)\n  (if (same-variable? (variable p1) (variable p2))\n    (make-poly (variable p1)\n               (mul-terms (term-list p1)\n                          (term-list p2)))\n    (error \"Polys not in same var - - MUL-POLY\"\n           (list p1 p2)))) ","2.5.3#p7":"\n    To incorporate polynomials into our generic arithmetic system, we need\n    to supply them with type tags.  We'll use the tag\n    polynomial,\n    and install appropriate operations on tagged polynomials in the operation\n    table.\n    \n\tWe'll embed all our code in an installation procedure\n\tfor the polynomial package, similar to the ones in\n\tsection 2.5.1:\n\t\n(define (install-polynomial-package)\n  ;; internal procedures\n  ;; representation of poly\n  (define (make-poly variable term-list)\n    (cons variable term-list))\n  (define (variable p) (car p))\n  (define (term-list p) (cdr p))\n  $langle$ procedures same-variable? and variable? from section 2.3.2 $\\rangle$\n  ;; representation of terms and term lists\n  $langle$ procedures adjoin-term $\\ldots$ coeff from text below $langle$\n  (define (add-poly p1 p2) $\\ldots$)\n  $langle$ procedures used by add-poly $langle$\n  (define (mul-poly p1 p2) $\\ldots$)\n  $langle$procedures used by mul-poly $langle$\n  ;; interface to rest of the system\n  (define (tag p) (attach-tag 'polynomial p))\n  (put 'add '(polynomial polynomial) \n       (lambda (p1 p2) (tag (add-poly p1 p2))))\n  (put 'mul '(polynomial polynomial) \n       (lambda (p1 p2) (tag (mul-poly p1 p2))))\n  (put 'make 'polynomial\n       (lambda (var terms) (tag (make-poly var terms))))\n  'done)\n\t  ","2.5.3#p8":"\n    Polynomial addition is performed termwise.  Terms of the same order\n    (i.e., with the same power of the indeterminate) must be combined.\n    This is done by forming a new term of the same order whose coefficient\n    is the sum of the coefficients of the addends.  Terms in one addend\n    for which there are no terms of the same order in the other addend are\n    simply accumulated into the sum polynomial being constructed.\n  ","2.5.3#p9":"\n    In order to manipulate term lists, we will assume that we have a\n    constructor \n      the-empty-termlist\n    that returns an empty term list and a constructor \n      adjoin-term\n    that adjoins a new term to a term list.  We will also assume that we have\n    a predicate \n      empty-termlist?\n    that tells if a given term list is empty, a selector\n      first-term\n    that extracts the highest-order term from a term list, and a selector \n      rest-terms\n    that returns all but the highest-order term.  To manipulate terms,\n    we will suppose that we have a constructor \n      make-term\n    that constructs a term with given order and coefficient, and selectors \n      order and \n      coeff that return, respectively, the order\n    and the coefficient of the term.  These operations allow us to consider\n    both terms and term lists as data abstractions, whose concrete\n    representations we can worry about separately.\n  ","2.5.3#p10":"\n    Here is the\n    procedure\n    that constructs the term list for the sum of two\n    polynomials:(define (add-terms L1 L2)\n          (cond ((empty-termlist? L1) L2)\n          ((empty-termlist? L2) L1)\n          (else\n          (let ((t1 (first-term L1)) (t2 (first-term L2)))\n          (cond ((> (order t1) (order t2))\n          (adjoin-term\n          t1 (add-terms (rest-terms L1) L2)))\n          ((< (order t1) (order t2))\n          (adjoin-term\n          t2 (add-terms L1 (rest-terms L2))))\n          (else\n          (adjoin-term\n          (make-term (order t1)\n          (add (coeff t1) (coeff t2)))\n          (add-terms (rest-terms L1)\n          (rest-terms L2))))))))) \n    The most important point to note here is that we used the generic addition\n    procedureadd to add together the coefficients of the\n    terms being combined.  This has powerful consequences, as we will see below.\n  ","2.5.3#footnote-link-3":"3","2.5.3#p11":"\n    In order to multiply two term lists, we multiply each term of the first\n    list by all the terms of the other list, repeatedly using\n    mul-term-by-all-terms,\n    which multiplies a given term by all terms in a given term list.  The\n    resulting term lists (one for each term of the first list) are accumulated\n    into a sum.  Multiplying two terms forms a term whose order is the sum of\n    the orders of the factors and whose coefficient is the product of the\n    coefficients of the factors:\n    (define (mul-terms L1 L2)\n          (if (empty-termlist? L1)\n          (the-empty-termlist)\n          (add-terms (mul-term-by-all-terms (first-term L1) L2)\n          (mul-terms (rest-terms L1) L2))))\n\n          (define (mul-term-by-all-terms t1 L)\n          (if (empty-termlist? L)\n          (the-empty-termlist)\n          (let ((t2 (first-term L)))\n          (adjoin-term\n          (make-term (+ (order t1) (order t2))\n          (mul (coeff t1) (coeff t2)))\n          (mul-term-by-all-terms t1 (rest-terms L)))))) ","2.5.3#p12":"\n    This is really all there is to polynomial addition and multiplication.\n    Notice that, since we operate on terms using the generic\n    proceduresadd and mul,\n    our polynomial package is automatically able to handle any type of\n    coefficient that is known about by the generic arithmetic package.\n    If we include a \n    \n    coercion mechanism such as one of those discussed in\n    section 2.5.2,\n    then we also are automatically able to handle operations on\n    polynomials of different coefficient types, such as\n    \n      \\[\n      \\begin{array}{l}\n      {\\left[3x^2 +(2+3i)x+7\\right] \\cdot \\left[x^4 +\\frac{2}{3}x^2\n      +(5+3i)\\right]}\n      \\end{array}\n      \\]\n    ","2.5.3#p13":"\n    Because we installed the polynomial addition and multiplication\n    proceduresadd-poly\n    and\n    mul-poly\n    in the generic arithmetic system as the add\n    and mul operations for type\n    polynomial, our system is also automatically\n    able to handle polynomial operations such as\n    \n      \\[\n      \\begin{array}{l}\n      {\\left[ (y+1)x^2 +(y^2 +1)x+(y-1)\\right]\\cdot \\left[(y-2)x+(y^3 +7)\\right]}\n      \\end{array}\n      \\]\n    \n    The reason is that when the system tries to combine coefficients, it\n    will dispatch through add and\n    mul.  Since the coefficients are themselves\n    polynomials (in $y$), these will be combined\n    using\n    add-poly\n    and\n    mul-poly.\n    The result is a kind of \n    \"data-directed recursion\" in which, for example, a call to\n    mul-poly\n    will result in recursive calls to\n    mul-poly\n    in order to multiply the coefficients.  If the coefficients of the\n    coefficients were themselves polynomials (as might be used to represent\n    polynomials in three variables), the data direction would ensure that the\n    system would follow through another level of recursive calls, and so on\n    through as many levels as the structure of the data dictates.","2.5.3#footnote-link-4":"4","2.5.3#h2":"Representing term lists","2.5.3#p14":"\n    Finally, we must confront the job of implementing a good\n    representation for term lists.  A term list is, in effect, a set of\n    coefficients keyed by the order of the term.  Hence, any of the\n    methods for representing sets, as discussed in\n    section 2.3.3, can be applied to this\n    task.  On the other hand, our\n    proceduresadd-terms and mul-terms\n    always access term lists sequentially from highest to lowest order.\n    Thus, we will use some kind of ordered list representation.\n  ","2.5.3#p15":"\n    How should we structure the list that represents a term list?  One\n    consideration is the \"density\" of the polynomials we intend\n    to manipulate.  A polynomial is said to be \n    dense if it has nonzero coefficients in terms of most orders.\n    If it has many zero terms it is said to be \n    sparse.  For example,\n    \n      \\[ A:\\quad x^5 +2x^4 +3x^2 -2x -5 \\]\n    \n    is a dense polynomial, whereas\n    \n      \\[ B:\\quad x^{100} +2x^2 +1 \\]\n    \n    is sparse.\n  ","2.5.3#p16":"\n\tThe term lists of dense polynomials are most efficiently represented\n\tas lists of the coefficients.\n      \n    For example,\n    $A$ above would be nicely represented as\n    (1 2 0 3 -2 -5).\n    The order of a term in this representation is the length of the sublist\n    beginning with that term's coefficient, decremented by 1.  \n    This would be a terrible representation for a sparse polynomial such as\n    $B$: There would be a giant list of zeros\n    punctuated by a few lonely nonzero terms.  A more reasonable representation\n    of the term list of a sparse polynomial is as a list of the nonzero terms,\n    where each term is a list containing the order of the term and the\n    coefficient for that order.  In such a scheme, polynomial\n    $B$ is efficiently represented as\n    ((100 1) (2 2) (0 1)).\n    As most polynomial manipulations are performed on sparse polynomials, we\n    will use this method.  We will assume that term lists are represented as\n    lists of terms, arranged from highest-order to lowest-order term. Once we\n    have made this decision, implementing the selectors and constructors for\n    terms and term lists is straightforward:(define (adjoin-term term term-list)\n          (if (=zero? (coeff term))\n          term-list\n          (cons term term-list)))\n\n          (define (the-empty-termlist) '())\n          (define (first-term term-list) (car term-list))\n          (define (rest-terms term-list) (cdr term-list))\n          (define (empty-termlist? term-list) (null? term-list))\n\n          (define (make-term order coeff) (list order coeff))\n          (define (order term) (car term))\n          (define (coeff term) (cadr term)) \n    where\n    =zero?\n    is as defined in exercise 2.80.  (See also\n    exercise 2.87 below.)\n  ","2.5.3#footnote-link-5":"5","2.5.3#footnote-link-6":"6","2.5.3#p17":"\n    Users of the polynomial package will create (tagged) polynomials by means\n    of the\n    procedure:(define (make-polynomial var terms)\n          ((get 'make 'polynomial) var terms)) ","2.5.3#ex-2.87":"\n    Install\n    =zero?\n    for polynomials in the generic arithmetic package.  This will allow\n    adjoin-term\n    to work for polynomials with coefficients that are themselves polynomials.\n    ","2.5.3#ex-2.88":"\n    Extend the polynomial system to include\n    \n    subtraction of polynomials.\n    (Hint: You may find it helpful to define a generic negation operation.)\n    ","2.5.3#ex-2.89":"Define procedures\n    that implement the term-list representation described above as\n    appropriate for dense polynomials.\n  ","2.5.3#ex-2.90":"\n    Suppose we want to have a polynomial system that is efficient for both\n    sparse and dense polynomials.  One way to do this is to allow both\n    kinds of term-list representations in our system.  The situation is\n    analogous to the complex-number example of\n    section 2.4, where we allowed both\n    rectangular and polar representations. To do this we must distinguish\n    different types of term lists and make the operations on term lists\n    generic.  Redesign the polynomial system to implement this generalization.\n    This is a major effort, not a local change.\n  ","2.5.3#ex-2.91":"\n    A univariate polynomial can be divided by another one to produce a\n    \n    polynomial quotient and a polynomial remainder.  For example,\n    \n      \\[\n      \\begin{array}{lll}\n      \\dfrac{x^5-1}{x^2 -1} & = & x^3 +x,\\ \\text{remainder }x-1\n      \\end{array}\n      \\]\n    \n    Division can be performed via long division.\n    That is, divide the highest-order term of the dividend by\n    the highest-order term of the divisor.  The result is the first term of the\n    quotient.  Next, multiply the result by the divisor, subtract that\n    from the dividend, and produce the rest of the answer by recursively\n    dividing the difference by the divisor.  Stop when the order of the\n    divisor exceeds the order of the dividend and declare the dividend to\n    be the remainder.  Also, if the dividend ever becomes zero, return\n    zero as both quotient and remainder.\n    \n    We can design a\n    div-polyprocedure\n    on the model of\n    add-poly\n    and\n    mul-poly.\n    The\n    procedure\n    checks to see if the two polys have the same variable.  If so,\n    div-poly\n    strips off the variable and passes the problem to\n    div-terms,\n    which performs the division operation on term lists.\n    Div-poly\n    finally reattaches the variable to the result supplied by\n    div-terms.\n    It is convenient to design\n    div-terms\n    to compute both the quotient and the remainder of a division.\n    div-terms\n    can take two term lists as arguments and return a list of the quotient\n    term list and the remainder term list.\n    \n    Complete the following definition of\n    div-terms\n    by filling in the missing\n    \n\texpressions.\n      \n    Use this to implement\n    div-poly,\n    which takes two polys as arguments and returns a list of the quotient and\n    remainder polys.\n    \n(define (div-terms L1 L2)\n  (if (empty-termlist? L1)\n    (list (the-empty-termlist) (the-empty-termlist))\n      (let ((t1 (first-term L1))\n            (t2 (first-term L2)))\n        (if (> (order t2) (order t1))\n          (list (the-empty-termlist) L1)\n          (let ((new-c (div (coeff t1) (coeff t2)))\n            (new-o (- (order t1) (order t2))))\n    (let ((rest-of-result\n           compute rest of result recursively\n          ))\n      form complete result\n            ))))))\n      ","2.5.3#h3":"Hierarchies of types in symbolic algebra","2.5.3#p18":"\n    Our polynomial system illustrates how objects of one type\n    (polynomials) may in fact be complex objects that have objects of many\n    different types as parts.  This poses no real difficulty in defining\n    generic operations.  We need only install appropriate generic operations\n    for performing the necessary manipulations of the parts of the\n    compound types.  In fact, we saw that polynomials form a kind of\n    \"recursive data abstraction,\" in that parts of a polynomial may\n    themselves be polynomials.  Our generic operations and our\n    data-directed programming style can handle this complication without\n    much trouble.\n  ","2.5.3#p19":"\n    On the other hand, polynomial algebra is a system for which the data\n    types cannot be naturally arranged in a tower.  For instance, it is\n    possible to have polynomials in $x$ whose\n    coefficients are polynomials in $y$.  It is also\n    possible to have polynomials in $y$ whose\n    coefficients are polynomials in $x$.  Neither of\n    these types is \"above\" the other in any natural way, yet it is\n    often necessary to add together elements from each set.  There are several\n    ways to do this.  One possibility is to convert one polynomial to the type\n    of the other by expanding and rearranging terms so that both polynomials\n    have the same principal variable.  One can impose a towerlike structure on\n    this by ordering the variables and thus always converting any polynomial\n    to a \n    \"canonical form\" with the highest-priority variable\n    dominant and the lower-priority variables buried in the coefficients.\n    This strategy works fairly well, except that the conversion may expand\n    a polynomial unnecessarily, making it hard to read and perhaps less\n    efficient to work with.  The tower strategy is certainly not natural\n    for this domain or for any domain where the user can invent new types\n    dynamically using old types in various combining forms, such as\n    trigonometric functions, power series, and integrals.\n  ","2.5.3#p20":"\n    It should not be surprising that controlling \n    \n    coercion is a serious problem in the design of large-scale\n    algebraic-manipulation systems. Much of the complexity of such systems is\n    concerned with relationships among diverse types.  Indeed, it is fair to\n    say that we do not yet completely understand coercion.  In fact, we do not\n    yet completely understand the concept of a data type.  Nevertheless, what\n    we know provides us with powerful structuring and modularity principles to\n    support the design of large systems.\n  ","2.5.3#ex-2.92":"\n    By imposing an ordering on variables, extend the polynomial package so\n    that addition and multiplication of polynomials works for polynomials\n    in different variables.  (This is not easy!)\n    ","2.5.3#h4":"Extended exercise: Rational functions","2.5.3#p21":"\n    We can extend our generic arithmetic system to include rational\n    functions.  These are \"fractions\" whose numerator and\n    denominator are polynomials, such as\n    \n      \\[\n      \\begin{array}{l}\n      \\dfrac{x+1}{x^3 -1}\n      \\end{array}\n      \\]\n    \n    The system should be able to add, subtract, multiply, and divide\n    rational functions, and to perform such computations as\n    \n      \\[\n      \\begin{array}{lll}\n      \\dfrac{x+1}{x^3 -1}+\\dfrac{x}{x^2 -1} & = & \\dfrac{x^3 +2x^2 +3x +1}{x^4 +\n      x^3 -x-1}\n      \\end{array}\n      \\]\n    \n    (Here the sum has been simplified by removing common factors.\n    Ordinary \"cross multiplication\" would have produced a \n    fourth-degree polynomial over a fifth-degree polynomial.)\n  ","2.5.3#p22":"\n    If we modify our rational-arithmetic package so that it uses generic\n    operations, then it will do what we want, except for the problem\n    of reducing fractions to lowest terms.\n  ","2.5.3#ex-2.93":" \n    Modify the rational-arithmetic package to use generic operations, but\n    change\n    make-rat\n    so that it does not attempt to reduce fractions to lowest terms.  Test\n    your system by calling\n    make-rational\n    on two polynomials to produce a rational function\n    (define p1 (make-polynomial 'x '((2 1)(0 1))))\n            (define p2 (make-polynomial 'x '((3 1)(0 1))))\n            (define rf (make-rational p2 p1))\n    Now add rf to itself, using\n    add. You will observe that this addition\n    procedure\n    does not reduce fractions to lowest terms.\n    ","2.5.3#p23":"\n    We can reduce polynomial fractions to lowest terms using the same idea\n    we used with integers: modifying\n    make-rat\n    to divide both the numerator and the denominator by their greatest common\n    divisor. The notion of \n    \"greatest common divisor\" makes sense for polynomials.  In\n    fact, we can compute the GCD of two polynomials using essentially the\n    same Euclid's Algorithm that works for integers.  The\n    integer version is\n    (define (gcd a b)\n          (if (= b 0)\n          a\n          (gcd b (remainder a b))))\n    Using this, we could make the obvious modification to define a GCD\n    operation that works on term lists:\n    (define (gcd-terms a b)\n          (if (empty-termlist? b)\n          a\n          (gcd-terms b (remainder-terms a b))))\n    where\n    remainder-terms\n    picks out the remainder component of the list returned by the term-list\n    division operation\n    div-terms\n    that was implemented in exercise 2.91.\n  ","2.5.3#footnote-link-7":"7","2.5.3#ex-2.94":"\n    Using\n    div-terms,\n    implement the\n    procedureremainder-terms\n    and use this to define\n    gcd-terms\n    as above.  Now write a\n    proceduregcd-poly\n    that computes the polynomial GCD of two polys. (The\n    procedure\n    should signal an error if the two polys are not in the same variable.)\n    Install in the system a generic operation\n    greatest-common-divisor\n    that reduces to\n    gcd-poly\n    for polynomials and to ordinary gcd for\n    ordinary numbers.  As a test, try\n    (define p1 (make-polynomial 'x '((4 1) (3 -1) (2 -2) (1 2))))\n            (define p2 (make-polynomial 'x '((3 1) (1 -1))))\n            (greatest-common-divisor p1 p2)\n    and check your result by hand.\n    ","2.5.3#ex-2.95":" \n    Define $P_{1}$,\n    $P_{2}$, and\n    $P_{3}$ to be the polynomials\n    $P_{1}$:\n\t$x^2 - 2x + 1$$P_{2}$:\n\t$11x^2 + 7$$P_{3}$:\n\t$13x + 5$\n    Now define $Q_1$ to be the product of\n    $P_1$ and $P_2$ and\n    $Q_2$ to be the product of\n    $P_1$ and $P_3$, and\n    use\n    greatest-common-divisor\n    (exercise 2.94) to compute the GCD of\n    $Q_1$ and $Q_2$.\n    Note that the answer is not the same as $P_1$.\n    This example introduces noninteger operations into the computation, causing\n    difficulties with the GCD\n    \n\talgorithm.\n    To understand what is happening, try tracing\n    gcd-terms\n    while computing the GCD or try performing the division by hand.\n    ","2.5.3#footnote-link-8":"8","2.5.3#p24":"\n    We can solve the problem exhibited in\n    exercise 2.95 if\n    we use the following modification of the GCD algorithm (which really\n    works only in the case of polynomials with integer coefficients).\n    Before performing any polynomial division in the GCD computation, we\n    multiply the dividend by an integer constant factor, chosen to\n    guarantee that no fractions will arise during the division process.\n    Our answer will thus differ from the actual GCD by an integer constant\n    factor, but this does not matter in the case of reducing rational\n    functions to lowest terms; the GCD will be used to divide both the\n    numerator and denominator, so the integer constant factor will cancel\n    out.\n  ","2.5.3#p25":"\n    More precisely, if $P$ and\n    $Q$ are polynomials, let\n    $O_1$ be the order of\n    $P$ (i.e., the order of the largest term of\n    $P$) and let $O_2$\n    be the order of $Q$.  Let\n    $c$ be the leading coefficient of\n    $Q$.  Then it can be shown that, if we multiply\n    $P$ by the\n    integerizing factor$c^{1+O_{1} -O_{2}}$, the resulting polynomial\n    can  be divided by $Q$ by using the\n    div-terms\n    algorithm without introducing any fractions.  The operation of multiplying\n    the dividend by this constant and then dividing is sometimes called the \n    pseudodivision of $P$ by\n    $Q$.  The remainder of the division is\n    called the\n    pseudoremainder.\n  ","2.5.3#ex-2.96":"\n        Implement the\n        procedurepseudoremainder-terms,\n\twhich is just like\n        remainder-terms\n\texcept that it multiplies the dividend by the integerizing factor\n\tdescribed above before calling\n\tdiv-terms.\n        Modify\n\tgcd-terms\n\tto use\n\tpseudoremainder-terms,\n\tand verify that\n\tgreatest-common-divisor\n\tnow produces an answer with integer coefficients on the example in\n\texercise 2.95.\n      \n        The GCD now has integer coefficients, but they are larger than those\n        of $P_1$.  Modify\n\tgcd-terms\n\tso that it removes common factors from the coefficients of the answer\n\tby dividing all the coefficients by their (integer) greatest common\n\tdivisor.\n      ","2.5.3#p26":"\n    Thus, here is how to reduce a rational function to lowest terms:\n    \n\tCompute the GCD of the numerator and denominator, using\n\tthe version of\n\tgcd-terms\n\tfrom exercise 2.96.\n      \n\tWhen you obtain the GCD, multiply both numerator and\n\tdenominator by the same integerizing factor before dividing through by\n\tthe GCD, so that division by the GCD will not introduce any noninteger\n\tcoefficients.  As the factor you can use the leading coefficient of\n\tthe GCD raised to the power\n\t$1+O_{1} -O_{2}$, where\n\t$O_{2}$ is the order of the GCD and\n\t$O_{1}$ is the maximum of the orders of the\n\tnumerator and denominator.  This will ensure that dividing the\n\tnumerator and denominator by the GCD will not introduce any fractions.\n      \n\tThe result of this operation will be a numerator and denominator\n\twith integer coefficients.  The coefficients will normally be very\n\tlarge because of all of the integerizing factors, so the last step is\n\tto remove the redundant factors by computing the (integer) greatest\n\tcommon divisor of all the coefficients of the numerator and the\n\tdenominator and dividing through by this factor.\n      ","2.5.3#ex-2.97":"\n\tImplement this algorithm as a\n\tprocedurereduce-terms\n\tthat takes two term lists n and\n\td as arguments and returns a list\n\tnn, dd,\n\twhich are n and\n\td reduced to lowest terms via the\n\talgorithm given above. Also write a\n\tprocedurereduce-poly,\n\tanalogous to\n\tadd-poly,\n\tthat checks to see if the two polys have the same variable.  If so,\n\treduce-poly\n\tstrips off the variable and passes the problem to\n\treduce-terms,\n\tthen reattaches the variable to the two term lists supplied by\n\treduce-terms.\n        Define a\n        procedure\n        analogous to\n\treduce-terms\n\tthat does what the original\n\tmake-rat\n\tdid for integers:\n        (define (reduce-integers n d)\n                (let ((g (gcd n d)))\n                (list (/ n g) (/ d g))))\n        and define\n        reduce as a generic operation that calls\n\tapply-generic\n\tto dispatch\t\n\t\n\t     to either\n\t     reduce-poly\n\t(for polynomial arguments) or\n\treduce-integers\n\t(for\n\tscheme-number\n\targuments). You can now easily make the rational-arithmetic package\n\treduce fractions to lowest terms by having\n\tmake-rat\n\tcall reduce before combining the given\n        numerator and denominator to form a rational number. The system now\n        handles rational expressions in either integers or polynomials.\n        To test your program, try the example at the beginning of this\n        extended exercise:\n        (define p1 (make-polynomial 'x '((1 1)(0 1))))\n                (define p2 (make-polynomial 'x '((3 1)(0 -1))))\n                (define p3 (make-polynomial 'x '((1 1))))\n                (define p4 (make-polynomial 'x '((2 1)(0 -1))))\n\n                (define rf1 (make-rational p1 p2))\n                (define rf2 (make-rational p3 p4))\n\n                (add rf1 rf2)\n        See if you get the correct answer, correctly reduced to lowest terms.\n      ","2.5.3#p27":"\n    The GCD computation is at the heart of any system that does operations\n    on rational functions.  The algorithm used above, although\n    mathematically straightforward, is extremely slow.  The slowness is\n    due partly to the large number of division operations and partly to\n    the enormous size of the intermediate coefficients generated by the\n    pseudodivisions.\n    \n    One of the active areas in the development of\n    algebraic-manipulation systems is the design of better algorithms for\n    computing polynomial GCDs.","2.5.3#footnote-link-9":"9","2.5.3#footnote-1":"On the other hand, we will\n      allow polynomials whose coefficients are themselves polynomials in other\n      variables.  This will give us essentially the same representational\n      power as a full multivariate system, although it does lead to coercion\n      problems, as discussed below.","2.5.3#footnote-2":"For univariate\n    polynomials, giving the value of a polynomial at a given set of points can\n    be a particularly good representation.  This makes polynomial arithmetic\n    extremely simple.  To obtain, for example, the sum of two polynomials\n    represented in this way, we need only add the values of the\n    polynomials at corresponding points.  To transform back to a more\n    familiar representation, we can use the \n    \n    Lagrange interpolation formula, which shows how to recover the coefficients\n    of a polynomial of degree $n$ given the values\n    of the polynomial at $n+1$ points.","2.5.3#footnote-3":"This operation is very much like the ordered\n    union-set\n    operation we developed in exercise 2.62.\n    In fact, if we think of the terms of the polynomial as a set ordered\n    according to the power of the indeterminate, then the program that\n    produces the term list for a sum is almost identical to\n    union-set.","2.5.3#footnote-4":"To\n    make this work completely smoothly, we should also add to our generic\n    arithmetic system the ability to coerce a \"number\" to a\n    polynomial by regarding it as a polynomial of degree zero whose coefficient\n    is the number.  This is necessary if we are going to perform operations\n    such as\n    \n      \\[\n      {\\left[ x^2 +(y+1)x+5\\right]+ \\left[ x^2 +2x+1\\right]}\n      \\]\n    \n    which requires adding the coefficient $y+1$ to\n    the coefficient 2.","2.5.3#footnote-5":"In\n    these polynomial examples, we assume that we have implemented the generic\n    arithmetic system using the type mechanism suggested in\n    exercise 2.78. Thus, coefficients\n    that are ordinary numbers will be represented as the numbers themselves\n    rather than as pairs whose\n    car\n    is the\n    symbol scheme-number.","2.5.3#footnote-6":"Although we are assuming\n    that term lists are ordered, we have implemented\n    adjoin-term\n    to simply\n    cons\n\tthe new term onto the existing term list.\n      \n    We can get away with this so\n    long as we guarantee that the\n    procedures\n    (such as\n    add-terms)\n    that use\n    adjoin-term\n    always call it with a higher-order term than appears in the list.  If we\n    did not want to make such a guarantee, we could have implemented\n    adjoin-term\n    to be similar to the\n    adjoin-set\n    constructor for the ordered-list\n    representation of sets\n    (exercise 2.61).\n    ","2.5.3#footnote-7":"The fact\n    that \n    \n    Euclid's Algorithm works for polynomials is formalized in algebra\n    by saying that polynomials form a kind of algebraic domain called a\n    Euclidean ring.  A Euclidean ring is a domain that admits\n    addition, subtraction, and commutative multiplication, together with a\n    way of assigning to each element $x$ of the\n    ring a positive integer\n    \"measure\"$m(x)$ with the\n    properties that $m(xy)\\geq m(x)$ for any nonzero\n    $x$ and $y$ and that,\n    given any $x$ and $y$,\n    there exists a $q$ such that\n    $y=qx+r$ and either\n    $r=0$ or\n    $m(r) < m(x)$.  From an abstract point of\n    view, this is what is needed to prove that Euclid's Algorithm works.\n    For the domain of integers, the measure $m$ of an\n    integer is the absolute value of the integer itself.  For the domain of\n    polynomials, the measure of a polynomial is its degree.","2.5.3#footnote-8":"In an implementation like MIT Scheme, this produces\n\ta polynomial that is indeed a divisor of\n\t$Q_1$ and $Q_2$,\n\tbut with rational coefficients. In many other Scheme systems, in which\n\tdivision of integers can produce limited-precision decimal numbers, we\n\tmay fail to get a valid divisor.","2.5.3#footnote-9":"One extremely efficient and\n    elegant method for computing \n    \n    polynomial GCDs was discovered by \n    \n    Richard Zippel (1979).  The method is a probabilistic algorithm, as is the\n    fast test for primality that we discussed in chapter 1.\n    Zippel's book (1993) describes this method, together with other ways\n    to compute polynomial GCDs.","3#p1":"\n    The preceding chapters introduced the basic elements from which\n    programs are made.  We saw how primitive\n    procedures\n    and primitive data are combined to construct compound entities, and we\n    learned that abstraction is vital in helping us to cope with the complexity\n    of large systems.  But these tools are not sufficient for designing\n    programs.  Effective program synthesis also requires organizational\n    principles that can guide us in formulating the overall design of a\n    program.  In particular, we need strategies to help us structure large\n    systems so that they will be\n    modular, that is, so that they can\n    be divided \"naturally\" into coherent parts that can be\n    separately developed and maintained.\n  ","3#p2":"\n    One powerful design strategy, which is particularly appropriate to the\n    construction of programs for\n    \n    modeling physical systems, is to base the\n    structure of our programs on the structure of the system being\n    modeled.  For each object in the system, we construct a corresponding\n    computational object.  For each system action, we define a symbolic\n    operation in our computational model.  Our hope in using this strategy\n    is that extending the model to accommodate new objects or new actions\n    will require no strategic changes to the program, only the addition of\n    the new symbolic analogs of those objects or actions.  If we have been\n    successful in our system organization, then to add a new feature or\n    debug an old one we will have to work on only a localized part of the\n    system.\n  ","3#p4":"\n    Both the object-based approach and the stream-processing approach\n    raise significant linguistic issues in programming.\n    With objects, we must be concerned with how a computational object can\n    change and yet maintain its identity.  This will force us to abandon\n    our old substitution model of computation\n    (section 1.1.5) in favor of a more\n    mechanistic but less theoretically tractable\n    environment model of\n    computation.  The difficulties of dealing with objects, change, and\n    identity are a fundamental consequence of the need to grapple with\n    time in our computational models.  These difficulties become even\n    greater when we allow the possibility of concurrent execution of\n    programs.  The stream approach can be most fully exploited when we\n    decouple simulated time in our model from the order of the events that\n    take place in the computer during evaluation.  We will accomplish this\n    using a technique known as\n    delayed evaluation.\n  ","3.1":"3.1  Assignment and Local State","3.1#p1":"\n      We ordinarily view the world as populated by independent objects, each\n      of which has a state that changes over time.  An object is said to\n      \"have state\" if its behavior is influenced by its history.\n      A bank account, for example, has state in that the answer to the question\n      \"Can I withdraw $100?\"  depends upon the history of\n      deposit and withdrawal transactions.  We can characterize an\n      object's state by one or more \n      state variables, which among them maintain enough\n      information about history to determine the object's current behavior.\n      In a simple banking system, we could characterize the state of an\n      account by a current balance rather than by remembering the entire\n      history of account transactions.\n    ","3.1#p2":"\n      In a system composed of many objects, the objects are rarely\n      completely independent.  Each may influence the states of others\n      through interactions, which serve to couple the state variables of one\n      object to those of other objects.  Indeed, the view that a system is\n      composed of separate objects is most useful when the state variables\n      of the system can be grouped into closely coupled subsystems that are\n      only loosely coupled to other subsystems.\n    ","3.1#p3":"\n      This view of a system can be a powerful framework for organizing\n      computational models of the system.  For such a model to be modular,\n      it should be decomposed into computational objects that model the\n      actual objects in the system.  Each computational object must have its\n      own local state variables describing the actual object's\n      state. Since the states of objects in the system being modeled change over\n      time, the state variables of the corresponding computational objects\n      must also change.  If we choose to model the flow of time in the\n      system by the elapsed time in the computer, then we must have a way to\n      construct computational objects whose behaviors change as our programs\n      run.  In particular, if we wish to model state variables by ordinary\n      symbolic names in the programming language, then the language must\n      provide an \n      assignment operator\n      to enable us to change the value\n      associated with a name.\n    ","3.1.1":"3.1.1  \n    Local State Variables","3.1.1#p1":"\n    To illustrate what we mean by having a computational object with\n    \n    time-varying state, let us model the situation of withdrawing money\n    from a\n    \n    bank account.  We will do this using a\n    procedurewithdraw, which takes as argument an\n    amount to be withdrawn.\n    If there is enough money in the account to accommodate the withdrawal,\n    then withdraw should return the balance\n    remaining after the withdrawal.  Otherwise,\n    withdraw should return the message\n    Insufficient funds. For example, if we begin with $100\n    in the account, we should obtain the following sequence of responses\n    using\n    withdraw:\n    (withdraw 25) (withdraw 25) (withdraw 60) (withdraw 15) \n    Observe that the expression\n    (withdraw 25),\n    evaluated twice, yields different values.  This is a new kind of\n    behavior for a\n    procedure.\n    Until now, all our\n    procedures\n    could be viewed as specifications for computing mathematical functions.\n    A call to a\n    procedure\n    computed the value of the function applied to the given arguments,\n    and two calls to the same\n    procedure\n    with the same arguments always produced the same\n    result.","3.1.1#footnote-link-1":"1","3.1.1#p2":"\n        To implement withdraw, we can use a\n\tvariable balance to indicate the balance of\n\tmoney in the account and define withdraw\n\tas a\n        procedure\n        that accesses balance.\n      \n    The withdrawprocedure\n    checks to see if balance is at least as large\n    as the requested amount.  If so,\n    withdraw decrements\n    balance by amount\n    and returns the new value of balance. Otherwise,\n    withdraw returns the Insufficient funds\n    message. Here are the\n    definitions\n    of balance and\n    withdraw:\n    (define balance 100)\n\n(define (withdraw amount)\n   (if (>= balance amount)\n     (begin (set! balance (- balance amount))\n            balance)\n     \"Insufficient funds\")) \n    Decrementing balance is accomplished by the \n    expression(set! balance (- balance amount))\n        This uses the set! special form, whose\n\tsyntax is\n\t\n(set! $\\langle \\textit{name} \\rangle$ $\\langle \\textit{new-value}\\rangle$)\n\t  \n    Here\n    $\\langle \\textit{name} \\rangle$\n\tis a symbol \n      \n    and\n    $\\langle \\textit{new-value} \\rangle$\n    is any expression.  \n    Set! \n    changes\n    $\\langle \\textit{name} \\rangle$\n    so that its value is the\n    result obtained by evaluating\n    $\\langle \\textit{new-value}\\rangle$.\n      \n    In the case at hand, we are changing balance so\n    that its new value will be the result of subtracting\n    amount from the previous value of\n    balance.","3.1.1#footnote-link-2":"2","3.1.1#p3":"Withdraw also uses the\n\tbegin \n\tspecial form to cause two expressions to be evaluated in the case where\n\tthe if test is true: first decrementing \n\tbalance and then returning the value of \n\tbalance. In general, evaluating the\n\texpression\n\t(begin $\\textit{exp}_{1}$ $\\textit{exp}_{2}$ $\\ldots$ $\\textit{exp}_{k}$)\n\tcauses the expressions $\\textit{exp}_{1}$\n\tthrough $\\textit{exp}_{k}$ to be evaluated in\n\tsequence and the value of the final expression\n\t$\\textit{exp}_{k}$ to be returned as the\n\tvalue of the entire begin\n\tform.","3.1.1#footnote-link-3":"3","3.1.1#p4":"\n    Although withdraw works as desired, the\n    variable balance presents a problem.  As\n    specified above, balance is a name defined\n    in the\n    global environment and is freely accessible to be examined or\n      modified by any\n      procedure.\n      It would be much better if we could somehow make\n      balance internal to\n      withdraw, so that\n      withdraw would be the only\n      procedure\n      that could access balance directly and\n      any other\n      procedure\n      could access balance only indirectly\n      (through calls to withdraw).  This would\n      more accurately model the notion that\n      balance is a local state variable used by\n      withdraw to keep track of the state of the\n      account.\n  ","3.1.1#p5":"\n    We can make balance internal to\n    withdraw by rewriting the definition as\n    follows:\n    (define new-withdraw\n  (let ((balance 100))\n    (lambda (amount)\n      (if (>= balance amount)\n        (begin (set! balance (- balance amount))\n               balance)\n        \"Insufficient funds\")))) \n        What we have done here is use let to\n\testablish an environment with a local variable\n\tbalance, bound to the initial value 100.\n        Within this local environment, we use\n\tlambda to create a procedure that takes\n\tamount as an argument and behaves like our\n        previous withdraw procedure.  This\n        procedure—returned as the result of evaluating the\n\tlet expression—is\n\tnew-withdraw,\n        which behaves in precisely the same way as\n\twithdraw but whose variable\n\tbalance is not accessible by any other\n        procedure.","3.1.1#footnote-link-4":"4","3.1.1#p6":"\n    Combining \n    set! \n        with local variables \n      \n    is the general programming\n    technique we will use for constructing computational objects with\n    local state.  Unfortunately, using this technique raises a serious\n    problem: When we first introduced\n    procedures,\n    we also introduced the substitution model of evaluation\n    (section 1.1.5) to provide an\n    interpretation of what\n    procedure\n    application means.  We said that applying a\n    procedure\n    should be interpreted as evaluating the\n    body of the procedure\n    with the\n    formal\n    parameters replaced by their values.\n    \n    The trouble is that,\n    as soon as we introduce assignment into our language, substitution is no\n    longer an adequate model of\n    procedure\n    application.  (We will see why this is so in\n    section 3.1.3.)  As a consequence, we\n    technically have at this point no way to understand why the\n    new-withdrawprocedure\n    behaves as claimed above.  In order to really understand a\n    procedure\n    such as\n    new-withdraw,\n    we will need to develop a new model of\n    procedure\n    application.  In section 3.2 we will\n    introduce such a model, together with an explanation of \n    set! and local variables.\n    First, however, we examine some variations on the theme established by\n    new-withdraw.","3.1.1#p7":"\n    The following\n    \n\tprocedure, make-withdraw,\n      \n    creates \"withdrawal processors.\"\n    The formal parameter\n    balance in\n    make-withdraw\n    specifies the initial amount of money in the\n    account.(define (make-withdraw balance)\n   (lambda (amount)\n      (if (>= balance amount)\n         (begin (set! balance (- balance amount))\n                balance)\n         \"Insufficient funds\"))) Make-withdraw\n    can be used as follows to create two objects W1\n    and W2:\n    (define W1 (make-withdraw 100))\n(define W2 (make-withdraw 100)) (W1 50) (W2 70) (W2 40) (W1 40) \n    Observe that W1 and \n    W2 are completely independent objects, each\n    with its own local state variable balance.\n    Withdrawals from one do not affect the other.\n  ","3.1.1#footnote-link-5":"5","3.1.1#p8":"\n    We can also create objects that handle\n    \n    deposits as well as\n    withdrawals, and thus we can represent simple bank accounts.  Here is\n    a\n    procedure\n    that returns a \"bank-account object\" with a specified initial\n    balance:\n    (define (make-account balance)\n   (define (withdraw amount)\n      (if (>= balance amount)\n         (begin (set! balance (- balance amount))\n                balance)\n            \"Insufficient funds\"))\n   (define (deposit amount)\n      (set! balance (+ balance amount))\n      balance)\n   (define (dispatch m)\n      (cond ((eq? m 'withdraw) withdraw)\n            ((eq? m 'deposit) deposit)\n            (else (error \"Unknown request - - MAKE-ACCOUNT\"\n                   m))))\n   dispatch) \n    Each call to make_account sets up an\n    environment with a local state variable balance.\n    Within this environment, make_account defines\n    proceduresdeposit and\n    withdraw that access\n    balance and an additional\n    proceduredispatch\n    that takes a \"message\" as input and returns one of the two local\n    procedures.\n    The dispatchprocedure\n    itself is returned as the value that represents the bank-account object.\n    This is precisely the \n    message-passing style of programming that we saw in\n    section 2.4.3, although here we are using\n    it in conjunction with the ability to modify local variables.\n  ","3.1.1#p9":"Make-account\n    can be used as follows:\n    (define acc (make-account 100)) ((acc 'withdraw) 50) ((acc 'withdraw) 60) ((acc 'deposit) 40) ((acc 'withdraw) 60) Each call to acc returns the locally defined\n    deposit or withdrawprocedure,\n    which is then applied to the specified amount.\n    As was the case with\n    make-withdraw, another\n\tcall to make-account(define acc2 (make-account 100)) \n    will produce a completely separate account object, which maintains its\n    own local balance.\n  ","3.1.1#ex-3.1":"\n    An \n    accumulator is a\n    procedure\n    that is called repeatedly with a single numeric argument and accumulates its\n    arguments into a sum. Each time it is called, it returns the currently\n    accumulated sum. Write a\n    proceduremake-accumulator\n    that generates accumulators, each maintaining an independent sum.  The\n    input to \n    make-accumulator\n    should specify the initial value of the sum; for example\n    (define A (make-accumulator 5)) (A 10) (A 10) ","3.1.1#ex-3.2":"\n    In software-testing applications, it is useful to be able to count the\n    number of times a given\n    procedure\n    is called during the course of a computation.  Write a\n    proceduremake-monitored\n    that takes as input a\n    procedure,f, that itself takes one input.  The result\n    returned by \n    make-monitored\n    is a third\n    procedure,\n    say mf, that keeps track of the number of times\n    it has been called by maintaining an internal counter. If the input to\n    mf is the\n    special symbol how-many-calls,\n      \n    then mf returns the value of the counter.  If\n    the input is the \n    special symbol reset-count,\n    then mf resets the counter to zero.  For any\n    other input, mf returns the result of calling\n    f on that input and increments the counter.\n    For instance, we could make a monitored version of the\n    sqrtprocedure:(define s (make-monitored sqrt)) (s 100) (s 'how-many-calls?) ","3.1.1#ex-3.3":"\n    Modify the \n    make-accountprocedure\n    so that it creates\n    \n    password-protected accounts.  That is, \n    make-account\n    should take a\n    symbol\n    as an additional argument, as in\n    (define acc (make-account 100 'secret-password)) \n    The resulting account object should process a request only if it is\n    accompanied by the password with which the account was created, and\n    should otherwise return a complaint:\n    ((acc 'secret-password 'withdraw) 40) ((acc 'some-other-password 'deposit) 50) ","3.1.1#ex-3.4":"\n    Modify the \n    make-accountprocedure\n    of exercise 3.3 by adding another\n    local state variable so that, if an account is accessed more than seven\n    consecutive times with an incorrect password, it invokes the\n    procedurecall-the-cops.","3.1.1#footnote-1":"Actually, this is not quite true. One exception was the \n    \n    random-number generator\n    in section 1.2.6.  Another exception\n    involved the\n    \n    operation/type tables we introduced in\n    section 2.4.3, where the values of two\n    calls to get with the same arguments\n    depended on intervening calls to put.\n    On the other hand, until we introduce assignment, we have no way to\n    create such\n    procedures\n    ourselves.","3.1.1#footnote-2":"\n\tThe value of a set! expression is\n\timplementation-dependent. Set! should be\n\tused only for its effect, not for its value.\n\t \n\tThe name set! reflects a naming convention\n\tused in Scheme: Operations that change the values of variables (or that\n\tchange data structures, as we will see in\n\tsection 3.3) are given names that end\n\twith an exclamation point.  This is similar to the convention of\n\tdesignating predicates by names that end with a question mark.\n      ","3.1.1#footnote-3":"We have already used \n\tcond and in \n\tbegin implicitly in our programs, because in\n\tScheme the body of a procedure can be a sequence of expressions.  Also,\n\tthe consequent part of each clause in a\n\tcond expression can be a sequence of\n\texpressions rather than a single expression.","3.1.1#footnote-4":"In programming-language jargon, the variable\n\tbalance is said to be \n        encapsulated within the\n\tnew-withdraw\n        procedure. Encapsulation reflects the general system-design principle\n\tknown as the \n        hiding principle: One can make a system more modular and robust\n\tby protecting parts of the system from each other; that is, by providing\n\tinformation access only to those parts of the system that have a\n\t\"need to know.\"","3.1.1#footnote-5":"\n    In contrast with\n    new-withdraw\n    above, we do not have to use\n    let\n    to make balance a local variable, since\n    formal\n    parameters are already\n    local. This will be clearer after the discussion of the environment\n    model of evaluation in\n    section 3.2.\n    (See also\n    exercise 3.10.)","3.1.2":"3.1.2  \n    The Benefits of Introducing Assignment","3.1.2#p1":"\n    As we shall see, introducing assignment into our programming language\n    leads us into a thicket of difficult conceptual issues.  Nevertheless,\n    viewing systems as\n    \n    collections of objects with local state is a\n    powerful technique for maintaining a\n    \n    modular design.  As a simple\n    example, consider the design of a\n    procedurerand that, whenever\n    it is called, returns an integer chosen at random.\n  ","3.1.2#p2":"\n    It is not at all clear what is meant by \"chosen at random.\"\n    What we presumably want is for successive calls to\n    rand to produce a sequence of numbers that has\n    statistical properties of uniform distribution.  We will not discuss methods\n    for generating suitable sequences here.  Rather, let us assume that we have a\n    procedurerand-update \n    that has the property that if we start with a given number\n    $x_{1}$ and form\n    \n = (rand-update )\n = (rand-update )\n\t  \n    then the sequence of values\n    $x_1, x_2, x_3, \\ldots$, will have the desired\n    statistical properties.","3.1.2#footnote-link-1":"1","3.1.2#p3":"\n    We can implement rand as a\n    procedure\n    with a local state variable x that is\n    initialized to some fixed value \n    random-init.\n    Each call to rand computes \n    rand-update \n    of the current value of x, returns this as the\n    random number, and also stores this as the new value of\n    x.\n    (define rand\n  (let ((x random-init))\n    (lambda ()\n      (set! x (rand-update x))\n      x))) ","3.1.2#p4":"\n    Of course, we could generate the same sequence of random numbers\n    without using assignment by simply calling \n    rand-update \n    directly. However, this would mean that any part of our program that used\n    random numbers would have to explicitly remember the current value of\n    x to be passed as an argument to \n    rand-update.\n    To realize what an annoyance this would be, consider using random numbers\n    to implement a technique called \n    Monte Carlo simulation.\n  ","3.1.2#p5":"\n    The Monte Carlo method consists of choosing sample experiments at random\n    from a large set and then making deductions on the basis of the\n    probabilities estimated from tabulating the results of those experiments.\n    For example, we can approximate \n    $\\pi$ using the fact that\n    $6/\\pi^2$ is the probability that two integers\n    chosen at random will have no factors in common; that is, that their\n    greatest common divisor will be 1.\n    To obtain the approximation to $\\pi$, we perform\n    a large number of experiments. In each experiment we choose two integers at\n    random and perform a test\n    \n    to see if their GCD is 1.  The fraction of times that the test is passed\n    gives us our estimate of $6/\\pi^2$, and from this\n    we obtain our approximation to $\\pi$.\n  ","3.1.2#footnote-link-2":"2","3.1.2#p6":"\n    The heart of our program is a\n    proceduremonte-carlo,\n    which takes as arguments the number of times to try an experiment, together\n    with the experiment, represented as a no-argument\n    procedure\n    that will return either true or false each time it is run.\n    Monte-carlo\n    runs the experiment for the designated number of trials and returns a\n    number telling the fraction of the trials in which the experiment was\n    found to be true.\n  ","3.1.2#p7":"(define (estimate-pi trials)\n  (sqrt (/ 6 (monte-carlo trials cesaro-test))))\n\n(define (cesaro-test)\n  (= (gcd (rand) (rand)) 1))\n\n(define (monte-carlo trials experiment)\n  (define (iter trials-remaining trials-passed)\n    (cond ((= trials-remaining 0)\n           (/ trials-passed trials))\n          ((experiment)\n           (iter (- trials-remaining 1) (+ trials-passed 1)))\n          (else\n           (iter (- trials-remaining 1) trials-passed))))\n  (iter trials 0)) ","3.1.2#p8":"\n    Now let us try the same computation using \n    rand-update \n    directly rather than rand, the way we would be\n    forced to proceed if we did not use assignment to model local state:\n    (define (estimate-pi trials)\n  (sqrt (/ 6 (random-gcd-test trials random-init))))\n\n(define (random-gcd-test trials initial-x)\n  (define (iter trials-remaining trials-passed x)\n    (let ((x1 (rand-update x)))\n      (let ((x2 (rand-update x1)))\n        (cond ((= trials-remaining 0)   \n               (/ trials-passed trials))\n              ((= (gcd x1 x2) 1)\n               (iter (- trials-remaining 1)\n                     (+ trials-passed 1)\n                     x2))\n              (else\n               (iter (- trials-remaining 1)\n                     trials-passed\n                     x2))))))\n  (iter trials 0 initial-x)) ","3.1.2#p9":"\n    While the program is still simple, it betrays some painful breaches of\n    modularity.  In our first version of the program, using\n    rand, we can express the Monte Carlo method\n    directly as a general \n    monte-carloprocedure\n    that takes as an argument an arbitrary\n    experimentprocedure.\n    In our second version of the program, with no local state for the\n    random-number generator, \n    random-gcd-test \n    must explicitly manipulate the random numbers\n    x1 and x2 and\n    recycle x2 through the iterative loop as the\n    new input to \n    rand-update.\n    This explicit handling of the random numbers intertwines the structure of\n    accumulating test results with the fact that our particular experiment uses\n    two random numbers, whereas other Monte Carlo experiments might use one\n    random number or three.  Even the top-level\n    procedureestimate-pi\n    has to be concerned with supplying an initial random number.  The fact that\n    the random-number generator's insides are leaking out into other parts\n    of the program makes it difficult for us to isolate the Monte Carlo idea so\n    that it can be applied to other tasks.  In the first version of the program,\n    assignment encapsulates the state of the random-number generator within the\n    randprocedure,\n    so that the details of random-number generation remain independent of the\n    rest of the program.\n  ","3.1.2#p10":"\n    The general phenomenon illustrated by the Monte Carlo example is this: From\n    the point of view of one part of a complex process, the other parts appear\n    to change with time.  They have hidden time-varying local state.  If we wish\n    to write computer programs whose structure reflects this decomposition, we\n    make computational objects (such as bank accounts and random-number\n    generators) whose behavior changes with time.  We model state with local\n    state variables, and we model the changes of state with assignments to those\n    variables.\n  ","3.1.2#p11":"\n    It is tempting to conclude this discussion by saying that, by introducing\n    assignment and the technique of hiding state in local variables, we are able\n    to structure systems in a more modular fashion than if all state had to be\n    manipulated explicitly, by passing additional parameters.  Unfortunately,\n    as we shall see, the story is not so simple.\n  ","3.1.2#ex-3.5":"Monte Carlo integration\n    is a method of estimating definite\n    integrals by means of Monte Carlo simulation.  Consider computing the\n    area of a region of space described by a predicate\n    $P(x, y)$ that is true for points\n    $(x, y)$ in the region and false for points not\n    in the region.  For example, the region contained within a circle of radius\n    $3$ centered at\n    $(5, 7)$ is described by the predicate that tests\n    whether $(x-5)^2 + (y-7)^2\\leq 3^2$.  To estimate\n    the area of the region described by such a predicate, begin by choosing a\n    rectangle that contains the region.  For example, a rectangle with diagonally\n    opposite corners at $(2, 4)$ and\n    $(8, 10)$ contains the circle above. The desired\n    integral is the area of that portion of the rectangle that lies in the\n    region.  We can estimate the integral by picking, at random, points\n    $(x, y)$ that lie in the rectangle, and testing\n    $P(x, y)$ for each point to determine whether the\n    point lies in the region. If we try this with many points, then the fraction\n    of points that fall in the region should give an estimate of the proportion\n    of the rectangle that lies in the region.  Hence, multiplying this fraction\n    by the area of the entire rectangle should produce an estimate of the\n    integral.\n    \n    Implement Monte Carlo integration as a\n    procedureestimate-integral\n    that takes as arguments a predicate P, upper\n    and lower bounds x1,\n    x2, y1, and\n    y2 for the rectangle, and the number of trials\n    to perform in order to produce the estimate.  Your\n    procedure\n    should use the same \n    monte-carloprocedure\n    that was used above to estimate $\\pi$.  Use your \n    estimate-integral\n    to produce an estimate of $\\pi$ by measuring the\n    area of a unit circle.\n    \n    You will find it useful to have a\n    procedure\n    that returns a number chosen at random from a given range.  The following \n    random-in-rangeprocedure\n    implements this in terms of the\n    random\n\tprocedure\n      \n    used in section 1.2.6, which returns a\n    nonnegative number less \n    \n\tthan its input.(define (random-in-range low high)\n  (let ((range (- high low)))\n    (+ low (random range)))) ","3.1.2#footnote-link-3":"3","3.1.2#ex-3.6":"\n    It is useful to be able to\n    \n    reset a random-number generator to produce\n    a sequence starting from a given value.  Design a new\n    randprocedure\n    that is called with an argument that is either the\n    \n\tsymbol generate or the symbol\n\treset\n    and behaves as follows:\n    (rand 'generate)\n    produces a new random number; \n    ((rand 'reset)$\\langle \\textit{new-value} \\rangle$)\n    resets the internal state variable to the designated new-value. Thus, by resetting the\n    state, one can generate repeatable sequences.  These are very handy to have\n    when testing and debugging programs that use random numbers.\n    ","3.1.2#footnote-1":"One common way to implement\n    rand-update \n    is to use the rule that $x$ is updated to\n    $ax+b$ modulo $m$,\n    where $a$, $b$, and\n    $m$ are appropriately chosen integers.\n    Chapter 3 of\n    Knuth 1997b includes an extensive\n    discussion of techniques for generating sequences of random numbers and\n    establishing their statistical properties.  Notice that the \n    rand-updateprocedure\n    computes a mathematical function: Given the same input twice, it\n    produces the same output.  Therefore, the number sequence produced by\n    rand-update \n    certainly is not \"random,\" if by \"random\" we\n    insist that each number in the sequence is unrelated to the preceding\n    number.  The relation between \"real randomness\" and so-called \n    pseudo-random sequences, which are produced by well-determined\n      computations and yet have suitable statistical properties, is a\n      complex question involving difficult issues in mathematics and\n      philosophy.  \n      \n      Kolmogorov,\n      \n      Solomonoff, and\n      \n      Chaitin have made great\n      progress in clarifying these issues; a discussion can be found in\n      Chaitin 1975.","3.1.2#footnote-2":"This theorem is due to G. \n    \n    Lejeune Dirichlet.  See section 4.5.2 of\n    Knuth 1997b for a discussion and a proof.","3.1.2#footnote-3":"\n\tMIT Scheme\n\t\n\tprovides such a\n\tprocedure.\n\tIf \n\trandom is given an exact integer (as in\n\tsection 1.2.6) it returns an exact integer,\n\tbut if it is given a decimal value (as in this exercise) it returns a\n\tdecimal value.","3.1.3":"3.1.3  \n    The Costs of Introducing Assignment","3.1.3#p1":"\n    As we have seen,\n    \n\tthe set! operation\n      \n    enables us to model objects\n    that have local state.  However, this advantage comes at a price.  Our\n    programming language can no longer be interpreted in terms of the\n    substitution model of\n    procedure\n    application that we introduced in\n    section 1.1.5.  Moreover, no simple\n    model with \"nice\" mathematical properties can be an adequate\n    framework for dealing with objects and assignment in programming languages.\n  ","3.1.3#p2":"\n    So long as we do not use assignments, two evaluations of the same\n    procedure\n    with the same arguments will produce the same result, so that\n    procedures\n    can be viewed as computing mathematical functions. Programming without any\n    use of assignments, as we did throughout the first two chapters of this\n    book, is accordingly known as\n    functional programming.\n  ","3.1.3#p3":"\n    To understand how assignment complicates matters, consider a simplified\n    version of the make_withdrawprocedure\n    of section 3.1.1 that does not\n    bother to check for an insufficient amount:\n    (define (make-simplified-withdraw balance)\n  (lambda (amount)\n    (set! balance (- balance amount))\n    balance)) (define W (make-simplified-withdraw 25)) (W 20) (W 10) \n    Compare this\n    procedure\n    with the following make_decrementerprocedure,\n    which does not use\n    set!:(define (make-decrementer balance)\n            (lambda (amount)\n            (- balance amount))) make-decrementer\n    returns a\n    procedure\n    that subtracts its input from a designated amount\n    balance, but there is no accumulated effect\n    over successive calls, as with\n    make_simplified_withdraw:\n    (define D (make-decrementer 25)) (D 20) (D 10) \n    We can use the substitution model to explain how\n    make_decrementer works.  For instance, let us\n    analyze the evaluation of the expression\n    ((make-decrementer 25) 20) \n    We first simplify the\n    \n\toperator of the combination\n      \n    by substituting\n    $25$ for balance in\n    the body of\n    make-decrementer.\n      \n    This reduces the\n    expression to\n    ((lambda (amount) (- 25 amount)) 20) \n    Now we apply the\n    \n\toperator\n      \n    by substituting 20 for\n    amount in the body of the\n    lambda\n    expression:\n    (- 25 20) \n    The final answer is 5. ","3.1.3#p4":"\n    Observe, however, what happens if we attempt a similar substitution analysis\n    with make_simplified_withdraw:\n    ((make-simplified-withdraw 25) 20) \n    We first simplify the\n    \n\toperator\n      \n    by substituting 25 for\n    balance in\n    the body ofmake_simplified_withdraw. This reduces the\n    \n\texpression\n      \n    to((lambda (amount) (set! balance (- 25 amount)) 25) 20)\n    Now we apply the\n    operator\n    by substituting 20 for amount\n    in the body of the\n    lambda expression:\n      (set! balance (- 25 20)) 25\n    If we adhered to the substitution model, we would have to say that the\n    meaning of the\n    procedure\n    application is to first set balance to 5 and\n    then return 25 as the value of the expression.  This gets the wrong answer.\n    In order to get the correct answer, we would have to somehow distinguish the\n    first occurrence of balance (before the effect\n    of the\n    set!)\n    from the second occurrence of balance\n    (after the effect of the\n    set!),\n    and the substitution model cannot do this.\n  ","3.1.3#footnote-link-1":"1","3.1.3#p5":"\n    The trouble here is that substitution is based ultimately on the notion that\n    \n\tthe symbols in our language are essentially\n\tnames for values.\n      \n\tBut as soon as we introduce set! and the\n\tidea that the value of a variable can change, a variable can no longer\n\tbe simply a name.  Now a variable somehow refers to a place where a\n\tvalue can be stored, and the value stored at this place can change. \n        \n    In section 3.2 we will see how\n    environments play this role of \"place\" in our computational\n    model.    \n  ","3.1.3#h1":"Sameness and change","3.1.3#p6":"\n    The issue surfacing here is more profound than the mere breakdown of a\n    particular model of computation.  As soon as we introduce change into\n    our computational models, many notions that were previously\n    straightforward become problematical.  Consider the concept of two\n    things being \"the same.\"","3.1.3#p7":"\n    Suppose we call \n    make-decrementer\n    twice with the same argument to create two\n    procedures:(define D1 (make-decrementer 25))\n\n(define D2 (make-decrementer 25)) \n    Are \n    D1\n    and \n    D2\n    the same?  An acceptable answer is yes, because \n    D1\n    and \n    D2\n    have the same computational behavior—each is a\n    procedure\n    that subtracts its input from 25.  In fact, \n    D1\n    could be substituted for \n    D2\n    in any computation without changing the result.\n  ","3.1.3#p8":"\n    Contrast this with making two calls to \n    make-simplified-withdraw:(define W1 (make-simplified-withdraw 25))\n\n(define W2 (make-simplified-withdraw 25)) \n    Are W1 and \n    W2 the same?  Surely not, because calls to \n    W1 and W2\n    have distinct effects, as shown by the following sequence of interactions:\n    (W1 20) (W1 20) (W2 20) \n    Even though W1 and \n    W2 are \"equal\" in the sense that\n    they are both created by evaluating the same expression, \n    (make-simplified-withdraw 25),\n      \n    it is not true that \n    W1 could be substituted for \n    W2 in any expression without changing the\n    result of evaluating the expression.\n  ","3.1.3#p9":"\n    A language that supports the concept that \"equals can be substituted\n    for equals\" in an expression without changing the value of the\n    expression is said to be\n    referentially transparent.  Referential transparency is violated\n    when we include\n    set!\n    in our computer language.  This makes it tricky to determine when we can\n    simplify expressions by substituting equivalent expressions.  Consequently,\n    reasoning about programs that use assignment becomes drastically more\n    difficult.\n  ","3.1.3#p10":"\n    Once we forgo referential transparency, the notion of what it means for\n    computational objects to be \"the same\" becomes difficult to\n    capture in a formal way.  Indeed, the meaning of \"same\" in the\n    real world that our programs model is hardly clear in itself.  In general,\n    we can determine that two apparently identical objects are indeed\n    \"the same one\" only by modifying one object and then observing\n    whether the other object has changed in the same way.  But how can we tell\n    if an object has \"changed\" other than by observing the\n    \"same\" object twice and seeing whether some property of the\n    object differs from one observation to the next?  Thus, we cannot determine\n    \"change\" without some a priori notion of\n    \"sameness,\" and we cannot determine sameness without observing\n    the effects of change.\n  ","3.1.3#p11":"\n    As an example of how this issue arises in programming, consider the\n    situation where Peter and Paul have a\n    \n    bank account with $100 in\n    it.  There is a substantial difference between modeling this as\n    (define peter-acc (make-account 100))\n(define paul-acc (make-account 100)) \n    and modeling it as\n    (define peter-acc (make-account 100))\n(define paul-acc peter-acc) In the first situation, the two bank accounts are distinct.\n    Transactions made by Peter will not affect Paul's account, and vice\n    versa.  In the second situation, however, we have defined \n    paul-acc\n    to be the same thing as \n    peter-acc.\n    In effect, Peter and Paul now have a joint bank account, and if Peter makes\n    a withdrawal from \n    peter-acc\n    Paul will observe less money in \n    paul-acc.\n    These two similar but distinct situations can cause confusion in building\n    computational models.  With the shared account, in particular, it can be\n    especially confusing that there is one object (the bank account) that has\n    two different names \n    \n\t(peter-acc and\n\tpaul-acc);\n      \n    if we are searching for all the places in our program where \n    paul-acc\n    can be changed, we must remember to look also at things that change \n    peter-acc.","3.1.3#footnote-link-2":"2","3.1.3#p12":"\n    With reference to the above remarks on \"sameness\" and\n    \"change,\" observe that if Peter and Paul could only examine\n    their bank balances, and could not perform operations that changed the\n    balance, then the issue of whether the two accounts are distinct would be\n    moot.  In general, so long as we never modify data objects, we can regard a\n    compound data object to be precisely the totality of its pieces.  For\n    example, a rational number is determined by giving its numerator and\n    its denominator.  But this view is no longer valid in the presence of\n    change, where a compound data object has an \"identity\" that is\n    something different from the pieces of which it is composed.  A bank\n    account is still \"the same\" bank account even if we change the\n    balance by making a withdrawal; conversely, we could have two\n    different bank accounts with the same state information.  This\n    complication is a consequence, not of our programming language, but of\n    \n    our perception of a bank account as an object.  We do not, for\n    example, ordinarily regard a rational number as a changeable object\n    with identity, such that we could change the numerator and still have\n    \"the same\" rational number.\n    ","3.1.3#h2":"Pitfalls of imperative programming","3.1.3#p13":"\n    In contrast to functional programming, programming that makes extensive use\n    of assignment is known as \n    imperative programming.  In addition to raising complications about\n    computational models, programs written in imperative style are susceptible\n    to bugs that cannot occur in functional programs.  For example, recall the\n    iterative factorial program from\n    \n\tsection 1.2.1:\n      (define (factorial n)\n  (define (iter product counter)\n    (if (> counter n)\n        product\n        (iter (* counter product)\n              (+ counter 1))))\n  (iter 1 1)) \n    Instead of passing arguments in the internal iterative loop, we could\n    adopt a more imperative style by using explicit assignment\n    to update the values of the variables product\n    and counter:\n    (define (factorial n)\n  (let ((product 1)\n        (counter 1))\n    (define (iter)\n      (if (> counter n)\n          product\n          (begin (set! product (* counter product))\n                 (set! counter (+ counter 1))\n                 (iter))))\n    (iter))) \n    This does not change the results produced by the program, but it does\n    introduce a subtle trap.  How do we decide the order of the assignments?\n    As it happens, the program is correct as written.  But writing the\n    assignments in the opposite order\n    (set! counter (+ counter 1))\n(set! product (* counter product))\n    would have produced a different,\n    \n    incorrect result. In general, programming\n    with assignment forces us to carefully consider the relative orders of the\n    assignments to make sure that each statement is using the correct version\n    of the variables that have been changed.  This issue simply does not arise\n    in functional programs.","3.1.3#footnote-link-3":"3","3.1.3#p14":"\n    The complexity of imperative programs becomes even worse if we consider\n    applications in which several processes execute concurrently.  We will\n    return to this in section 3.4.\n    First, however, we will address the issue of providing a computational\n    model for expressions that involve assignment, and explore the uses of\n    objects with local state in designing simulations.\n  ","3.1.3#ex-3.7":"\n    Consider the bank account objects created by \n    make-account,\n    with the password modification described in\n    exercise 3.3.  Suppose that our\n    banking system requires the ability to make\n    \n    joint accounts.  Define a\n    proceduremake-joint\n    that accomplishes this.  \n    Make-joint\n    should take three arguments.  The first is a password-protected account.\n    The second argument must match the password with which the account was\n    defined in order for the \n    make-joint\n    operation to proceed.  The third argument is a new password.  \n    Make-joint\n    is to create an additional access to the original account using the new\n    password.  For example, if \n    peter-acc\n    is a bank account with\n    password \n    open-sesame,\n    then\n    (define paul-acc\n  (make-joint peter-acc 'open-sesame 'rosebud)) \n    will allow one to make transactions on \n    peter-acc\n    using the name\n    paul-acc\n    and the password\n    rosebud.\n      \n    You may wish to modify your solution to\n    exercise 3.3 to accommodate this\n    new feature.\n    ","3.1.3#ex-3.8":"\n    When we defined the evaluation model in\n    section 1.1.3, we said that the\n    \n    first step in evaluating an expression is to evaluate its subexpressions.\n    But we never specified the order in which the subexpressions should be\n    evaluated (e.g., left to right or right to left).\n    \n\tWhen we introduce\n\tassignment, the order in which the arguments to a procedure\n\tare evaluated can make a difference to the result.\n      \n    Define a simple\n    proceduref such that evaluating\n    (+ (f 0) (f 1))\n    will return 0 if the\n    arguments to +\n    are evaluated from left to right but will return 1 if the\n    arguments\n    are evaluated from right to left.\n    ","3.1.3#footnote-1":"We don't substitute for the occurrence of\n    balance in the\n    set! expression\n    because the name in\n    a set!\n    is not evaluated. If we did substitute for it, we would get\n    (set! 25 (- 25 amount)),\n    which makes no sense.","3.1.3#footnote-2":"The\n    phenomenon of a single computational object being accessed by more than one\n    name is known as \n    aliasing.  The joint bank account situation illustrates a very\n    simple example of an alias.  In section 3.3\n    we will see much more complex examples, such as \"distinct\"\n    compound data structures that share parts.  Bugs can occur in our programs if\n    we forget that a change to an object may also, as a\n    \"side effect,\" change a \"different\" object because\n    the two \"different\" objects are actually a single object\n    appearing under different aliases.  These so-called side-effect\n    bugs are so difficult to locate and to analyze that some people have\n    proposed that programming languages be designed in such a way as to not\n    allow side effects or aliasing\n    \n    (Lampson et al. 1981; \n    Morris, Schmidt, and Wadler 1980).","3.1.3#footnote-3":"In view of this, it is ironic that\n    introductory programming is most often taught in a highly imperative style.\n    This may be a vestige of a belief, common throughout the 1960s and 1970s,\n    that programs that call\n    procedures\n    must inherently be less efficient than programs that perform assignments.  \n    (Steele (1977)\n    debunks this argument.)  Alternatively it may reflect a view that\n    step-by-step assignment is easier for beginners to visualize than\n    procedure\n    call. Whatever the reason, it often saddles beginning programmers with\n    \"should I set this variable before or after that one\" concerns\n    that can complicate programming and obscure the important ideas.","3.2":"3.2  The Environment Model of Evaluation","3.2#p1":"\n    When we introduced compound\n    procedures\n    in chapter 1, we used the\n    \n    substitution model of evaluation\n    (section 1.1.5) to define what is\n    meant by applying a\n    procedure\n    to arguments:\n    To apply a compound\n      procedure\n      to arguments, evaluate the \n      body of the procedure\n      with each\n      \n\t  formal\n\t\n      parameter replaced by the corresponding\n      argument.\n      ","3.2#p2":"\n    Once we admit assignment into our programming language, such a\n    definition is no longer adequate.  In particular,\n    section 3.1.3 argued that, in the\n    presence of assignment,\n    \n\ta variable cannot be considered to be merely a name for \n\ta value.  Rather, a variable must somehow designate a\n\t\"place\" in which values can be stored.\n      \n    In our new model of\n    evaluation, these places will be maintained in structures called \n    environments.\n  ","3.2#p3":"\n    An environment is a sequence of \n    frames.  Each frame is a table (possibly empty) of \n    bindings, which associate\n    \n\tvariable names\n      \n    with their corresponding\n    values.\n    \n\t(A single frame may contain at most one binding for any variable.)\n      \n    Each frame also has a pointer to its \n    enclosing environment, unless, for the purposes of discussion, the\n    frame is considered to be \n    global.  The \n    value of a variable\n    with respect to an environment is the value given by the binding of\n    the\n    \n\tvariable\n      \n    in the first frame in the environment that contains a\n    binding for that\n    \n\tvariable.\n      \n    If no frame in the sequence specifies a\n    binding for the\n    \n\tvariable,\n      \n    then the\n    \n\tvariable\n      \n    is said to be \n    unbound in the environment.\n    ","3.2#fig-":"","3.2#p4":"\n\tFigure \n    shows a simple environment\n    structure consisting of three frames, labeled I, II, and III.  In the\n    diagram, A, B, C, and D are pointers to environments.  C and D point\n    to the same environment.  The\n    \n\tvariables\n      z and\n    x are bound in frame II, while\n    y and x are bound\n    in frame I.  The value of x in environment D\n    is 3.  The value of x with respect to\n    environment B is also 3.  This is determined as follows: We examine the\n    first frame in the sequence (frame III) and do not find a binding for\n    x, so we proceed to the enclosing environment\n    D and find the binding in frame I.  On the other hand, the value of\n    x in environment A is 7, because the first\n    frame in the sequence (frame II) contains a binding of\n    x to 7.  With respect to environment A, the\n    binding of x to 7 in frame II is said to \n    shadow the binding of x to 3 in\n    frame I.\n  ","3.2#p5":"\n    The environment is crucial to the evaluation process, because it determines\n    the context in which an expression should be evaluated.  Indeed, one could\n    say that expressions in a programming language do not, in themselves, have\n    any meaning.  Rather, an expression acquires a meaning only with respect to\n    some environment in which it is evaluated.  \n    \n\tEven the interpretation of an expression as straightforward as\n\t(+ 1 1) depends on an\n\tunderstanding that one is operating in a context in which\n\t+ is the symbol for addition. \n      \n    Thus, in our model of evaluation we will always speak of evaluating an\n    expression with respect to some environment.  To describe interactions with\n    the\tinterpreter, we will suppose that there is a \n    \n    global environment, consisting of a single frame (with no enclosing\n    environment) that includes values for the\n    \n\tsymbols\n      \n    associated with the\n    primitive\n    \n\tprocedures.\n      \n    For example, the idea that\n    + is the symbol for addition is captured\n\tby saying that the symbol +\n    is bound in the global environment to the primitive\n    addition procedure.","3.2.1":"3.2.1  \n    The Rules for Evaluation","3.2.1#p1":"\n    The overall specification of how the interpreter\n    \n    evaluates a\n    \n\tcombination\n      \n    remains the same as when we first introduced it in\n    section 1.1.4:\n    \n        To evaluate\n\t\n\t    a combination:\n\t  \n\t    Evaluate the subexpressions\n\t    of the\n\t    \n\t\tcombination.\n\t    Apply the value of the\n\t    \n\t\toperator\n\t      \n\t    subexpression\n\t    to the values of the\n\t    \n\t\toperand\n\t      \n\t    subexpressions.\n\t  \n    The environment model of evaluation replaces the substitution model in\n    specifying what it means to apply a compound\n    procedure\n    to arguments.\n  ","3.2.1#footnote-link-1":"1","3.2.1#p2":"\n    In the environment model of evaluation, a\n    procedure\n    is always a pair consisting of some code and a pointer to an environment. \n    Procedures\n    are created in one way only: by evaluating a\n    lambda\t\n    expression.\n    \n    This produces a\n    procedure\n    whose code is obtained from the text of the\n    lambda\t\n    expression and whose environment is the environment in which the\n    lambda\t\n    expression was evaluated to produce the\n    procedure.\n    For example, consider the\n    procedure definition(define (square x)\n  (* x x)) \n    evaluated in the\n    \n\tglobal\n      \n    environment.  The\n    procedure definition\n    syntax is\n    just syntactic sugar for\n    an underlying implicit\n    lambda\n    expression.  It would have been equivalent to have used(define square\n  (lambda (x) (* x x))) \n    which evaluates\n    (lambda (x) (* x x))\n    and binds square to the resulting value, all\n    in the\n    \n\tglobal\n      \n    environment.\n  ","3.2.1#footnote-link-2":"2","3.2.1#p3":"\n\tFigure \n    shows the result of evaluating this\n    define expression.\n    The\n    procedure\n    object is a pair whose code specifies that the\n    procedure\n    has one\n    formal\n    parameter, namely x, and a\n    procedure\n    body\n    (* x x).\n    The environment part of the\n    procedure\n    is a pointer to the program environment, since that is the environment in\n    which the\n    lambda\n    expression was evaluated to produce the\n    procedure.\n    A new binding, which associates the\n    procedure\n    object with the\n    \n\tsymbol\n      square, has been added\n    to the program frame. \n    \n        In general, define creates definitions by\n\tadding bindings to frames.\n      ","3.2.1#fig-":"","3.2.1#p4":"\n    Now that we have seen how\n    procedures\n    are created, we can describe how\n    procedures\n    are applied.  The environment model specifies: To apply a\n    procedure\n    to arguments, create a new environment containing a frame that binds the\n    parameters to the values of the arguments.  The enclosing environment of\n    this frame is the environment specified by the\n    procedure.\n    Now, within this new environment, evaluate the\n    procedure\n    body.\n  ","3.2.1#p5":"\n    To show how this rule is followed,\n    \n\tfigure \n    illustrates the environment structure created by evaluating the\n    expression\n    (square 5)\n    in the\n    global\n    environment, where square is the\n    procedure\n    generated in\n    \n\tfigure .\n      \n    Applying the\n    procedure\n    results in the creation of a new environment, labeled E1 in the figure, that\n    begins with a frame in which x, the\n    formal\n    parameter for the\n    procedure,\n    is bound to the argument 5.\n    \n    The pointer leading upward from this frame shows that the\n    frame's enclosing environment is the\n    global\n    environment.  The\n    global\n    environment is chosen here, because this is the environment that is\n    indicated as part of the squareprocedure\n    object.  Within E1, we evaluate the body of the\n    procedure,(* x x).\n    Since the value of x in E1 is 5, the result is\n    (* 5 5),\n    or 25.\n    ","3.2.1#p6":"\n    The environment model of\n    procedure\n    application can be summarized by two\n    rules:\n    \n\tA\n\tprocedure\n\tobject is applied to a set of arguments by constructing a frame, \n\tbinding the formal parameters of the procedure\n\tto the arguments of the call, and then evaluating the body of the\n\tprocedure\n\tin the context of the new environment constructed.  The new frame has as\n\tits enclosing environment the environment part of the\n\tprocedure\n\tobject being applied.\n\tThe result of the application is the result of evaluating\n\tthe return expression of the first return statement encountered\n\twhile evaluating the function body.\n      \n\tA\n\tprocedure\n\tis created by evaluating a \n\tlambda\n\texpression relative to a given environment.  The resulting\n\tprocedure\n\tobject is a pair consisting of the text of the\n\tlambda\n\texpression and a pointer to the environment in which the\n\tprocedure\n\twas created.\n      ","3.2.1#p7":"\n\tWe also specify that defining a symbol using\n\tdefine\n\tcreates a binding in the current environment frame and assigns to the symbol\n\tthe indicated value.\n      \n\tEvaluating the expression\n\t(set! variable value)\n\tin some environment locates the binding of the variable in the\n\tenvironment. For this, one finds the first frame in the environment that\n\tcontains a binding for the variable and modifies that frame. If the\n\tvariable is unbound in the environment, then\n\tset! signals an error.\n      ","3.2.1#p8":"\n    These evaluation rules, though considerably more complex than the\n    substitution model, are still reasonably straightforward.  Moreover,\n    the evaluation model, though abstract, provides a correct description\n    of how the interpreter evaluates expressions.  In chapter 4 we shall\n    see how this model can serve as a blueprint for implementing a working\n    interpreter.  The following sections elaborate the details of the\n    model by analyzing some illustrative programs.\n    ","3.2.1#footnote-1":"Assignment introduces a subtlety into step\n\t\t1 of the evaluation rule.  As shown in\n\t\texercise 3.8, the\n\t\tpresence of assignment allows us to write expressions that will\n\t\tproduce different values depending on the\n\t\t\n\t\torder in which the subexpressions in a combination\n\t\tare evaluated. Thus, to be precise, we should specify an\n\t\tevaluation order in step 1 (e.g., left to right or right to\n\t\tleft). However, this order should always be considered to be an\n\t\timplementation detail, and one should never write programs that\n\t\tdepend on some particular order. For instance, a sophisticated\n\t\tcompiler might optimize a program by varying the order in which\n\t\tsubexpressions are evaluated. The ECMAScript standard specifies\n\t\tevaluation of subexpressions from left to right.","3.2.1#footnote-2":"\n\tFootnote 2 in chapter 1\n\tmentions subtle differences between the two in full JavaScript, which\n\twe will ignore in this book.","3.2.2":"3.2.2  \n    Applying Simple","3.2.2#p1":"\n    When we introduced the substitution model in\n    section 1.1.5 we showed how the\n    combination (f 5)\n    evaluates to 136, given the following\n    procedure definitions:(define (square x)\n  (* x x))\n\n(define (sum-of-squares x y)\n  (+ (square x) (square y)))\n\n(define (f a)\n  (sum-of-squares (+ a 1) (* a 2))) \n    We can analyze the same example using the environment model.\n    Figure 3.4 shows the three\n    procedure\n    objects created by evaluating the definitions of\n    f, square, and\n    sum-of-squares\n    in the\n    global\n    environment.  Each\n    procedure\n    object consists of some code, together with a pointer to the\n    global\n    environment.\n    ","3.2.2#fig-":"","3.2.2#p2":"\n\n    In\n    figure \n    we see the environment structure created by evaluating the expression\n    (f 5).\n    The call to f creates a new environment, E1,\n    beginning with a frame in which a, the\n    formal parameter of\n    f, is bound to the argument 5.  In E1, we\n    evaluate the body of f:\n    (sum-of-squares (+ a 1) (* a 2))To evaluate\n    \n    this combination,\n    we first evaluate the subexpressions.\n      \n    The first subexpression,\n    sum-of-squares,\n    has a value that is a\n    procedure\n    object.  (Notice how this value is found: We first look in the first frame\n    of E1, which contains no binding for\n    sum-of-squares.\n    Then we proceed to the enclosing environment, i.e., the\n    global\n    environment, and find the binding shown in\n    \n        figure .)\n      \n    The other two subexpressions are evaluated by applying the primitive\n    operations + and *\n    to evaluate the two combinations\n    (+ a 1)\n    and\n    (* a 2)\n    to obtain 6 and 10, respectively.\n  ","3.2.2#p3":"\n    Now we apply the\n    procedure\n    object\n    sum-of-squares\n    to the arguments 6 and 10.  This results in a new environment, E2, in which\n    the formal parameters\n    x and y are bound\n    to the arguments. Within E2 we evaluate \n    \n\tthe combination\t\n\t(+ (square x) (square y)).\n      \n    This leads us to evaluate\n    (square x),\n    where square is found in the\n    global\n    frame and x is 6.  Once again, we set up a\n    new environment, E3, in which x is bound to 6,\n    and within this we evaluate the body of square,\n    which is\n    (* x x).\n    Also as part of applying\n    sum-of-squares,\n    we must evaluate the subexpression\n    (square y),\n    where y is 10.  This second call to\n    square creates another environment, E4, in\n    which x, the\n    formal parameter of\n    square, is bound to 10.  And within E4 we must\n    evaluate\n    (* x x).","3.2.2#p4":"\n    The important point to observe is that each call to\n    square creates a new environment containing a\n    binding for x.  We can see here how the\n    different frames serve to keep separate the different local variables all\n    named x.  Notice that each frame created by\n    square points to the\n    global\n    environment, since this is the environment indicated by the\n    squareprocedure\n    object.\n  ","3.2.2#p5":"\n    After the subexpressions are evaluated, the results are returned.  The\n    values generated by the two calls to square are\n    added by\n    sum-of-squares,\n    and this result is returned by f.\n    Since our focus here is on the environment structures, we will not\n    dwell on how these returned values are passed from call to call;\n    however, this is also an important aspect of the evaluation process,\n    and we will return to it in detail in chapter 5.\n    ","3.2.2#ex-3.9":"\n    In section 1.2.1\n    we used the substitution model to analyze two\n    procedures\n    for computing\n    \n    factorials, a recursive version\n    (define (factorial n)\n  (if (= n 1)\n      1\n      (* n (factorial (- n 1))))) \n    and an iterative version\n    (define (factorial n)\n  (fact-iter 1 1 n))\n\n(define (fact-iter product counter max-count)\n  (if (> counter max-count)\n      product\n      (fact-iter (* counter product)\n                 (+ counter 1)\n                 max-count))) \n    Show the environment structures created by evaluating\n    (factorial 6)\n    using each version of the factorialprocedure.","3.2.2#footnote-link-1":"1","3.2.2#footnote-1":"\n    The environment model will not clarify our claim in\n    section 1.2.1 that the\n    interpreter can execute a\n    procedure\n    such as\n    fact-iter\n    in a constant amount of space using tail recursion.  We will discuss\n    \n    tail recursion when we\n    deal with the control structure of the interpreter in\n    section 5.4.","3.2.3":"3.2.3  \n    Frames as the Repository of Local State","3.2.3#p1":"\n    We can turn to the environment model to see how\n    procedures\n    and assignment can be used to represent objects with local state.  As an\n    example, consider the\n    \"withdrawal processor\" from\n    section 3.1.1 created by calling the\n    procedure(define (make-withdraw balance)\n  (lambda (amount)\n    (if (>= balance amount)\n        (begin (set! balance (- balance amount))\n               balance)\n        \"Insufficient funds\"))) \n    Let us describe the evaluation of\n    (define W1 (make-withdraw 100)) \n    followed by\n    (W1 50) \n\tFigure \n    shows the result of\n    defining the\n      make-withdrawprocedure\n    in the\n    global\n    environment.  This produces a\n    procedure\n    object that contains a pointer to the\n    global\n    environment. So far, this is no different from the examples we have already\n    seen, except that\n    \n\tthe body of the\n\tprocedure is itself a\n\tlambda\n\texpression.\n      ","3.2.3#fig-":"","3.2.3#p2":"\n    The interesting part of the computation happens when we apply the \n    proceduremake-withdraw\n    to an argument:\n    (define W1 (make-withdraw 100)) \n    We begin, as usual, by setting up an environment E1 in which the\n    formal parameter\n    balance is bound to the argument 100.  Within\n    this environment, we evaluate the body of\n    make-withdraw,\n    namely the\n    lambda expression. This\n    constructs a new\n    procedure\n    object, whose code is as specified by the\n    lambda\n    and whose environment is E1, the environment in which the\n    lambda\n    was evaluated to produce the\n    procedure.\n    The resulting\n    procedure\n    object is the value returned by the call to\n    make-withdraw.\n    This is bound to W1 in the\n    global\n    environment, since the\n    define\n    itself is being evaluated in the\n    global\n    environment.\n    \n\tFigure \n    shows the resulting environment structure.\n    ","3.2.3#p3":"\n    Now we can analyze what happens when W1\n    is applied to an argument:\n    (W1 50)\n    We begin by constructing a frame in which\n    amount, the\n    formal parameter of\n    W1, is bound to the argument 50.  The crucial\n    point to observe is that this frame has as its enclosing environment not the\n    global\n    environment, but rather the environment E1, because this is the\n    environment that is specified by the W1procedure\n    object. Within this new environment, we evaluate the body of the\n    procedure:(if (>= balance amount)\n    (begin (set! balance (- balance amount))\n           balance)\n    \"Insufficient funds\")\n    The resulting environment structure is shown in\n    \n\tfigure .\n      \n    The expression being evaluated references\n    both amount and\n    balance.\n    Amount\n    will be found in the first frame in the environment, and\n    balance will be found by following the\n    enclosing-environment pointer to E1.\n    ","3.2.3#p4":"\n    When the\n    set!\n    is executed, the binding of balance in E1 is\n    changed.  At the completion of the call to\n    W1, balance is 50,\n    and the frame that contains balance is still\n    pointed to by the\n    procedure\n    object W1.  The frame that binds\n    amount (in which we executed the code that\n    changed balance) is no longer relevant, since\n    the\n    procedure\n    call that constructed it has terminated, and there are no pointers to that\n    frame from other parts of the environment.  The next time\n    W1 is called, this will build a new frame that\n    binds amount and whose enclosing environment is\n    E1. We see that E1 serves as the \"place\" that holds the local\n    state variable for the\n    procedure\n    object W1.\n    \n\tFigure \n    shows the situation after the call to W1.\n    ","3.2.3#p5":"\n    Observe what happens when we create a second \"withdraw\" object\n    by making another call to\n    make_withdraw:(define W2 (make-withdraw 100)) \n    This produces the environment structure of\n    \n\tfigure ,\n      \n    which shows\n    that W2 is a\n    procedure\n    object, that is, a pair with some code and an environment.  The environment\n    E2 for W2 was created by the call to\n    make-withdraw.\n    It contains a frame with its own local binding for\n    balance.  On the other hand,\n    W1 and W2 have the\n    same code: the code specified by the\n    lambda\n    expression in the body of\n    make-withdraw. We see\n    here why W1 and W2\n    behave as independent objects.  Calls to\n    W1 reference the state variable\n    balance stored in E1, whereas calls to \n    W2 reference the\n    balance stored in E2. Thus, changes to the\n    local state of one object do not affect the other object.\n    ","3.2.3#footnote-link-1":"1","3.2.3#ex-3.10":"\n    In the\n    make-withdrawprocedure,\n    the local variable balance is created as a\n    parameter of\n    make-withdraw.\n    We could also create the local state variable\n    \n\texplicitly,\n      \n    using \n    let,\n    as follows:\n    (define (make-withdraw initial-amount)\n  (let ((balance initial-amount))\n        (lambda (amount)\n    (if (>= balance amount)\n        (begin (set! balance (- balance amount))\n               balance)\n        \"Insufficient funds\")))) \n        Recall from section 1.3.2 that \n\tlet is simply syntactic sugar for a\n        procedure\n        call:\n        (let ((var exp)) body)\n        is interpreted as an alternate syntax for\n        ((lambda (var) body) exp)\n    Use the environment model to analyze this alternate version of\n    make_withdraw, drawing figures like the ones\n    above to illustrate the interactions\n    (define W1 (make-withdraw 100))\n\n(W1 50)\n\n(define W2 (make-withdraw 100)) \n    Show that the two versions of\n    make-withdraw\n    create objects with the same behavior.  How do the environment structures\n    differ for the two versions?\n    ","3.2.3#footnote-1":"Whether\n    W1 and W2 share\n    the same physical code stored in the computer, or whether they each keep a\n    copy of the code, is a detail of the implementation.  For the interpreter we\n    implement in chapter 4, the code is in fact shared.","3.2.4":"3.2.4  \n    Internal","3.2.4#p1":"\n    Section 1.1.8 introduced the idea that\n    procedures\n    can have internal\n    \n\tdefinitions,\n      \n    thus leading to a block structure as in the\n    \n    following\n    procedure\n    to compute square roots:\n    (define (sqrt x)\n  (define (good-enough? guess)\n    (< (abs (- (square guess) x)) 0.001))\n  (define (improve guess)\n    (average guess (/ x guess)))\n  (define (sqrt-iter guess)\n    (if (good-enough? guess)\n        guess\n        (sqrt-iter (improve guess))))\n  (sqrt-iter 1.0)) \n    Now we can use the environment model to see why these internal\n    \n\tdefinitions\n      \n    behave as desired.\n    \n    Figure \n    shows the point in the evaluation of the expression\n    (sqrt 2)\n    where the internal\n    proceduregood-enough?\n    has been called for the first time with\n    guess equal to 1.\n    ","3.2.4#fig-":"","3.2.4#p2":"\n    Observe the structure of the environment.\n    Sqrt is a\n\tsymbol in the\n\tglobal\n\tenvironment that is bound\n      \n    to a\n    procedure\n    object whose associated environment is the\n    global\n    environment.  When sqrt was called, a new\n    environment, E1, was formed, subordinate to the\n    global\n    environment, in which the parameter x is bound\n    to 2. The body of sqrt was then\n    evaluated in E1. \n    \n\tSince the first expression\n\tin the body of sqrt is\n      (define (good-enough? guess)\n  (< (abs (- (square guess) x)) 0.001))\n\tevaluating this expression defined the procedure\n\tgood-enough?\n\tin the environment E1.\n      \n\tTo be more precise, the symbol \n\tgood-enough? was added to the first frame\n\tof E1, bound to a procedure\n\tobject whose associated environment is E1.\n      \n    Similarly,\n    improve and \n    sqrt-iter\n    were defined as\n    procedures in E1.\n    For conciseness,\n    \n\tfigure \n    shows only the\n    procedure\n    object for \n    good-enough?.","3.2.4#p3":"\n    After the local\n    procedures\n    were defined, the expression\n    (sqrt-iter 1.0)\n    was evaluated, still in environment\n    E1.\n    So the\n    procedure\n    object bound to \n    sqrt-iter\n      in E1 was called with 1 as an argument.  This created an environment E2\n      in which\n      guess, the parameter of \n    sqrt-iter,\n    is bound to 1.  \n    sqrt-iter\n    in turn called \n    good-enough?\n    with the value of guess\n\t(from E2) as the argument for \n\tgood-enough?.\n      \n    This set up another environment,\n    \n\tE3, in which\n\tguess (the parameter of \n\tgood-enough?)\n      \n    is bound to 1. Although \n    sqrt-iter\n    and \n    good-enough?\n    both have a parameter named guess, these are two\n    distinct local variables located in different frames.\n    \n\tAlso, E2 and E3 both have E1 as their enclosing environment, because the \n\tsqrt-iter and \n\tgood-enough? procedures\n\tboth have E1 as their environment part.\n      \n    One consequence of this is that the\n    \n\tsymbol\n      x that appears in the body of \n    good-enough?\n    will reference the binding of x that appears in\n    E1, namely the value of x with which the\n    original sqrtprocedure\n    was called.\n    ","3.2.4#p4":"\n    The environment model thus explains the two key properties that make local\n    procedure definitions\n    a useful technique for modularizing programs:\n    \n\tThe names of the local\n\tprocedures\n\tdo not interfere with\n\tnames external to the enclosing\n\tprocedure,\n\tbecause the local\n\tprocedure\n\tnames will be bound in the frame that the\n\tprocedure creates when it is run,\n\trather than being bound in the\n\tglobal\n\tenvironment.\n      \n\tThe local\n\tprocedures\n\tcan access the arguments of the enclosing\n\tprocedure,\n\tsimply by using parameter names as free\n\tvariables.\n\tThis is\n\tbecause the body of the local\n\tprocedure\n\tis evaluated in an environment that is subordinate to the\n\tevaluation environment for the enclosing\n\tprocedure.","3.2.4#ex-3.11":"\n    In section 3.2.3 we saw how the\n    environment model described the behavior of\n    procedures\n    with local state.  Now we have seen how internal\n    \n\tdefinitions\n      \n    work.\n    \n    A typical message-passing\n    procedure\n    contains both of these aspects.  Consider the\n    \n    bank account\n    procedure\n    of section 3.1.1:\n    (define (make-account balance)\n  (define (withdraw amount)\n    (if (>= balance amount)\n        (begin (set! balance (- balance amount))\n               balance)\n        \"Insufficient funds\"))\n  (define (deposit amount)\n    (set! balance (+ balance amount))\n    balance)\n  (define (dispatch m)\n    (cond ((eq? m 'withdraw) withdraw)\n          ((eq? m 'deposit) deposit)\n          (else (error \"Unknown request - - MAKE-ACCOUNT\"\n                       m))))\n  dispatch) \n    Show the environment structure generated by the sequence of\n    interactions\n    (define acc (make-account 50))((acc 'deposit) 40)((acc 'withdraw) 60)\n    Where is the local state for acc kept?\n    Suppose we define another account\n    (define acc2 (make-account 100))\n    How are the local states for the two accounts kept distinct?  Which parts\n    of the environment structure are shared between\n    acc and acc2?\n    ","3.2.4#h1":"More about blocks","3.2.4#p5":"\n    As we saw, the scope of the names declared in\n    sqrt is the whole body of\n    sqrt. This explains why\n    mutual recursion works, as in this (quite\n    wasteful) way of checking whether a nonnegative\n    integer is even.\n    \n      \n      \n    \n    At the time when \n    is_even is called during a call to\n    f, the environment diagram looks\n    like the one in figure 3.11 when\n    sqrt_iter is called. The functions\n    is_even and\n    is_odd are bound in E2 to function objects\n    that point to E2 as the environment in which to evaluate calls to those\n    functions. Thus\n    is_odd in the body of\n    is_even refers to the right function.\n    Although\n    is_odd\n    is defined after\n    is_even,\n    this is no different from how in the body of\n    sqrt_iter\n    the name\n    improve\n    and the name\n    sqrt_iter\n    itself refer to the right functions.\n  ","3.2.4#p6":"\n    Equipped with a way to handle declarations within blocks, we can\n    revisit declarations of names at the top level. In\n    section 3.2.1, we saw\n    that the names declared at the top level are added to the program\n    frame. A better explanation is that the whole program is placed in\n    an implicit block, which is evaluated in the global environment.\n    The treatment of blocks described above then handles the top\n    level:\n    The global environment is extended by a frame that contains the\n    bindings of all names declared in the implicit block. That frame is\n    the program frame and the resulting\n    environment is the\n    \n    program environment.\n  ","3.2.4#p7":"\n    We said that a block's body is evaluated in an environment that\n    contains all names declared directly in the body of the block.\n    A locally declared name is put into the environment when the block is\n    entered, but without an associated value. The evaluation of its\n    declaration during evaluation of the block body then assigns to the\n    name the result of evaluating the expression to the right of the\n    =, as if the declaration were\n    an assignment. Since the addition of the name to the environment is\n    separate from the evaluation of the declaration, and the whole block\n    is in the scope of the name, an erroneous program could attempt to\n    \n    access the value of a name before its declaration is evaluated;\n    the evaluation of an unassigned name signals an error.","3.2.4#footnote-link-1":"1","3.2.4#footnote-1":"\n    This explains why the program in\n    footnote 4 of chapter 1 goes wrong.\n    The time between creating the binding for a name and evaluating\n    the declaration of the name is called the\n    temporal dead zone (TDZ).","3.2.5":"3.2.5  \n    CSE Machine","3.3":"3.3  Modeling with Mutable Data","3.3#p1":"\n    Chapter 2 dealt with compound data as a means for constructing\n    computational objects that have several parts, in order to model\n    real-world objects that have several aspects.  In that chapter we\n    introduced the discipline of data abstraction, according to which data\n    structures are specified in terms of constructors, which create data\n    objects, and selectors, which access the parts of compound data\n    objects.  But we now know that there is another aspect of data that\n    chapter 2 did not address.  The desire to model systems composed of\n    objects that have changing state leads us to the need to modify\n    compound data objects, as well as to construct and select from them.\n    In order to model compound objects with changing state, we will design\n    data abstractions to include, in addition to selectors and\n    constructors, operations called \n    mutators, which modify data\n    objects.  For instance, modeling a banking system requires us to\n    change account balances.  Thus, a data structure for representing bank\n    accounts might admit an operation\n\n    \n(set-balance! $account$ $new$-$value$)\n      \n    that changes the balance of the designated account to the designated\n    new value.  Data objects for which mutators are defined are known as\n    mutable data objects.\n  ","3.3#p2":"\n    Chapter 2 introduced pairs as a general-purpose \"glue\"\n    for synthesizing compound data.  We begin this section by defining basic\n    mutators for pairs, so that pairs can serve as building blocks for\n    constructing mutable data objects.  These mutators greatly enhance the\n    representational power of pairs, enabling us to build data structures\n    other than the sequences and trees that we worked with in\n    section 2.2.  We also present some\n    examples of simulations in which complex systems are modeled as collections\n    of objects with local state.\n  ","3.3.1":"3.3.1  \n    Mutable List Structure","3.3.1#p1":"\n    The basic operations on\n    \n\tpairs—cons,\n      car,\n    and\n    cdr—can\n    be used to construct list structure and to select parts\n    from list structure, but they are incapable of modifying list\n    structure.  The same is true of the list operations we have used so\n    far, such as append and\n    list, since these can be defined in terms of\n    cons,car,\n    and\n    cdr.\n    To modify list structures we need new operations.\n  ","3.3.1#p2":"\n    The primitive mutators for pairs are\n    set-car!\n    and\n    set-cdr!.Set-car!\n    takes two arguments, the first of which must be a pair.  It modifies this\n    pair, replacing the\n    car\n    pointer by a pointer to the second argument of\n    set-car!.","3.3.1#footnote-link-1":"1","3.3.1#p3":"\n    As an example, suppose that x is bound to \n    \n\tthe list\n\t((a b) c d)\n    and y to\n    \n\tthe list\n\t(e f)\n    as illustrated in\n    \n        figure .\n      \n    Evaluating the expression\n    (set-car! x y)\n    modifies the pair to which x is bound,\n    replacing its\n    car\n    by the value of y.  The result of the operation\n    is shown in\n    \n        figure .\n      \n    The structure x has been modified and\n    \n\twould now be printed as\n\t((e f) c d).\n      \n    The pairs representing the list\n    (a b),\n    identified by the pointer that was replaced, are now detached from the\n    original structure.","3.3.1#footnote-link-2":"2","3.3.1#fig-":"","3.3.1#p4":"\n    Compare\n    \n        figure \n    with\n    \n        figure ,\n      \n    which illustrates the result of executing\n    (define z (cons y (cdr x))) \n    with x and y\n    bound to the original lists of\n    \n        figure .\n      \n    The\n    \n\tvariable\n      z is now bound to a\n    new pair created by the\n    cons\n    operation; the list to which x is bound is\n    unchanged.\n  ","3.3.1#p5":"\n    The\n    set-cdr!\n    operation is similar to\n    set-car!.\n    The only difference is that the\n    cdr\n    pointer of the pair, rather than the\n    car\n    pointer, is replaced.  The effect of executing\n    (set-cdr! x y)\n    on the lists of\n    \n        figure \n    is shown in\n    \n        figure .\n      \n    Here the\n    cdr\n    pointer of\n    x has been replaced by the pointer to\n    (e f).\n    Also, the list\n    (c d),\n    which used to be the\n    cdr\n    of x, is now detached from the structure.\n  ","3.3.1#p6":"Cons\n    builds new list structure by creating new pairs, \n    while set-car!\n    and\n    set-cdr!\n    modify existing pairs.\n    Indeed, we could\n    \n    implement\n    cons\n    in terms of the two mutators, together with a\n    procedureget-new-pair,\n    which returns a new pair that is not part of any existing list structure.\n    We obtain the new pair, set its\n    car\n    and\n    cdr\n    pointers to the designated objects, and return the new pair as the result of\n    the\n    cons.(define (cons x y)\n  (let ((new (get-new-pair)))\n    (set-car! new x)\n    (set-cdr! new y)\n    new)) ","3.3.1#footnote-link-3":"3","3.3.1#ex-3.12":"\n    The following\n    procedure\n    for appending lists was introduced in\n    section 2.2.1:\n    (define (append x y)\n  (if (null? x)\n    y\n    (cons (car x) (append (cdr x) y)))) Append\n    forms a new list by successively\n    consing\n\tthe elements of x onto\n\ty.\n      \n    The\n    procedureappend!\n    is similar to append, but it is a mutator\n    rather than a constructor. It appends the lists by splicing them together,\n    modifying the final pair of x so that its\n    cdr\n    is now y. (It is an error to call\n    append!\n    with an empty x.)\n    (define (append! x y)\n  (set-cdr! (last-pair x) y)\n            x) \n    Here\n    last-pair\n    is a\n    procedure\n    that returns the last pair in its argument:\n    (define (last-pair x)\n  (if (null? (cdr x))\n      x\n      (last-pair (cdr x)))) \n    Consider the interaction\n    (define x (list 'a 'b)) (define y (list 'c 'd)) (define z (append x y)) z (cdr x) (define w (append! x y)) w (cdr x) \n    What are the missing responses?\n    Draw box-and-pointer diagrams to explain your answer.\n    ","3.3.1#ex-3.13":"\n    Consider the following\n    make-cycleprocedure,\n    which uses the\n    last-pairprocedure\n    defined in exercise 3.12:\n    (define (make-cycle x)\n  (set-cdr! (last-pair x) x)\n            x) \n    Draw a box-and-pointer diagram that shows the structure\n    z created by\n    (define z (make-cycle (list 'a 'b 'c))) \n    What happens if we try to compute\n    (last-pair z)?","3.3.1#ex-3.14":"\n    The following\n    procedure\n    is quite useful, although obscure:\n    (define (mystery x)\n  (define (loop x y)\n    (if (null? x)\n        y\n        (let ((temp (cdr x)))\n          (set-cdr! x y)\n          (loop temp x))))\n  (loop x '())) Loop\n    uses the \"temporary\"\n\tvariable\n      temp\n    to hold the old value of the\n    cdr\n    of x, since the\n    set-cdr!\n    on the next line destroys the\n    cdr.\n    Explain what mystery does in general.  Suppose\n    v is defined by\n    (define v (list 'a 'b 'c 'd)) \n    Draw the box-and-pointer diagram that represents the list to which\n    v is bound.  Suppose that we now evaluate\n    (define w (mystery v)) \n    Draw box-and-pointer diagrams that show the structures\n    v and w after\n    evaluating this\n    expression.\n    What would be printed as the values of v\n    and w?\n    ","3.3.1#h1":"Sharing and identity","3.3.1#p7":"\n    We mentioned in section 3.1.3 the\n    theoretical issues of\n    \"sameness\" and \"change\"\n    raised by the introduction of assignment.  These issues arise in practice\n    when individual pairs are shared among different data objects.\n    For example, consider the structure formed by\n    (define x (list 'a 'b))\n(define z1 (cons x x)) \n    As shown in\n    \n        figure ,\n      z1 is a pair whose\n    car\n    and\n    cdr\n    both point to the same pair x.  This sharing\n    of x by the\n    car\n    and\n    cdr\n    of z1 is a consequence of the straightforward\n    way in which\n    cons\n    is implemented.  In general, using\n    cons\n    to construct lists will result in an interlinked structure of pairs in\n    which many individual pairs are shared by many different structures.\n    ","3.3.1#p8":"\n    In contrast to\n    \n\tfigure ,\n\tfigure \n    shows\n    the structure created by\n    (define z2 (cons (list 'a 'b) (list 'a 'b))) \n    In this structure, the pairs in the two\n    (a b)\n    lists are distinct, although\n    \n\tthe actual symbols are shared.\n\t","3.3.1#footnote-link-4":"4","3.3.1#p9":"\n    When thought of as a list, z1 and\n    z2 both represent \"the same\" list:\n    ((a b) a b) \n    In general, sharing is completely undetectable if we operate on lists using\n    only\n    cons,car,\n    and\n    cdr.\n    However, if we allow mutators on list structure, sharing becomes\n    significant.  As an example of the difference that sharing can make,\n    consider the following\n    procedure,\n    which modifies the\n    car\n    of the structure to which it is applied:\n    (define (set-to-wow! x)\n  (set-car! (car x) 'wow)\n  x) \n    Even though z1 and\n    z2 are \"the same\" structure,\n    applying\n    set-to-wow!\n    to them yields different results.  With z1,\n    altering the\n    car\n    also changes the\n    cdr,\n    because in z1 the\n    car\n    and the\n    cdr\n    are the same pair.  With z2, the\n    car\n    and\n    cdr\n    are distinct, so\n    set-to-wow!\n    modifies only the\n    car:z1 (set-to-wow! z1) z2 (set-to-wow! z2) ","3.3.1#p10":"\n\tOne way to detect sharing in list structures is to use the predicate \n\teq?, which we introduced in\n\tsection  as a way to test whether two \n\tsymbols are equal. More generally, (eq? x y)\n\ttests whether x and\n\ty are the same object (that is, whether\n\tx and y\n\tare equal as pointers).\n      \n    Thus, with z1 and\n    z2 as defined in\n    \n\tfigure \n\tand ,\n      (eq? (car z1) (cdr z1))\n    is true and\n    (eq? (car z2) (cdr z2))\n    is false.\n  ","3.3.1#p11":"\n    As will be seen in the following sections, we can exploit sharing to\n    greatly extend the repertoire of data structures that can be\n    represented by pairs.  On the other hand, sharing can also be\n    \n    dangerous, since modifications made to structures will also affect\n    other structures that happen to share the modified parts.  The mutation\n    operations\n    set-car!\n    and\n    set-cdr!\n    should be used with care; unless we have a good understanding of how our\n    data objects are shared, mutation can have unanticipated\n    results.","3.3.1#footnote-link-5":"5","3.3.1#ex-3.15":"\n    Draw box-and-pointer diagrams to explain the effect of\n    set-to-wow!\n    on the structures z1 and\n    z2 above.\n  ","3.3.1#ex-3.16":"\n    Ben Bitdiddle decides to write a\n    procedure\n    to count the number of pairs in any list structure.\n    \"It's easy,\" he reasons.  \"The number of pairs in\n    any structure is the number in the\n    car\n    plus the number in the\n    cdr\n    plus one more to count the current pair.\" So Ben writes the following\n    procedure:(define (count-pairs x)\n  (if (not (pair? x))\n      0\n      (+ (count-pairs (car x))\n         (count-pairs (cdr x))\n         1))) \n    Show that this\n    procedure\n    is not correct.  In particular, draw box-and-pointer diagrams representing\n    list structures made up of exactly three pairs for which Ben's\n    procedure\n    would return 3; return 4; return 7; never return at all.\n    ","3.3.1#ex-3.17":"\n    Devise a correct version of the\n    count-pairsprocedure\n    of exercise 3.16 that returns the number of\n    distinct pairs in any structure.  (Hint: Traverse the structure, maintaining\n    an auxiliary data structure that is used to keep track of which pairs have\n    already been counted.)\n    ","3.3.1#ex-3.18":"\n    Write a\n    procedure\n    that examines a list and\n    \n    determines whether it contains a cycle, that is,\n    whether a program that tried to find the end of the list by taking\n    successive\n    cdrs\n    would go into an infinite loop. Exercise 3.13\n    constructed such lists.\n    ","3.3.1#ex-3.19":"\n    Redo exercise 3.18 using an algorithm that\n    takes only a constant amount of space.  (This requires a very clever idea.)\n    ","3.3.1#h2":"Mutation is just assignment","3.3.1#p12":"\n    When we introduced compound data, we observed in\n    section 2.1.3 that pairs can be represented purely\n    in terms of\n    procedures:(define (cons x y)\n  (define (dispatch m)\n    (cond ((eq? m 'car) x)\n          ((eq? m 'cdr) y)\n          (else (error \"Undefined operation - - CONS\" m))))\n  dispatch)\n\n(define (car z) (z 'car))\n\n(define (cdr z) (z 'cdr)) \n    The same observation is true for mutable data.  We can implement\n    mutable data objects as\n    procedures\n    using assignment and local state. For instance, we can extend the above\n    pair implementation to handle\n    set-car!\n    and\n    set-cdr!\n    in a manner analogous to the way we implemented bank accounts using\n    make-account\n    in section 3.1.1:\n    (define (cons x y)\n  (define (set-x! v) (set! x v))\n  (define (set-y! v) (set! y v))\n  (define (dispatch m)\n    (cond ((eq? m 'car) x)\n          ((eq? m 'cdr) y)\n          ((eq? m 'set-car!) set-x!)\n          ((eq? m 'set-cdr!) set-y!)\n          (else (error \"Undefined operation - - CONS\" m))))\n  dispatch)\n\n(define (car z) (z 'car))\n\n(define (cdr z) (z 'cdr))\n\n(define (set-car! z new-value)\n  ((z 'set-car!) new-value)\n  z)\n\n(define (set-cdr! z new-value)\n  ((z 'set-cdr!) new-value)\n  z) ","3.3.1#p13":"\n    Assignment is all that is needed, theoretically, to account for the\n    behavior of mutable data.  As soon as we admit\n    set!\n    to our language, we raise all the issues, not only of assignment, but of\n    mutable data in general.","3.3.1#footnote-link-6":"6","3.3.1#ex-3.20":"\n    Draw environment diagrams to illustrate the evaluation of the sequence\n    of\n    \n\texpressions\n      (define x (cons 1 2))\n(define z (cons x x))\n(set-car! (cdr z) 17)\n\n(car x) (car x) \n    using the \n    \n            procedural \n      \n    implementation of pairs given above.  (Compare\n    exercise 3.11.)\n    ","3.3.1#footnote-1":"Set-car! and\n\tset-cdr! return\n\timplementation-dependent\n\t\n\tvalues.  Like set!, they should be used\n\tonly for their effect.\n      ","3.3.1#footnote-2":"We see from this that mutation operations on\n    lists can create \"garbage\" that is not part of any accessible\n    structure. We will see in section 5.3.2 that\n    Lisp\n    memory-management systems include a \n    garbage collector, which identifies and recycles the memory\n    space used by unneeded pairs.","3.3.1#footnote-3":"Get-new-pair\n\tis one of the operations that must be implemented as part of the memory\n\tmanagement required by a Lisp implementation.\n\tWe will discuss this in\n\tsection 5.3.1.\n      ","3.3.1#footnote-4":"The two pairs are distinct because each call to\n\tcons returns a new pair.  The\n        symbols are shared; in Scheme there is a\n        \n\tunique symbol with any given\n        name.  Since Scheme provides no way to mutate a symbol, this sharing is\n        undetectable.  Note also that the sharing is what enables us to\n        compare symbols using eq?, which simply checks equality of\n        pointers.","3.3.1#footnote-5":"The subtleties of dealing with sharing of mutable data\n    objects reflect the underlying issues of \"sameness\" and\n    \"change\" that were raised in\n    section 3.1.3.  We mentioned there\n    that admitting change to our language requires that a compound object must\n    have an \"identity\" that is something different from the pieces\n    from which it is composed.  In\n    Lisp,\n    we consider this \"identity\" to be the quality that is tested by\n    eq?,\n    i.e., by equality of pointers.  Since in most\n    Lisp\n    implementations a pointer is essentially a memory address, we are\n    \"solving the problem\" of defining the identity of objects by\n    stipulating that a data object \"itself\" is the information\n    stored in some particular set of memory locations in the computer.  This\n    suffices for simple\n    Lisp\n    programs, but is hardly a general way to resolve the issue of\n    \"sameness\" in computational models.","3.3.1#footnote-6":"On the other hand, from the viewpoint of\n    implementation, assignment requires us to modify the environment, which is\n    itself a mutable data structure.  Thus, assignment and mutation are\n    equipotent: Each can be implemented in terms of the other.","3.3.2":"3.3.2  \n    Representing Queues","3.3.2#p1":"\n    The mutators\n    set-car!\n    and\n    set-cdr!\n    enable us to use pairs to construct data structures that cannot be built\n    with\n    cons,car,\n    and\n    cdr\n    alone.  This section shows how to use pairs to represent a data structure\n    called a queue.  Section 3.3.3 will show how to\n    represent data structures called tables.\n  ","3.3.2#p2":"\n    A queue is a sequence in which items are inserted at one end\n    (called the\n    rear of the queue) and deleted from the other end (the\n    front).\n    Figure \n    shows an initially empty queue in which the items\n    a and b are\n    inserted.  Then a is removed,\n    c and d are\n    inserted, and b is removed.  Because items are\n    always removed in the order in which they are inserted, a queue is\n    sometimes called a\n    FIFO (first in, first out) buffer.\n    ","3.3.2#fig-":"","3.3.2#p3":"\n    In terms of\n    \n    data abstraction, we can regard a queue as defined by the\n    following set of operations:\n    \n\ta constructor:\n\t(make-queue)\n          returns an empty queue (a queue containing no items).\n\t  \n        a predicate:\n        (empty-queue? queue)\n          tests if the queue is empty.\n\t  \n        a selector:\n        (front-queue queue)\n          returns the object at the front of the queue, signaling an error if\n\t  the queue is empty; it does not modify the queue.\n\t  \n        two mutators:\n\t(insert-queue! queue item)\n          inserts\n           \n\t  the item at the rear of the queue and returns the modified\n          queue as its value.(delete-queue! queue)\n          removes\n           \n\t  the item at the front of the queue and returns the modified\n\t  queue as its value, signaling an error if the queue is empty before\n\t  the deletion.\n      ","3.3.2#p4":"\n    Because a queue is a sequence of items, we could certainly represent\n    it as an ordinary list; the front of the queue would be the\n    car\n    of the list, inserting an item in the queue would amount to appending\n    a new element at the end of the list, and deleting an item from the\n    queue would just be taking the\n    cdr\n    of the list.  However, this representation is inefficient, because in order\n    to insert an item we must scan the list until we reach the end.  Since the\n    only method we have for scanning a list is by successive\n    cdr\n    operations, this scanning requires $\\Theta(n)$\n    steps for a list of $n$ items.  A simple\n    modification to the list representation overcomes this disadvantage by\n    allowing the queue operations to be implemented so that they require\n    $\\Theta(1)$ steps; that is, so that the number\n    of steps needed is independent of the length of the queue.\n  ","3.3.2#p5":"\n    The difficulty with the list representation arises from the need to\n    scan to find the end of the list.  The reason we need to scan is that,\n    although the standard way of representing a list as a chain of pairs\n    readily provides us with a pointer to the beginning of the list, it\n    gives us no easily accessible pointer to the end.  The modification\n    that avoids the drawback is to represent the queue as a list, together\n    with an additional pointer that indicates the final pair in the list.\n    That way, when we go to insert an item, we can consult the rear\n    pointer and so avoid scanning the list.\n  ","3.3.2#p6":"\n    A queue is represented, then, as a pair of pointers,\n    front-ptr\n    and\n    rear-ptr,\n    which indicate, respectively, the first and last pairs in an ordinary list.\n    Since we would like the queue to be an identifiable object, we can use\n    cons\n    to combine the two pointers.  Thus, the queue itself will be the\n    cons\n    of the two pointers.\n    Figure \n    illustrates this representation.\n    ","3.3.2#p7":"\n    To define the queue operations we use the following\n    procedures,\n    which enable us to select and to modify the front and rear pointers of a\n    queue:\n    (define (front-ptr queue) (car queue))\n\n(define (rear-ptr queue) (cdr queue))\n\n(define (set-front-ptr! queue item) (set-car! queue item))\n\n(define (set-rear-ptr! queue item) (set-cdr! queue item)) ","3.3.2#p8":"\n    Now we can implement the actual queue operations.  We will consider a\n    queue to be empty if its front pointer is the empty list:\n    (define (empty-queue? queue) (null? (front-ptr queue))) \n    The\n    make-queue\n    constructor returns, as an initially empty queue, a pair whose\n    car\n    and\n    cdr\n    are both the empty list:\n    (define (make-queue) (cons '() '())) \n    To select the item at the front of the queue, we return the\n    car\n    of the pair indicated by the front pointer:\n    (define (front-queue queue)\n  (if (empty-queue? queue)\n      (error \"FRONT called with an empty queue\" queue)\n      (car (front-ptr queue)))) ","3.3.2#p9":"\n    To insert an item in a queue, we follow the method whose result is\n    indicated in\n    figure 3.20.\n    We first create a new\n    pair whose\n    car\n    is the item to be inserted and whose\n    cdr\n    is the empty list.  If the queue was initially empty, we set the front and\n    rear pointers of the queue to this new pair.  Otherwise, we modify the\n    final pair in the queue to point to the new pair, and also set the\n    rear pointer to the new pair.\n    (define (insert-queue! queue item)\n  (let ((new-pair (cons item '())))\n    (cond ((empty-queue? queue)\n           (set-front-ptr! queue new-pair)\n           (set-rear-ptr! queue new-pair)\n           queue)\n          (else\n           (set-cdr! (rear-ptr queue) new-pair)\n           (set-rear-ptr! queue new-pair)\n           queue)))) ","3.3.2#p10":"\n    To delete the item at the front of the queue, we merely modify the\n    front pointer so that it now points at the second item in the queue,\n    which can be found by following the\n    cdr\n    pointer of the first item (see\n    figure ):(define (delete-queue! queue)\n  (cond ((empty-queue? queue)\n         (error \"DELETE! called with an empty queue\" queue))\n        (else\n         (set-front-ptr! queue (cdr (front-ptr queue)))\n         queue))) ","3.3.2#footnote-link-1":"1","3.3.2#ex-3.21":"\n    Ben Bitdiddle decides to test the queue implementation described\n    above.  He types in the\n    procedures\n    to the\n    Lisp\n    interpreter and proceeds to try them out:\n    (define q1 (make-queue)) (insert-queue! q1 'a) (insert-queue! q1 'b) (delete-queue! q1) (delete-queue! q1) \"It's all wrong!\" he complains.\n    \"The interpreter's response shows that the last item is inserted\n    into the queue twice.  And when I delete both items, the second\n    b is still there, so the queue isn't\n    empty, even though it's supposed to be.\"  Eva Lu Ator suggests\n    that Ben has misunderstood what is happening.  \"It's not that the\n    items are going into the queue twice,\" she explains.\n    \"It's just that the standard\n    Lisp\n    printer doesn't know how to make sense of the queue representation.\n    If you want to see the queue printed correctly, you'll have to define\n    your own print\n    procedure\n    for queues.\" Explain what Eva Lu is talking about.  In particular,\n    show why Ben's examples produce the printed results that they do.\n    Define a\n    procedureprint-queue\n    that takes a queue as input and prints the sequence of items in the queue.\n    ","3.3.2#ex-3.22":"\n    Instead of representing a queue as a pair of pointers, we can build a\n    queue as a\n    procedure\n    with local state.  The local state will consist of pointers to the\n    beginning and the end of an ordinary list.  Thus, the\n    make-queueprocedure\n    will have the form\n    \n(define (make-queue)\n  (let ((front-ptr $\\ldots$ )\n        (rear-ptr $\\ldots$ ))\n    definitions of internal procedures\n    (define (dispatch m) $\\ldots$)\n    dispatch))\n      \n    Complete the definition of\n    make-queue\n    and provide implementations of the queue operations using this\n    representation.\n    ","3.3.2#ex-3.23":"\n    A deque\n    (\"double-ended queue\") is a sequence in which\n    items can be inserted and deleted\n    \n\tat either the front or\n      \n    the rear.\n    Operations on deques are the constructor\n    make-deque,\n    the predicate\n    empty-deque,\n    selectors\n    front-deque\n    and\n    rear-deque,\n    and mutators\n    front-insert-deque!,front-delete-deque!,rear-insert-deque!,\n    and\n    rear-delete-deque.\n    Show how to represent deques using pairs, and give implementations of the\n    operations.\n    All operations should be accomplished in\n    $\\Theta(1)$ steps.\n    ","3.3.2#footnote-link-2":"2","3.3.2#footnote-1":"If the first item is\n    the final item in the queue, the front pointer will be the empty list after\n    the deletion, which will mark the queue as empty; we needn't worry\n    about updating the rear pointer, which will still point to the deleted\n    item, because\n    empty-queue?\n    looks only at the front pointer.","3.3.2#footnote-2":"Be careful not to make the interpreter try to print a\n    structure that contains cycles.  (See\n    exercise 3.13.)","3.3.3":"3.3.3  \n    Representing Tables","3.3.3#p1":"\n    When we studied various ways of representing sets in chapter 2, we\n    mentioned in section 2.3.3 the task of\n    maintaining a table of records\n    \n    indexed by identifying keys.  In the\n    implementation of data-directed programming in\n    section 2.4.3, we made extensive use of\n    two-dimensional tables, in which information is stored and retrieved\n    using two keys.  Here we see how to build tables as mutable list\n    structures.\n  ","3.3.3#p2":"\n    We first consider a\n    \n    one-dimensional table, in which each value is\n    stored under a single key.  We implement the table as a list of\n    records, each of which is implemented as a pair consisting of a key\n    and the associated value. The records are glued together to form a\n    list by pairs whose\n    cars\n    point to successive records.  These gluing pairs are called the \n    backbone of the table.  In order to have a place that we can\n    change when we add a new record to the table, we build the table as a \n    headed list.  A headed list has a special backbone pair at the\n    beginning, which holds a dummy \"record\"—in this case\n    the arbitrarily chosen \n    symbol *table*.\n\tFigure \n    shows the box-and-pointer diagram for the table\n    a:  1\nb:  2\nc:  3","3.3.3#fig-":"","3.3.3#p3":"\n    To extract information from a table we use the\n    lookupprocedure,\n    which takes a key as argument and returns the associated value (or\n    false\n    if\n    there is no value stored under that key).\n    Lookup\n    is defined in terms of the assoc operation,\n    which expects a key and a list of records as arguments. Note that\n    assoc never sees the dummy record.  \n    Assoc\n    returns the record that has the given key as its\n    car.Lookup\n    then checks to see that the resulting record returned by\n    assoc is not\n    false,\n    and returns the value (the\n    cdr)\n    of the record.\n\n    (define (lookup key table)\n  (let ((record (assoc key (cdr table))))\n    (if record\n        (cdr record)\n        false)))\n\n(define (assoc key records)\n  (cond ((null? records) false)\n        ((equal? key (caar records)) (car records))\n        (else (assoc key (cdr records))))) ","3.3.3#footnote-link-1":"1","3.3.3#p4":"\n    To insert a value in a table under a specified key, we first use\n    assoc to see if there is already a record in\n    the table with this key. If not, we form a new record by\n    consing\n    the key with the value, and insert this at the head of the table's\n    list of records, after the dummy record.  If there already is a record with\n    this key, we set the\n    cdr\n    of this record to the designated new value.  The header of the table\n    provides us with a fixed location to modify in order to insert the new\n    record.(define (insert! key value table)\n  (let ((record (assoc key (cdr table))))\n    (if record\n        (set-cdr! record value)\n        (set-cdr! table\n                  (cons (cons key value) (cdr table)))))\n  'ok) ","3.3.3#footnote-link-2":"2","3.3.3#p5":"\n    To construct a new table, we simply create a list containing\n    \n\tthe symbol *table*:\n      (define (make-table)\n  (list '*table*)) ","3.3.3#h1":"Two-dimensional tables","3.3.3#p6":"\n    In a two-dimensional table, each value is indexed by two keys.  We can\n    construct such a table as a one-dimensional table in which each key\n    identifies a subtable.\n    \n        Figure \n    shows the box-and-pointer diagram for the table\n    math:\n    +:  43\n    -:  45\n    *:  42\nletters:\n    a:  97\n    b:  98\n    which has two subtables.  (The subtables don't need a special header\n    \n\tsymbol,\n      \n    since the key that identifies the subtable serves this purpose.)\n    ","3.3.3#p7":"\n    When we look up an item, we use the first key to identify the correct\n    subtable.  Then we use the second key to identify the record within the\n    subtable.\n    (define (lookup key-1 key-2 table)\n  (let ((subtable (assoc key-1 (cdr table))))\n    (if subtable\n        (let ((record (assoc key-2 (cdr subtable))))\n          (if record\n              (cdr record)\n              false))\n        false))) ","3.3.3#p8":"\n    To insert a new item under a pair of keys, we use\n    assoc to see if there is a subtable stored\n    under the first key.  If not, we build a new subtable containing the single\n    record\n    (key-2,value) and insert it into the table under the\n    first key.  If a subtable already exists for the first key, we insert the\n    new record into this subtable, using the insertion method for\n    one-dimensional tables described above:\n    (define (insert! key-1 key-2 value table)\n  (let ((subtable (assoc key-1 (cdr table))))\n        (if subtable\n            (let ((record (assoc key-2 (cdr subtable))))\n              (if record\n                  (set-cdr! record value)\n                  (set-cdr! subtable\n                            (cons (cons key-2 value)\n                                  (cdr subtable)))))\n            (set-cdr! table\n                      (cons (list key-1\n                                  (cons key-2 value))\n                            (cdr table)))))\n  'ok) ","3.3.3#h2":"Creating local tables","3.3.3#p9":"\n    The lookup and \n    insert!\n    operations defined above take the table as an argument.  This enables us to\n    use programs that access more than one table.  Another way to deal with\n    multiple tables is to have separate lookup and \n    insert!procedures\n    for each table.  We can do this by representing a table procedurally, as an\n    object that maintains an internal table as part of its local state. When\n    sent an appropriate message, this \"table object\" supplies the\n    procedure\n    with which to operate on the internal table.  Here is a generator for\n    two-dimensional tables represented in this fashion:\n    (define (make-table)\n  (let ((local-table (list '*table*)))\n    (define (lookup key-1 key-2)\n      (let ((subtable (assoc key-1 (cdr local-table))))\n        (if subtable\n            (let ((record (assoc key-2 (cdr subtable))))\n              (if record\n                  (cdr record)\n                  false))\n            false)))\n    (define (insert! key-1 key-2 value)\n      (let ((subtable (assoc key-1 (cdr local-table))))\n        (if subtable\n            (let ((record (assoc key-2 (cdr subtable))))\n              (if record\n                  (set-cdr! record value)\n                  (set-cdr! subtable\n                            (cons (cons key-2 value)\n                                  (cdr subtable)))))\n            (set-cdr! local-table\n                      (cons (list key-1\n                                  (cons key-2 value))\n                            (cdr local-table)))))\n      'ok)    \n    (define (dispatch m)\n      (cond ((eq? m 'lookup-proc) lookup)\n            ((eq? m 'insert-proc!) insert!)\n            (else (error \"Unknown operation - - TABLE\" m))))\n  dispatch)) ","3.3.3#p10":"\n    Using\n    make-table,\n    we could\n    \n    implement the get and\n    put operations used in\n    section 2.4.3 for data-directed\n    programming, as follows:\n    (define operation-table (make-table))\n(define get (operation-table 'lookup-proc))\n(define put (operation-table 'insert-proc!)) Get\n    takes as arguments two keys, and put takes\n    as arguments two keys and a value.  Both operations access the same\n    local table, which is encapsulated within the object created by the\n    call to \n    make-table.","3.3.3#ex-3.24":" \n    In the table implementations above, the keys are\n    \n    tested for equality using \n    equal?\n    (called by assoc).  This is not always the\n    appropriate test.  For instance, we might have a table with numeric keys in\n    which we don't need an exact match to the number we're looking\n    up, but only a number within some tolerance of it. Design a table\n    constructor \n    make-table\n    that takes as an argument a \n    same-key?procedure\n    that will be used to test \"equality\" of keys.  \n    Make-table\n    should return a dispatchprocedure\n    that can be used to access appropriate \n    lookup and \n    insert!procedures\n    for a local table.\n    ","3.3.3#ex-3.25":" \n    Generalizing one- and two-dimensional tables, show how to implement a\n    table in which values are stored under an\n    \n    arbitrary number of keys and\n    different values may be stored under different numbers of keys.\n    The\n    lookup and \n    insert!procedures\n    should take as input a list of keys used to access the table.\n    ","3.3.3#ex-3.26":"\n    To search a table as implemented above, one needs to scan through the\n    list of records.  This is basically the unordered list representation of\n    section 2.3.3.  For large tables, it\n    may be more efficient to structure the table in a different manner.\n    Describe a table implementation where the (key, value) records are organized\n    using a\n    \n    binary tree, assuming that keys can be ordered in some way\n    (e.g., numerically or alphabetically).  (Compare\n    exercise 2.66 of chapter 2.)\n    ","3.3.3#ex-3.27":"Memoization\n    (also called tabulation) is a technique that\n    enables a\n    procedure\n    to record, in a local table, values that have previously been computed.\n    This technique can make a vast difference in the performance of a program.\n    A memoized\n    procedure\n    maintains a table in which values of previous calls are stored\n    using as keys the arguments that produced the values.  When the\n    memoized\n    procedure\n    is asked to compute a value, it first checks the table to see if the value\n    is already there and, if so, just returns that value.  Otherwise, it\n    computes the new value in the ordinary way and stores this in the table.\n    As an example of memoization, recall from\n    section 1.2.2 the exponential process for\n    computing Fibonacci numbers:\n    (define (fib n)\n  (cond ((= n 0) 0)\n        ((= n 1) 1)\n        (else (+ (fib (- n 1))\n                 (fib (- n 2)))))) \n    The memoized version of the same\n    procedure\n    is\n    (define memo-fib\n  (memoize (lambda (n)\n             (cond ((= n 0) 0)\n                   ((= n 1) 1)\n                   (else (+ (memo-fib (- n 1))\n                            (memo-fib (- n 2)))))))) \n    where the memoizer is defined as\n    (define (memoize f)\n  (let ((table (make-table)))\n    (lambda (x)\n      (let ((previously-computed-result (lookup x table)))\n        (or previously-computed-result\n            (let ((result (f x)))\n              (insert! x result table)\n              result)))))) \n    Draw an environment diagram to analyze the computation of\n    (memo-fib 3).\n    Explain why \n    memo-fib\n    computes the $n$th Fibonacci number in a number\n    of steps proportional to $n$. Would the scheme\n    still work if we had simply defined \n    memo-fib\n    to be\n    (memoize fib)?","3.3.3#footnote-1":"\n\tBecause\n\tassoc uses\n\tequal?, it can recognize keys that are\n\tsymbols, numbers, or list structure.\n      ","3.3.3#footnote-2":"Thus, the first backbone pair is the object that represents\n    the table \"itself\"; that is, a pointer to the table is a\n    pointer to this pair.  This same backbone pair always starts the table.\n    If we did not arrange things in this way,\n    insert!\n    would have to return a new value for the start of the table\n    when it added a new record.","3.3.4":"3.3.4  \n    A Simulator for Digital Circuits","3.3.4#p1":"\n    Designing complex digital systems, such as computers, is an important\n    engineering activity.  Digital systems are constructed by\n    interconnecting simple elements.  Although the behavior of these\n    individual elements is simple, networks of them can have very complex\n    behavior.  Computer simulation of proposed circuit designs is an\n    important tool used by digital systems engineers.  In this section we\n    design a system for performing digital logic simulations.  This system\n    typifies a kind of program called an \n    event-driven simulation, in\n    which actions (\"events\") trigger further events that happen\n    at a later time, which in turn trigger more events, and so on.\n  ","3.3.4#p2":"\n    Our computational model of a circuit will be composed of objects that\n    correspond to the elementary components from which the circuit is\n    constructed.  There are \n    wires, which carry \n    digital signals.  A digital signal may at any moment have only one of two\n    possible values,\n    0 and 1.  There are also various types of digital \n    function boxes, which connect wires carrying input signals to other output\n    wires.  Such boxes produce output signals computed from their input\n    signals.  The output signal is \n    \n    delayed by a time that depends on the\n    type of the function box.  For example, an \n    inverter is a\n    primitive function box that inverts its input.  If the\n    input signal to an inverter changes to 0, then one inverter-delay\n    later the inverter will change its output signal to 1.  If the input\n    signal to an inverter changes to 1, then one inverter-delay later\n    the inverter will change its output signal to 0.  We draw an inverter\n    symbolically as in figure 3.24.  An \n    and-gate,\n    also shown in figure 3.24, is a primitive\n    function box with two inputs and one output.  It drives its output signal\n    to a value that is the \n    logical and of the inputs.  That is, if both\n    of its input signals become 1, then one and-gate-delay time\n    later the and-gate will force its output signal to be 1; otherwise the\n    output will be 0.  An \n    or-gate is a similar two-input primitive function\n    box that drives its output signal to a value that is the \n    logical or of the inputs.  That is, the output will become 1 if at least one\n    of the input signals is 1; otherwise the output will become 0.\n  ","3.3.4#fig-3.24":"","3.3.4#p3":"\n    We can connect primitive functions together to construct more complex\n    functions.  To accomplish this we wire the outputs of some\n    function boxes to the inputs of other function boxes.  For example,\n    the \n    half-adder circuit shown in\n    figure 3.25 consists of an\n    or-gate, two and-gates, and an inverter.  It takes two input signals,\n    $A$ and $B$, and has\n    two output signals, $S$ and $C$.\n    $S$ will become 1\n    whenever precisely one of $A$ and $B$\n    is 1, and $C$ will become 1 whenever\n    $A$ and $B$ are both 1.  We can see\n    from the figure that, because of the\n    delays involved, the outputs may be generated at different times.\n    Many of the difficulties in the design of digital circuits arise from\n    this fact.\n    ","3.3.4#fig-3.25":"","3.3.4#p4":"\n    We will now build a program for modeling the digital logic circuits we\n    wish to study.  The program will construct computational objects\n    modeling the wires, which will \"hold\" the signals.  Function\n    boxes will be modeled by\n    procedures\n    that enforce the correct relationships among the signals.\n  ","3.3.4#p5":"\n    One basic element of our simulation will be a\n    proceduremake-wire,\n    which constructs wires.  For example, we can construct six wires as follows:\n    (define a (make-wire))\n(define b (make-wire))\n(define c (make-wire))\n(define d (make-wire))\n(define e (make-wire))\n(define s (make-wire)) \n    We attach a function box to a set of wires by calling a\n    procedure\n    that constructs that kind of box.  The arguments to the constructor\n    procedure\n    are the wires to be attached to the box.  For example, given\n    that we can construct and-gates, or-gates, and inverters, we can wire\n    together the half-adder shown in figure 3.25:\n    (or-gate a b d) (and-gate a b c) (inverter c e) (and-gate d e s) ","3.3.4#p6":"\n    Better yet, we can explicitly name this operation by defining a\n    procedurehalf-adder\n    that constructs this circuit, given the four\n    external wires to be attached to the half-adder:\n    (define (half-adder a b s c)\n  (let ((d (make-wire)) (e (make-wire)))\n    (or-gate a b d)\n    (and-gate a b c)\n    (inverter c e)\n    (and-gate d e s)\n    'ok)) \n    The advantage of making this definition is that we can use half-adder itself as a building block in creating more complex\n    circuits.  Figure 3.26, for example, shows a\n    full-adder composed of two half-adders and an or-gate. We can construct a\n      full-adder as follows:\n    (define (full-adder a b c-in sum c-out)\n  (let ((s (make-wire))\n        (c1 (make-wire))\n        (c2 (make-wire)))\n    (half-adder b c-in s c1)\n    (half-adder a s sum c2)\n    (or-gate c1 c2 c-out)\n    'ok)) \n    Having defined\n    full-adder\n    as a\n    procedure,\n    we can now use it as a building block for creating still more complex\n    circuits.  (For example, see exercise 3.30.)\n  ","3.3.4#footnote-link-1":"1","3.3.4#fig-3.26":"","3.3.4#p7":"\n    In essence, our simulator provides us with the tools to construct a\n    language of circuits.  If we adopt the general perspective on\n    languages with which we approached the study of\n    Lisp\n    in section 1.1,\n    we can say that the primitive function boxes form the primitive\n    elements of the language, that wiring boxes together provides a means\n    of combination, and that specifying wiring patterns as\n    procedures\n    serves as a means of abstraction.\n  ","3.3.4#h1":"Primitive function boxes","3.3.4#p8":"\n    The primitive function boxes\n    \n    implement the \"forces\" by which a\n    change in the signal on one wire influences the signals on other\n    wires.  To build function boxes, we use the following operations on\n    wires:\n    (get-signal wire):\n\treturns the current value of the signal on the wire.\n      (set-signal! wire new-value):\n\t  \n        changes the value of the signal on the wire to the new value.\n      (add-action! wire procedure-of-no-arguments):\n\t  \n        asserts that the designated\n        procedure\n        should be run whenever the signal on the wire changes value.  Such\n        procedures\n        are the vehicles by which changes in the signal value on the wire are\n\tcommunicated to other wires.\n      \n    In addition, we will make use of a\n    procedureafter-delay\n    that takes a time delay and a\n    procedure\n    to be run and executes the given\n    procedure\n    after the given delay.\n  ","3.3.4#p9":"\n    Using these\n    procedures,\n    we can define the primitive digital logic functions.  To connect an input\n    to an output through an inverter, we use\n    add-action!\n    to associate with the input wire a\n    procedure\n    that will be run whenever the signal on the input wire changes value.\n    The\n    procedure\n    computes the\n    logical-not\n    of the input signal, and then, after one\n    inverter-delay,\n    sets the output signal to be this new value:\n    (define (inverter input output)\n  (define (invert-input)\n    (let ((new-value (logical-not (get-signal input))))\n      (after-delay inverter-delay\n                   (lambda ()\n                     (set-signal! output new-value)))))\n  (add-action! input invert-input)\n  'ok)\n                \n(define (logical-not s)\n  (cond ((= s 0) 1)\n        ((= s 1) 0)\n        (else (error \"Invalid signal\" s)))) ","3.3.4#p10":"\n    An and-gate is a little more complex.  The action\n    procedure\n    must be run if\n    either of the inputs to the gate changes.  It computes the\n    logical-and\n      (using a procedure analogous to\n      logical-not)\n      \n    of the values of the signals on the input wires and sets up a change\n    to the new value to occur on the output wire after one\n    and-gate-delay.(define (and-gate a1 a2 output)\n  (define (and-action-procedure)\n    (let ((new-value\n          (logical-and (get-signal a1) (get-signal a2))))\n      (after-delay and-gate-delay\n                   (lambda ()\n                     (set-signal! output new-value)))))\n  (add-action! a1 and-action-procedure)\n  (add-action! a2 and-action-procedure)\n  'ok) ","3.3.4#ex-3.28":"\n    Define an\n    \n    or-gate as a primitive function box.  Your\n    or-gate\n    constructor should be similar to\n    and-gate.","3.3.4#ex-3.29":"\n    Another way to construct an\n    \n    or-gate is as a compound digital logic\n    device, built from and-gates and inverters.  Define a\n    procedureor-gate\n    that accomplishes this.  What is the delay time of the\n    or-gate in terms of\n    and-gate-delay\n    and\n    inverter-delay?","3.3.4#ex-3.30":"\n    Figure 3.27 shows a \n    ripple-carry adder formed by stringing\n    together $n$ full-adders.\n    This is the simplest form of parallel adder\n    for adding two $n$-bit binary numbers.\n    The inputs $A_{1}$,\n    $A_{2}$,\n    $A_{3}$, …,\n    $A_{n}$ and\n    $B_{1}$,\n    $B_{2}$,\n    $B_{3}$, …,\n    $B_{n}$\n    are the two binary numbers to be added (each\n    $A_{k}$ and\n    $B_{k}$\n    is a 0 or a 1).  The circuit generates\n    $S_{1}$,\n    $S_{2}$,\n    $S_{3}$,\n    …,\n    $S_{n}$,\n    the $n$ bits of the sum, and\n    $C$, the carry from\n    the addition.  Write a\n    procedureripple-carry-adder\n    that generates this circuit.  The\n    procedure\n    should take as arguments three lists of\n    $n$ wires each—the\n    $A_{k}$, the\n    $B_{k}$, and the\n    $S_{k}$—and\n    also another wire $C$.\n    The major drawback of the ripple-carry adder is the need to wait for the\n    carry signals to propagate.  What is the delay needed to obtain the\n    complete output from an $n$-bit ripple-carry\n    adder, expressed in terms of the delays for and-gates, or-gates, and\n    inverters?\n    ","3.3.4#fig-3.27":"","3.3.4#h2":"Representing wires","3.3.4#p11":"\n    A wire\n    \n    in our simulation will be a computational object with two local\n    state variables:\n    a signal-value\n    (initially taken to be 0) and a collection of\n    action-procedures\n    to be run when the signal changes value.  We implement the wire,\n    using\n    \n    message-passing style, as\n    a collection of local\n    procedures\n    together with a dispatchprocedure\n    that selects the appropriate local operation, just as we did\n    with the simple bank-account object in section\n     3.1.1:\n    (define (make-wire)\n  (let ((signal-value 0) (action-procedures '()))\n    (define (set-my-signal! new-value)\n      (if (not (= signal-value new-value))\n          (begin (set! signal-value new-value)\n            (call-each action-procedures))\n            'done))\n                \n(define (accept-action-procedure! proc)\n  (set! action-procedures (cons proc action-procedures))\n  (proc))\n               \n(define (dispatch m)\n  (cond ((eq? m 'get-signal) signal-value)\n        ((eq? m 'set-signal!) set-my-signal!)\n        ((eq? m 'add-action!) accept-action-procedure!)\n        (else (error \"Unknown operation -- WIRE\" m))))\n  dispatch)) \n    The local\n    procedureset-my-signal\n    tests whether the new signal value changes the signal on the wire.\n    If so, it runs each of the action\n    procedures,\n    using the following\n    procedurecall-each,\n    which calls each of the items in a list of no-argument\n    procedures:(define (call-each procedures)\n  (if (null? procedures)\n      'done\n      (begin\n        ((car procedures))\n        (call-each (cdr procedures))))) \n    The local\n    procedureaccept-action-procedure\n    adds the given\n    procedure\n    to the list of\n    procedures\n    to be run, and then runs the new\n    procedure\n    once.  (See exercise 3.31.)\n  ","3.3.4#p12":"\n    With the local dispatchprocedure\n    set up as specified, we can\n    provide the following\n    procedures\n    to access the local operations on\n    wires:(define (get-signal wire)\n  (wire 'get-signal))\n                \n(define (set-signal! wire new-value)\n  ((wire 'set-signal!) new-value))\n                \n\n(define (add-action! wire action-procedure)\n  ((wire 'add-action!) action-procedure)) ","3.3.4#footnote-link-2":"2","3.3.4#p13":"\n    Wires, which have time-varying signals and may be incrementally attached to\n    devices, are typical of mutable objects.  We have modeled them as\n    procedures\n    with local state variables that are modified by assignment.  When a new\n    wire is created, a new set of state variables is allocated (by the\n    let expression inmake-wire)\n    and a new dispatchprocedure\n    is constructed and returned, capturing\n    the environment with the new state variables.\n  ","3.3.4#p14":"\n    The wires are shared among the various devices that have been\n    connected to them.  Thus, a change made by an interaction with one\n    device will affect all the other devices attached to the wire.  The\n    wire communicates the change to its neighbors by calling the action\n    procedures\n    provided to it when the connections were established.\n  ","3.3.4#h3":"The agenda","3.3.4#p15":"\n    The only thing needed to complete the simulator is\n    after-delay.\n    The idea here is that we maintain a data structure, called an\n    agenda, that contains a schedule of things to do.\n    The following operations are defined for agendas:\n    (make-agenda):\n\treturns a new empty agenda.\n      (empty-agenda? agenda):\n\t  \n\tis true if the specified agenda is empty.\n      (first-agenda-item agenda):\n\t  \n\treturns the first item on the agenda.\n      (remove-first-agenda-item! agenda):\n\t  \n        modifies the agenda by removing the first item.\n      (add-to-agenda! time action agenda):\n\t  \n        modifies the agenda by adding the given action\n        procedure\n        to be run at the specified time.\n      (current-time agenda):\n\t  \n        returns the current simulation time.\n      ","3.3.4#p16":"\n    The particular agenda that we use is denoted by\n    the-agenda.\n    The\n    procedureafter-delay\n    adds new elements to\n    the-agenda:(define (after-delay delay action)\n  (add-to-agenda! (+ delay (current-time the-agenda))\n                  action\n                  the-agenda)) ","3.3.4#p17":"\n\tThe simulation is driven by the procedure\n\tpropagate, which operates on\n\tthe-agenda,\n\texecuting each procedure on the agenda in sequence.\n      \n    In general, as the simulation runs, new items\n    will be added to the agenda, and propagate\n    will continue the simulation as long as there are items on the agenda:\n    (define (propagate)\n  (if (empty-agenda? the-agenda)\n      'done\n      (let ((first-item (first-agenda-item the-agenda)))\n        (first-item)\n        (remove-first-agenda-item! the-agenda)\n        (propagate)))) ","3.3.4#h4":"A sample simulation","3.3.4#p18":"\n    The following\n    procedure,\n    which places a \"probe\" on a wire, shows the simulator in\n    action.  The probe tells the wire that, whenever its signal changes value,\n    it should print the new signal value, together with the current time and\n    a name that identifies the\n    wire:(define (probe name wire)\n  (add-action! wire\n               (lambda ()\n                 (newline)\n                 (display name)\n                 (display \" \")\n                 (display (current-time the-agenda))\n                 (display \"  New-value = \")\n                 (display (get-signal wire))))) ","3.3.4#p19":"\n    We begin by initializing the agenda and specifying delays for the\n    primitive function boxes:\n    (define the-agenda (make-agenda))\n(define inverter-delay 2)\n(define and-gate-delay 3)\n(define or-gate-delay 5) \n    Now we define four wires, placing probes on two of them:\n    (define input-1 (make-wire))\n(define input-2 (make-wire))\n(define sum (make-wire))\n(define carry (make-wire))\n    \n(probe 'sum sum) (probe 'carry carry) \n    Next we connect the wires in a half-adder circuit (as in\n    figure 3.25), set the signal on\n    input-1\n    to 1, and run the simulation:\n    (half-adder input-1 input-2 sum carry) (set-signal! input-1 1) (propagate) \n    The sum signal changes to 1 at time 8.\n    We are now eight time units from the beginning of the simulation.\n    At this point, we can set the signal on\n    input-2\n    to 1 and allow the values to propagate:\n    (set-signal! input-2 1) (propagate) The carry changes to 1 at time 11 and the\n    sum changes to 0 at time 16.\n  ","3.3.4#ex-3.31":"\n    The internal\n    procedureaccept-action-procedure!\n    defined in\n    make-wire\n    specifies that when a new action\n    procedure\n    is added to\n    a wire, the\n    procedure\n    is immediately run.  Explain why this initialization\n    is necessary.  In particular, trace through the half-adder example in\n    the paragraphs above and say how the system's response would differ\n    if we had defined\n    accept-action-procedure!\n    as\n    (define (accept-action-procedure! proc)\n  (set! action-procedures (cons proc action-procedures)))","3.3.4#h5":"Implementing the agenda","3.3.4#p20":"\n    Finally, we give details of the agenda data structure, which holds the\n    procedures\n    that are scheduled for future execution.\n  ","3.3.4#p21":"\n    The agenda is made up of \n    time segments.  Each time segment is a\n    pair consisting of a number (the time) and a \n    \n    queue (see\n    exercise 3.32) that holds the\n    procedures\n    that are scheduled to be run during that time segment.\n    (define (make-time-segment time queue)\n  (cons time queue))\n                \n(define (segment-time s) (car s))\n                \n(define (segment-queue s) (cdr s)) \n    We will operate on the time-segment queues using the queue operations\n    described in section 3.3.2.\n  ","3.3.4#p22":"\n    The agenda itself is a one-dimensional\n    \n    table of time segments.  It\n    differs from the tables described in section 3.3.3\n    in that the segments will be sorted in order of increasing time.  In\n    addition, we store the \n    current time (i.e., the time of the last action\n    that was processed) at the head of the agenda.  A newly constructed\n    agenda has no time segments and has a current time of 0:(define (make-agenda) (list 0))\n                \n(define (current-time agenda) (car agenda))\n                \n(define (set-current-time! agenda time)\n  (set-car! agenda time))\n                \n(define (segments agenda) (cdr agenda))\n                \n(define (set-segments! agenda segments)\n  (set-cdr! agenda segments))\n                \n(define (first-segment agenda) (car (segments agenda)))\n                \n(define (rest-segments agenda) (cdr (segments agenda))) ","3.3.4#footnote-link-3":"3","3.3.4#p23":"\n    An agenda is empty if it has no time segments:\n    (define (empty-agenda? agenda)\n  (null? (segments agenda))) ","3.3.4#p24":"\n    To add an action to an agenda, we first check if the agenda is empty.\n    If so, we create a time segment for the action and install this in\n    the agenda.  Otherwise, we scan the agenda, examining the time of each\n    segment.  If we find a segment for our appointed time, we add the\n    action to the associated queue.  If we reach a time later than the one\n    to which we are appointed, we insert a new time segment into the\n    agenda just before it.  If we reach the end of the agenda, we must\n    create a new time segment at the end.\n    (define (add-to-agenda! time action agenda)\n  (define (belongs-before? segments)\n    (or (null? segments)\n        (< time (segment-time (car segments)))))\n  (define (make-new-time-segment time action)\n    (let ((q (make-queue)))\n      (insert-queue! q action)\n      (make-time-segment time q)))\n  (define (add-to-segments! segments)\n     (if (= (segment-time (car segments)) time)\n         (insert-queue! (segment-queue (car segments))\n                        action)\n         (let ((rest (cdr segments)))\n           (if (belongs-before? rest)\n               (set-cdr!\n                segments\n                (cons (make-new-time-segment time action)\n                      (cdr segments)))\n               (add-to-segments! rest)))))\n  (let ((segments (segments agenda)))\n    (if (belongs-before? segments)\n        (set-segments!\n         agenda\n         (cons (make-new-time-segment time action)\n               segments))\n        (add-to-segments! segments)))) ","3.3.4#p25":"\n    The\n    procedure\n    that removes the first item from the agenda deletes the\n    item at the front of the queue in the first time segment.  If this\n    deletion makes the time segment empty, we remove it from the list of\n    segments:(define (remove-first-agenda-item! agenda)\n  (let ((q (segment-queue (first-segment agenda))))\n    (delete-queue! q)\n    (if (empty-queue? q)\n        (set-segments! agenda (rest-segments agenda))))) ","3.3.4#footnote-link-4":"4","3.3.4#p26":"\n    The first agenda item is found at the head of the queue in the first\n    time segment.  Whenever we extract an item, we also update the current\n    time:(define (first-agenda-item agenda)\n  (if (empty-agenda? agenda)\n      (error \"Agenda is empty -- FIRST-AGENDA-ITEM\")\n      (let ((first-seg (first-segment agenda)))\n        (set-current-time! agenda (segment-time first-seg))\n        (front-queue (segment-queue first-seg))))) ","3.3.4#footnote-link-5":"5","3.3.4#ex-3.32":"\n    The\n    procedures\n    to be run during each time segment of the agenda are kept in a queue.\n    Thus, the\n    procedures\n    for each segment are called in the order in which they were added to the\n    agenda (first in, first out).  Explain why this order must be used.  In\n    particular, trace the behavior of an and-gate whose inputs change from\n    0,1 to 1,0 in the same segment and say how the behavior would differ if\n    we stored a segment's\n    procedures\n    in an ordinary list, adding and removing\n    procedures\n    only at the front (last in, first out).\n    ","3.3.4#footnote-1":"A\n      full-adder is a basic circuit element used in adding two binary\n      numbers.  Here $A$ and $B$\n      are the bits at corresponding positions in the\n      two numbers to be added, and $C_{\\mathit{in}}$ is the\n      carry bit from the addition one place to the right.  The circuit generates\n      $\\mathit{SUM}$, which is the sum bit in the corresponding position, and\n      $C_{\\mathit{out}}$, which is the\n      carry bit to be propagated to the left.","3.3.4#footnote-2":"These\n    procedures\n    are simply syntactic sugar that allow\n    \n    us to use ordinary \n    \n        procedural \n      \n    syntax to access the local\n    procedures\n    of objects.  It is striking that we can interchange the role of\n    \"procedures\"\n    and\n    \"data\" in such a simple way.  For example, if we write\n    (wire 'get-signal)\n    we think of wire as a\n    procedure\n    that is called with the message\n    get-signal\n    as input. Alternatively, writing\n    (get-signal wire)\n    encourages us to think of wire as a data\n    object that is the input to a\n    procedureget-signal.\n    The truth of the matter is that, in a language in which we can deal with\n    procedures\n    as objects, there is no fundamental difference between\n    \"procedures\"\n    and \"data,\" and we can choose our syntactic sugar to allow us\n    to program in whatever style we choose.\n    ","3.3.4#footnote-3":"The\n    agenda is a \n    \n    headed list, like the tables in section 3.3.3,\n    but since the list is headed by the time, we do not need an additional\n    dummy header (such as the\n    *table* symbol\n      \n    used\n    with tables).","3.3.4#footnote-4":"Observe that the\n    if\n\texpression in this\n\tprocedure\n\thas no alternative expression.\n      \n    Such a\n    \"one-armed if expression\"\n    is used to decide whether to do something, rather than to select between two\n    \n\texpressions.\n\tAn if expression returns an\n\tunspecified value if the predicate is false and there is no\n\talternative.\n      ","3.3.4#footnote-5":"In this way, the current time will always be the time\n    of the action most recently processed.  Storing this time at the head\n    of the agenda ensures that it will still be available even if the\n    associated time segment has been deleted.","3.3.5":"3.3.5  \n    Propagation of Constraints","3.3.5#p1":"\n    Computer programs are traditionally organized as\n    one-directional computations, which perform operations on prespecified\n    arguments to produce desired outputs.  On the other hand, we often\n    model systems in terms of relations among quantities.  For example, a\n    mathematical model of a mechanical structure might include the\n    information that the deflection $d$ of a metal\n    rod is related to the force $F$ on the rod, the\n    length $L$ of the rod, the cross-sectional\n    area $A$, and the elastic modulus\n    $E$ via the equation\n    \n      \\[\n      \\begin{array}{lll}\n      d A E & = & F L\n      \\end{array}\n      \\]\n    \n    Such an equation is not one-directional.  Given any four of the\n    quantities, we can use it to compute the fifth.  Yet translating the\n    equation into a traditional computer language would force us to choose\n    one of the quantities to be computed in terms of the other four.\n    Thus, a\n    procedure\n    for computing the area $A$ could not be used to\n    compute the deflection $d$, even though the\n    computations of $A$ and\n    $d$ arise from the same\n    equation.","3.3.5#footnote-link-1":"1","3.3.5#p2":"\n    In this section, we sketch the design of a language that enables us to work\n    in terms of\n    \n    relations themselves.  The primitive elements of the language\n    are \n    primitive constraints, which state that certain relations hold\n    between quantities.  For example,\n    (adder a b c)\n    specifies that the quantities $a$,\n    $b$, and $c$ must be\n    related by the equation $a+b=c$,\n    (multiplier x y z)\n    expresses the constraint $xy = z$, and\n    (constant 3.14 x)\n    says that the value of $x$ must be 3.14.\n  ","3.3.5#p3":"\n    Our language provides a means of combining primitive constraints in order to\n    express more complex relations.  We combine constraints by constructing \n    constraint networks, in which constraints are joined by \n    connectors.  A connector is an object that \"holds\" a\n    value that may participate in one or more constraints.  For example, we know\n    that the relationship between Fahrenheit and Celsius temperatures is\n    \n      \\[\n      \\begin{array}{lll}\n      9C & = & 5(F - 32)\n      \\end{array}\n      \\]\n    \n    Such a constraint can be thought of as a network consisting of primitive\n    adder, multiplier, and constant constraints\n    (figure 3.28).  In the figure, we see on the\n    left a multiplier box with three terminals, labeled\n    $m_1$, $m_2$, and\n    $p$. These connect the multiplier to the rest of\n    the network as follows:\n    The $m_1$ terminal is linked to a connector\n    $C$, which will hold the Celsius temperature.\n    The $m_2$ terminal is linked to a connector\n    $w$, which is also linked to a constant box that\n    holds 9.  The $p$ terminal, which the multiplier\n    box constrains to be the product of $m_1$ and\n    $m_2$, is linked to the\n    $p$ terminal of another multiplier box, whose\n    $m_2$ is connected to a constant 5 and whose\n    $m_1$ is connected to one of the terms in a sum.\n    ","3.3.5#fig-3.28":"","3.3.5#p4":"\n    Computation by such a network proceeds as follows: When a connector is\n    given a value (by the user or by a constraint box to which it is\n    linked), it awakens all of its associated constraints (except for the\n    constraint that just awakened it) to inform them that it has a value.\n    Each awakened constraint box then polls its connectors to see if there\n    is enough information to determine a value for a connector.  If so,\n    the box sets that connector, which then awakens all of its associated\n    constraints, and so on.  For instance, in conversion between\n    Celsius and Fahrenheit, $w$,\n    $x$, and $y$ are\n    immediately set by the constant boxes to $9$, $5$, and $32$, respectively.  The\n    connectors awaken the multipliers and the adder, which determine that there\n    is not enough information to proceed.  If the user (or some other part of\n    the network) sets $C$ to a value (say 25), the\n    leftmost multiplier will be awakened, and it will set\n    $u$ to $25\\cdot 9=225$.\n    Then $u$ awakens the second multiplier, which sets\n    $v$ to $45$, and $v$\n    awakens the adder, which sets $F$ to $77$.\n  ","3.3.5#h1":"Using the constraint system","3.3.5#p5":"\n\tTo use the constraint system to carry out the temperature computation\n\toutlined above, we first create two connectors,\n\tC and F, by\n\tcalling the constructor\n\tmake-connector, and link\n\tC and F\n\tin an appropriate network:\n      (define C (make-connector))\n(define F (make-connector))\n(celsius-fahrenheit-converter C F) \n    The\n    procedure\n    that creates the network is defined as follows:\n    (define (celsius-fahrenheit-converter c f)\n  (let ((u (make-connector))\n        (v (make-connector))\n        (w (make-connector))\n        (x (make-connector))\n        (y (make-connector)))\n    (multiplier c w u)\n    (multiplier v x u)\n    (adder v y f)\n    (constant 9 w)\n    (constant 5 x)\n    (constant 32 y)\n    'ok)) \n    This\n    procedure\n    creates the internal connectors u,\n    v, w,\n    x, and y, and\n    links them as shown in figure 3.28 using the\n    primitive constraint constructors adder,\n    multiplier, and\n    constant.  Just as with the digital-circuit\n    simulator of section 3.3.4, expressing\n    these combinations of primitive elements in terms of\n    procedures\n    automatically provides our language with a means of abstraction for compound\n    objects.\n  ","3.3.5#p6":"\n    To watch the network in action, we can place probes on the connectors\n    C and F, using a\n    probeprocedure\n    similar to the one we used to monitor wires in\n    section 3.3.4. Placing a probe on a\n    connector will cause a message to be printed whenever the connector is\n    given a value:\n    (probe \"Celsius temp\" C)\n(probe \"Fahrenheit temp\" F) \n    Next we set the value of C to 25.  (The third\n    argument to\n    set-value!\n    tells C that this directive comes from the\n    user.)\n    (set-value! C 25 'user) \n    The probe on C awakens and reports the value.\n    C also\n    propagates its value through the network as described above.  This\n    sets F to 77, which is reported by the probe\n    on F.\n  ","3.3.5#p7":"\n    Now we can try to set F to a new value, say 212:\n    (set-value! F 212 'user) \n    The connector complains that it has sensed a contradiction: Its value\n    is 77, and someone is trying to set it to 212.  If we really want to\n    reuse the network with new values, we can tell\n    C to forget its old value:\n    (forget-value! C 'user) C finds that the\n    user,\n      \n    who set its value originally, is now retracting that value, so\n    C agrees to lose its value, as shown by the\n    probe, and informs the rest of the network of this fact. This information\n    eventually propagates to F, which now finds\n    that it has no reason for continuing to believe that its own\n    value is 77.  Thus, F also\n    gives up its value, as shown by the probe.\n  ","3.3.5#p8":"\n    Now that F has no value, we are free to set it\n    to 212:\n    (set-value! F 212 'user) \n    This new value, when propagated through the network, forces\n    C to have a value of 100, and this is\n    registered by the probe on C. Notice that the\n    very same network is being used to compute C\n    given F and to compute\n    F given C.\n    This nondirectionality of computation is the distinguishing feature of\n    constraint-based systems.\n  ","3.3.5#h2":"Implementing the constraint system","3.3.5#p9":"\n    The constraint system is implemented via procedural objects with local\n    state, in a manner very similar to the digital-circuit simulator of\n    section 3.3.4.  Although the primitive\n    objects of the constraint system are somewhat more complex, the overall\n    system is simpler, since there is no concern about agendas and logic delays.\n  ","3.3.5#p10":"\n    The basic\n    \n    operations on connectors are the following:\n    (has-value? connector):\n        tells whether the connector has a value.\n      (get-value connector):\n        returns the connector's current value.\n      (set-value! connector new-value informant):\n\t  \n        indicates that the informant is requesting the connector to set its\n        value to the new value.\n      (forget-value! connector retractor):\n\t  \n        tells the connector that the retractor is requesting it to forget its\n\tvalue.\n      (connect connector new-constraint):\n\t  \n        tells the connector to participate in the new constraint.\n      \n    The connectors communicate with the constraints by means of the\n    proceduresinform-about-value,\n      \n    which tells the given constraint that the connector has a value, and\n    inform-about-no-value,\n    which tells the constraint that the connector has lost its value.\n  ","3.3.5#p11":"Adder constructs an adder constraint among\n    summand connectors a1 and\n    a2 and a sum\n    connector.  An adder is implemented as a\n    procedure\n    with local state (the\n    procedureme below):\n    (define (adder a1 a2 sum)\n  (define (process-new-value)\n    (cond ((and (has-value? a1) (has-value? a2))\n            (set-value! sum\n                        (+ (get-value a1) (get-value a2))\n                        me))\n          ((and (has-value? a1) (has-value? sum))\n           (set-value! a2\n                       (- (get-value sum) (get-value a1))\n                       me))\n          ((and (has-value? a2) (has-value? sum))\n           (set-value! a1\n                       (- (get-value sum) (get-value a2))\n                       me))))\n  (define (process-forget-value)\n    (forget-value! sum me)\n    (forget-value! a1 me)\n    (forget-value! a2 me)\n    (process-new-value))\n  (define (me request)\n    (cond ((eq? request 'I-have-a-value)  \n           (process-new-value))\n          ((eq? request 'I-lost-my-value) \n            (process-forget-value))\n          (else \n            (error \"Unknown request - - ADDER\" request))))\n  (connect a1 me)\n  (connect a2 me)\n  (connect sum me)\n  me) Adder\n    connects the new adder to the designated\n    connectors and returns it as its value.  The\n    procedureme, which represents the adder, acts as a\n    dispatch to the local\n    procedures.\n    The following\n    \"syntax interfaces\" (see\n    footnote 2 in\n    section 3.3.4) are used in conjunction\n    with the dispatch:\n    (define (inform-about-value constraint)\n  (constraint 'I-have-a-value))\n\n(define (inform-about-no-value constraint)\n  (constraint 'I-lost-my-value)) \n    The adder's local\n    procedureprocess-new-value\n    is called when the adder is informed that one of its connectors has a value.\n    The adder first checks to see if both a1 and\n    a2 have values. If so, it tells\n    sum to set its value to the sum of the two\n    addends. The informant argument to\n    set-value!\n    is me, which is the adder object itself.  If\n    a1 and a2 do not\n    both have values, then the adder checks to see if perhaps\n    a1 and sum have\n    values.  If so, it sets a2 to the difference of\n    these two. Finally, if a2 and\n    sum have values, this gives the adder enough\n    information to set a1.  If the adder is told\n    that one of its connectors has lost a value, it requests that all of its\n    connectors now lose their values.  (Only those values that were set by\n    this adder are actually lost.)  Then it runs\n    process-new-value.\n    The reason for this last step is that one or more connectors may still\n    have a value (that is, a connector may have had a value that was not\n    originally set by the adder), and these values may need to be\n    propagated back through the adder.\n  ","3.3.5#p12":"\n    A multiplier is very similar to an adder. It will set its\n    product to 0 if either of the factors is 0,\n    even if the other factor is not known.\n    (define (multiplier m1 m2 product)\n  (define (process-new-value)\n            (cond ((or (and (has-value? m1) (= (get-value m1) 0))\n                       (and (has-value? m2) (= (get-value m2) 0)))\n                   (set-value! product 0 me))\n                  ((and (has-value? m1) (has-value? m2))\n                   (set-value! product\n                               (* (get-value m1) (get-value m2))\n                               me))\n                  ((and (has-value? product) (has-value? m1))\n                   (set-value! m2\n                               (/ (get-value product) (get-value m1))\n                               me))\n                  ((and (has-value? product) (has-value? m2))\n                   (set-value! m1\n                               (/ (get-value product) (get-value m2))\n                               me))))\n  (define (process-forget-value)\n    (forget-value! product me)\n    (forget-value! m1 me)\n    (forget-value! m2 me)\n    (process-new-value))\n  (define (me request)\n    (cond ((eq? request 'I-have-a-value)\n           (process-new-value))\n          ((eq? request 'I-lost-my-value)\n           (process-forget-value))\n          (else\n            (error \"Unknown request - - MULTIPLIER\" request))))\n  (connect m1 me)\n  (connect m2 me)\n  (connect product me)\n  me) \n    A constant constructor simply sets the value of\n    the designated connector.  Any\n    I-have-a-value\n    or\n    I-lost-my-value\n    message sent to the constant box will produce an error.\n    (define (constant value connector)\n  (define (me request)\n    (error \"Unknown request - - CONSTANT\" request))\n  (connect connector me)\n  (set-value! connector value me)\n  me) \n    Finally, a probe prints a message about the setting or unsetting of\n    the designated connector:\n    (define (probe name connector)\n  (define (print-probe value)\n    (newline)\n    (display \"Probe: \")\n    (display name)\n    (display \" = \")\n    (display value))\n  (define (process-new-value)\n    (print-probe (get-value connector)))\n  (define (process-forget-value)\n    (print-probe \"?\"))\n  (define (me request)\n    (cond ((eq? request 'I-have-a-value)\n           (process-new-value))\n          ((eq? request 'I-lost-my-value)\n           (process-forget-value))\n          (else\n           (error \"Unknown request - - PROBE\" request))))\n  (connect connector me)\n  me) ","3.3.5#h3":"Representing connectors","3.3.5#p13":"\n    A connector is represented as a procedural object with local state variables\n    value, the current value of the connector;\n    informant, the object that set the\n    connector's value; and constraints,\n    a list of the constraints in which the connector participates.\n    (define (make-connector)\n  (let ((value false) (informant false) (constraints '()))\n    (define (set-my-value newval setter)\n      (cond ((not (has-value? me))\n             (set! value newval)\n             (set! informant setter)\n             (for-each-except setter\n                              inform-about-value\n                              constraints))\n            ((not (= value newval))\n             (error \"Contradiction\" (list value newval)))\n            (else 'ignored)))\n    (define (forget-my-value retractor)\n      (if (eq? retractor informant)\n          (begin (set! informant false)\n            (for-each-except retractor\n                             inform-about-no-value\n                             constraints))\n          'ignored))\n    (define (connect new-constraint)\n      (if (not (memq new-constraint constraints))\n          (set! constraints \n                (cons new-constraint constraints)))\n      (if (has-value? me)\n          (inform-about-value new-constraint))\n      'done)\n    (define (me request)\n      (cond ((eq? request 'has-value?)\n             (if informant true false))\n            ((eq? request 'value) value)\n            ((eq? request 'set-value!) set-my-value)\n            ((eq? request 'forget) forget-my-value)\n            ((eq? request 'connect) connect)\n            (else (error \"Unknown operation - - CONNECTOR\"\n                         request))))\n    me)) ","3.3.5#p14":"\n    The connector's local\n    procedureset-my-value\n    is called when there is a request to set the connector's value.  If\n    the connector does not currently have a value, it will set its value and\n    remember as informant the constraint that\n    requested the value to be set. Then the connector will\n    notify all of its participating constraints except the constraint that\n    requested the value to be set. This is accomplished using the following\n    iterator, which applies a designated\n    procedure\n    to all items in a list except a given one:\n    (define (for-each-except exception procedure list)\n  (define (loop items)\n    (cond ((null? items) 'done)\n          ((eq? (car items) exception) (loop (cdr items)))\n          (else (procedure (car items))\n                (loop (cdr items)))))\n  (loop list)) ","3.3.5#footnote-link-2":"2","3.3.5#p15":"\n    If a connector is asked to forget its value, it runs\n    \n\tthe local procedure\n\tforget-my-value,\n\twhich\n      \n    first checks to make sure that the request is coming from the same\n    object that set the value originally.  If so, the connector informs its\n    associated constraints about the loss of the value.\n  ","3.3.5#p16":"\n    The local\n    procedureconnect adds the designated new constraint\n    to the list of constraints if it is not already in that\n    list.\n    Then, if the connector has a value, it informs the new constraint of this\n    fact.\n  ","3.3.5#p17":"\n    The connector's\n    procedureme serves as a dispatch to the other internal\n    procedures\n    and also represents the connector as an object. The following\n    procedures\n    provide a syntax interface for the dispatch:\n    (define (has-value? connector)\n  (connector 'has-value?))\n\n(define (get-value connector)\n  (connector 'value))\n\n(define (set-value! connector new-value informant)\n  ((connector 'set-value!) new-value informant))\n\n(define (forget-value! connector retractor)\n  ((connector 'forget) retractor))\n\n(define (connect connector new-constraint)\n  ((connector 'connect) new-constraint)) ","3.3.5#ex-3.33":"\n    Using primitive multiplier, adder, and constant constraints, define a\n    procedureaverager that takes three connectors\n    a, b,\n    and c as inputs and establishes the\n    constraint that the value of\n    c is the average of the values of\n    a and b.\n    ","3.3.5#ex-3.34":"\n    Louis Reasoner wants to build a\n    \n    squarer, a constraint device with two\n    terminals such that the value of connector\n    b on the second\n    terminal will always be the square of the value\n    a on the first\n    terminal.  He proposes the following simple device made from a\n    multiplier:\n    (define (squarer a b)\n  (multiplier a a b))\n    There is a serious flaw in this idea.  Explain.\n    ","3.3.5#ex-3.35":"\n    Ben Bitdiddle tells Louis that one way to avoid the trouble in\n    exercise 3.34 is to define a\n    \n    squarer as a new primitive constraint.  Fill in the missing\n    portions in Ben's outline for a\n    procedure\n    to implement such a constraint:\n    \n(define (squarer a b)\n  (define (process-new-value)\n    (if (has-value? b)\n        (if (< (get-value b) 0)\n            (error \"square less than 0 - - SQUARER\" (get-value b))\n            $\\langle alternative1\\rangle$)\n        $\\langle alternative2 \\rangle$))\n  (define (process-forget-value) $\\langle body1 \\rangle$)\n    (define (me request) $\\langle body2 \\rangle$)\n  $\\langle rest\\ of\\ definition\\rangle$\n  me)\n      ","3.3.5#ex-3.36":"\n    Suppose we evaluate the following sequence of\n    \n\texpressions\n      \n    in the\n    global\n    environment:\n    (define a (make-connector))\n(define b (make-connector))\n(set-value! a 10 'user)\n    At some time during evaluation of the\n    set-value!,\n    the following expression from the connector's local\n    procedure\n    is evaluated:\n    (for-each-except setter inform-about-value constraints)\n    Draw an environment diagram showing the environment in which the above\n    expression is evaluated.\n    ","3.3.5#ex-3.37":"\n    The\n    celsius-fahrenheit-converterprocedure\n    is cumbersome when\n    compared with a more expression-oriented style of definition, such as\n    (define (celsius-fahrenheit-converter x)\n  (c+ (c* (c/ (cv 9) (cv 5))\n          x)\n      (cv 32)))\n\n(define C (make-connector))\n(define F (celsius-fahrenheit-converter C))\n    Here\n    c+,c*,\n    etc. are the \"constraint\"\n    versions of the  arithmetic operations.  For example,\n    c+\n    takes two connectors as arguments and returns a connector that is\n    related to these by an adder constraint:\n    (define (c+ x y)\n  (let ((z (make-connector)))\n    (adder x y z)\n    z))\n    Define analogous\n    proceduresc-,c*,c/,\n    and\n    cv\n    (constant value) that enable us to define compound constraints as in\n    the converter example above.","3.3.5#footnote-link-3":"3","3.3.5#footnote-1":"Constraint propagation first appeared in the incredibly\n    forward-looking \n    \n    SKETCHPAD system of\n    \n    Ivan Sutherland (1963).  A beautiful constraint-propagation system based\n    on the \n    \n    Smalltalk language was developed by \n    \n    Alan Borning (1977) at \n    \n    Xerox Palo Alto Research Center.  Sussman, Stallman, and Steele\n    applied constraint propagation to electrical circuit analysis \n    \n    (Sussman and Stallman 1975; \n    Sussman and Steele 1980). \n    \n    TK!Solver\n    \n    (Konopasek and Jayaraman 1984) \n    is an extensive modeling environment based on constraints.","3.3.5#footnote-2":"The\n    setter might not be a constraint.  In our\n    temperature example, we used\n    user\n    as the\n    setter.","3.3.5#footnote-3":"The\n    \n    expression-oriented format\n    is convenient because it avoids the need to name the intermediate\n    expressions in a computation.  Our original formulation of the\n    constraint language is cumbersome in the same way that many languages\n    are cumbersome when dealing with operations on compound data.  For\n    example, if we wanted to compute the product\n    ${(a+b)}\\cdot{(c+d)}$, where the\n    variables represent vectors, we could work in\n    \"imperative style,\"\n    using\n    procedures\n    that set the values of designated vector arguments\n    but do not themselves return vectors as values:\n    (v-sum a b temp1)\n(v-sum c d temp2)\n(v-prod temp1 temp2 answer)\n    Alternatively, we could deal with expressions, using\n    procedures\n    that return vectors as values, and thus avoid\n    explicitly mentioning temp1 and\n    temp2:\n    (define answer (v-prod (v-sum a b) (v-sum c d)))\n    Since\n    Lisp\n    allows us to return compound objects as values of\n    procedures,\n    we can transform our imperative-style constraint language\n    into an expression-oriented style as shown in this exercise.\n    \n\tIn languages that are impoverished in handling compound objects, such\n\tas\n\t\n\tAlgol, Basic, and Pascal (unless one explicitly uses Pascal pointer\n\tvariables), one is usually stuck with the imperative style when\n\tmanipulating compound objects.\n      \n    Given the advantage of the\n    expression-oriented format, one might ask if there is any reason to\n    have implemented the system in imperative style, as we did in this\n    section.  One reason is that the non-expression-oriented constraint\n    language provides a handle on constraint objects (e.g., the value of\n    the adderprocedure)\n    as well as on connector objects.  This is\n    useful if we wish to extend the system with new operations that\n    communicate with constraints directly rather than only indirectly via\n    operations on connectors.  Although it is easy to implement the\n    expression-oriented style in terms of the imperative implementation,\n    it is very difficult to do the converse.","3.4":"3.4  Concurrency: Time Is of the Essence","3.4#p1":"\n    We've seen the power of computational objects with local state as\n    tools for modeling.  Yet, as\n    section 3.1.3\n    warned, this power extracts a price: the loss of referential\n    transparency, giving rise to a thicket of questions about sameness and\n    change, and the need to abandon the substitution model of evaluation in\n    favor of the more intricate environment model.\n  ","3.4#p2":"\n    The central issue lurking beneath the complexity of state, sameness,\n    and change is that by introducing assignment we are forced to admit\n    time into our computational models.  Before we introduced\n    assignment, all our programs were timeless, in the sense that any\n    expression that has a value always has the same value.  In contrast,\n    recall the example of modeling withdrawals from a bank account\n    and returning the resulting balance,\n    introduced at the beginning of\n    section 3.1.1:\n\n    (withdraw 25) (withdraw 25) \n    Here successive evaluations of the same expression yield different\n    values.  This behavior arises from the fact that the execution of\n    assignments (in this case, assignments to the variable\n    balance) delineates moments in time\n    when values change.  The result of evaluating an expression depends not\n    only on the expression itself, but also on whether the evaluation occurs\n    before or after these moments.  Building models in terms of computational\n    objects with local state forces us to confront time as an essential concept\n    in programming.\n  ","3.4#p3":"\n    We can go further in structuring computational models to match our\n    perception of the physical world.  Objects in the world do not change\n    one at a time in sequence.  Rather we perceive them as acting\n    concurrently—all at once.  So it is often natural to\n    model systems as collections of \n    computational processes \n    that execute concurrently.\n    Just as we can make our programs modular by\n    organizing models in terms of objects with separate local state, it is\n    often appropriate to divide computational models into parts that evolve\n    separately and concurrently.  Even if the programs are to be executed on\n    a sequential computer, the practice of writing programs as if they were\n    to be executed concurrently forces the programmer to avoid inessential\n    timing constraints and thus makes programs more modular.\n  ","3.4#p4":"\n    In addition to making programs more modular, concurrent computation\n    can provide a speed advantage over sequential computation.  Sequential\n    computers execute only one operation at a time, so the amount of time\n    it takes to perform a task is proportional to the total number of\n    operations performed.\n    However, if it is possible to decompose a problem into pieces that are\n    relatively independent and need to communicate only rarely, it may be\n    possible to allocate pieces to separate computing processors,\n    producing a speed advantage proportional to the number of processors\n    available.\n  ","3.4#footnote-link-1":"1","3.4#p5":"\n    Unfortunately, the complexities introduced by assignment become even\n    more problematic in the presence of concurrency.  The fact of\n    concurrent execution, either because the world operates in parallel or\n    because our computers do, entails additional complexity in our\n    understanding of time.\n  ","3.4#footnote-1":"Most real processors actually execute a few\n    operations at a time, following a strategy called \n    pipelining.  Although this technique greatly improves the\n    effective\n    utilization of the hardware, it is used only to speed up the execution\n    of a sequential instruction stream, while retaining the behavior of\n    the sequential program.","3.4.1":"3.4.1  \n    The Nature of Time in Concurrent Systems","3.4.1#p1":"\n    On the surface, time seems straightforward.  It\n    is an ordering imposed on events.\n    For any events $A$ and\n    $B$,\n    either $A$ occurs before\n    $B$,\n    $A$ and\n    $B$ are simultaneous, or\n    $A$ occurs after\n    $B$.  For instance,\n    returning to the bank account example, suppose that Peter withdraws\n    $10 and Paul withdraws $25 from a \n    \n    joint account that initially\n    contains $100, leaving $65 in the account.  Depending on the\n    order of the two withdrawals, the sequence of balances in the account is\n    either $\\$100 \\rightarrow \\$90 \\rightarrow\\$65$\n    or $\\$100 \\rightarrow \\$75 \\rightarrow\\$65$.\n    In a computer implementation of the banking system, this changing\n    sequence of balances could be  modeled by successive assignments to\n    a variable balance.\n  ","3.4.1#footnote-link-1":"1","3.4.1#p2":"\n    In complex situations, however, such a view can be problematic.\n    Suppose that Peter and Paul, and other people besides, are\n    accessing the same bank account through a network of banking machines\n    distributed all over the world.  The actual sequence of balances in\n    the account will depend critically on the detailed timing of the\n    accesses and the details of the communication among the machines.\n  ","3.4.1#p3":"\n    This\n    \n    indeterminacy in the order of events can pose serious problems in\n    the design of concurrent systems.  For instance, suppose that the\n    withdrawals made by Peter and Paul are implemented as two separate\n    processes \n    sharing a common variable balance, each \n    process \n    specified by the\n    procedure\n    given in section 3.1.1:\n    (define (withdraw amount)\n   (if (>= balance amount)\n     (begin (set! balance (- balance amount))\n            balance)\n     \"Insufficient funds\")) \n    If the two \n    processes \n    operate independently, then Peter might test the\n    balance and attempt to withdraw a legitimate amount.\n    \n    However, Paul\n    might withdraw some funds in between the time that Peter checks the\n    balance and the time Peter completes the withdrawal, thus invalidating\n    Peter's test.\n  ","3.4.1#p4":"\n    Things can be worse still.  Consider the\n    \n\texpression\n      (set! balance (- balance amount))\n    executed as part of each withdrawal process.  This consists of three\n    steps: (1) accessing the value of the balance\n    variable; (2) computing the new balance; (3) setting\n    balance to this new value.  If Peter and\n    Paul's withdrawals execute this statement concurrently, then the\n    two withdrawals might interleave the order in which they access\n    balance and set it to the new value.\n  ","3.4.1#fig-":"","3.4.1#p5":"\n    The timing diagram in\n    \n\tfigure \n    depicts\n    an order of events where balance starts at 100,\n    Peter withdraws 10, Paul withdraws 25, and yet the final value of\n    balance is 75.  As shown in the diagram,\n    the reason for this anomaly is that Paul's assignment of 75 to\n    balance is made under the assumption that\n    the value of balance to be decremented is 100.\n    That assumption, however, became invalid when Peter changed\n    balance to 90.  This is a catastrophic\n    failure for the banking system, because the total amount of money in the\n    system is not conserved.  Before the transactions, the total amount of\n    money was $100.  Afterwards, Peter has $10, Paul has\n    $25, and the bank has $75.","3.4.1#footnote-link-2":"2","3.4.1#p6":"\n    The general phenomenon illustrated\n    here is that several \n    processes \n    may\n    \n    share a common state variable.  What makes this complicated is that more than one \n    process \n    may be trying to manipulate the shared state at the same\n    time.  For the bank account example, during each transaction, each\n    customer should be able to act as if the other customers did not\n    exist.  When\n    a customer changes\n    the balance in a way that depends on\n    the balance,\n    he    \n    must be able to assume that, just before the moment of\n    change, the balance is still what\n    he    \n    thought it was.\n  ","3.4.1#h1":"Correct behavior of concurrent programs","3.4.1#p7":"\n    The above example typifies the subtle bugs that can creep into\n    concurrent programs.  The root of this complexity lies in the\n    assignments to variables that are shared among the different\n    processes.  \n    We already know that we must be careful in writing programs that use\n    set!,\n    because the results of a computation depend on the order in which the\n    assignments occur.\n    With concurrent \n    processes \n    we must be especially careful about\n    assignments, because we may not be able to control the order of the\n    assignments made by the different \n    processes.  \n    If several such changes\n    might be made concurrently (as with two depositors accessing a joint\n    account) we need some way to ensure that our system behaves correctly.\n    For example, in the case of withdrawals from a joint bank account, we\n    must ensure that money is conserved.\n    To make concurrent programs behave correctly, we may have to\n    place some restrictions on concurrent execution.\n  ","3.4.1#footnote-link-3":"3","3.4.1#p8":"\n    One possible restriction on concurrency would stipulate that no two\n    operations that change any shared state variables can occur at the same\n    time.  This is an extremely stringent requirement.  For distributed banking,\n    it would require the system designer to ensure that only one transaction\n    could proceed at a time. This would be both inefficient and overly\n    conservative.  Figure 3.30 shows\n    Peter and Paul sharing a bank account, where Paul has a private account\n    as well. The diagram illustrates two withdrawals from the shared account\n    (one by Peter and one by Paul) and a deposit to Paul's private\n    account.\n    The two withdrawals from the shared account must not be concurrent (since\n    both access and update the same account), and Paul's deposit and\n    withdrawal must not be concurrent (since both access and update the amount\n    in Paul's wallet). But there should be no problem permitting\n    Paul's deposit to his private account to proceed concurrently with\n    Peter's withdrawal from the shared account.\n    ","3.4.1#footnote-link-4":"4","3.4.1#fig-3.30":"","3.4.1#p9":"\n    A less stringent restriction on concurrency would ensure that a\n    concurrent system produces the same result as if the \n    processes \n    had run sequentially in some order. There are two important aspects to this\n    requirement. First, it does not require the \n    processes \n    to actually run sequentially, but only to produce results that are the same\n    as if they had run sequentially.  For the example in\n    figure 3.30, the designer of the\n    bank account system can safely allow Paul's deposit and Peter's\n    withdrawal to happen concurrently, because the net result will be the same as\n    if the two operations had happened sequentially.  Second, there may be more\n    than one possible \"correct\" result produced by a concurrent\n    program, because we require only that the result be the same as for\n    some sequential order. For example, suppose that Peter and\n    Paul's joint account starts out with $100, and Peter deposits\n    $40 while Paul concurrently withdraws half the money in the account.\n    Then sequential execution could result in the account balance being either\n    $70 or $90 (see\n    exercise 3.38).","3.4.1#footnote-link-5":"5","3.4.1#p10":"\n    There are still weaker requirements for correct execution of concurrent\n    programs.  A program for simulating \n    \n    diffusion (say, the flow of heat in an object) might consist of a large\n    number of \n    processes,\n    each one representing a small volume of space, that update their values\n    concurrently.  Each \n    process \n    repeatedly changes its value to the average of its own value and its\n    neighbors' values. This algorithm converges to the right answer\n    independent of the order in which the operations are done; there is no\n    need for any restrictions on concurrent use of the shared values.\n  ","3.4.1#ex-3.38":"\n    Suppose that\n    \n    Peter, Paul, and Mary share a joint bank account that\n    initially contains $100.  Concurrently, Peter deposits\n    $10, Paul withdraws $20, and Mary withdraws half the\n    money in the account, by executing the following commands:\n    Peter:(set! balance (+ balance 10))Paul:(set! balance (- balance 20))Mary:(set! balance (- balance (/ balance 2)))\n        List all the different possible values for\n        balance\n\tafter these three transactions have been completed,\n\tassuming that the banking system forces the three \n  processes \n  to run sequentially in some order.\n      \n        What are some other values\n        that could be produced if the system allows the \n        processes \n\tto be interleaved? Draw timing diagrams like the one in\n\tfigure 3.29 to explain\n\thow these values can occur.\n      ","3.4.1#footnote-1":"To quote some graffiti seen\n    on a \n    building wall in Cambridge,\n    Massachusetts: \"Time\n    \n    is a device that was invented to keep\n    everything from happening at once.\"","3.4.1#footnote-2":"An even worse\n    failure for this system could occur if the two\n    set! operations\n    attempt to change the balance simultaneously, in which case the actual data\n    appearing in memory might end up being a random combination of the\n    information being written by the two \n    processes.  \n    Most computers have interlocks on\n    the primitive memory-write operations, which protect against such\n    simultaneous access.  Even this seemingly simple kind of protection,\n    however, raises implementation challenges in the design of\n    multiprocessing computers, where elaborate \n    cache-coherence\n    protocols are required to ensure that the various processors will\n    maintain a consistent view of memory contents, despite the fact that\n    data may be replicated (\"cached\") among the different\n    processors to increase the speed of memory access.","3.4.1#footnote-3":"The factorial program in\n    section 3.1.3 illustrates this for\n    a single sequential \n    process.","3.4.1#footnote-4":"The columns show the contents of Peter's wallet,\n    the joint account (in Bank1), Paul's wallet, and Paul's\n    private account (in Bank2), before and after each withdrawal (W) and\n    deposit (D). Peter withdraws $10 from Bank1; Paul deposits\n    $5 in Bank2, then withdraws $25 from Bank1.","3.4.1#footnote-5":"A more formal way\n    to express this idea is to say that concurrent programs are inherently \n    nondeterministic. That is, they are described not by single-valued\n    functions, but by functions whose results are sets of possible values.\n    In section 4.3 we will\n    study a language for expressing nondeterministic computations.\n    ","3.4.2":"3.4.2  \n    Mechanisms for Controlling Concurrency","3.4.2#p1":"\n    We've seen that the difficulty in dealing with concurrent \n    processes \n    is rooted in the need to consider the interleaving of the order of events\n    in the different \n    processes.  \n    For example, suppose we have two\n    processes, \n    one with three ordered events $(a,b,c)$\n    and one with three ordered events $(x,y,z)$.\n    If the two \n    processes \n    run concurrently, with no constraints on how their execution is\n    interleaved, then there are 20 different possible orderings for the events\n    that are consistent with the individual orderings for the two \n    processes:\n    \n      \\[ \\begin{array}{cccc}\n      (a,b,c,x,y,z) & (a,x,b,y,c,z) & (x,a,b,c,y,z) & (x,a,y,z,b,c)\\\\\n      (a,b,x,c,y,z) & (a,x,b,y,z,c) & (x,a,b,y,c,z) & (x,y,a,b,c,z)\\\\\n      (a,b,x,y,c,z) & (a,x,y,b,c,z) & (x,a,b,y,z,c) & (x,y,a,b,z,c)\\\\\n      (a,b,x,y,z,c) & (a,x,y,b,z,c) & (x,a,y,b,c,z) & (x,y,a,z,b,c)\\\\\n      (a,x,b,c,y,z) & (a,x,y,z,b,c) & (x,a,y,b,z,c) & (x,y,z,a,b,c)\n      \\end{array} \\]\n    \n    As programmers designing this system, we would have to consider the\n    effects of each of these 20 orderings and check that each behavior is\n    acceptable.  Such an approach rapidly becomes unwieldy as the numbers of \n    processes and events increase.\n  ","3.4.2#p2":"\n    A more practical approach to the design of concurrent systems is to\n    devise general mechanisms that allow us to constrain the interleaving\n    of concurrent \n    processes \n    so that we can be sure that the program\n    behavior is correct.  Many mechanisms have been developed for this\n    purpose.  In this section, we describe one of them, the \n    serializer.\n  ","3.4.2#h1":"Serializing access to shared state","3.4.2#p3":"\n    Serialization implements the following idea: \n    Processes \n    will execute concurrently, but there will be certain collections of\n    procedures\n    that cannot be executed concurrently.  More precisely, serialization\n    creates distinguished sets of\n    procedures\n    such that only one execution of a\n    procedure\n    in each serialized set is permitted to happen at a time. If some\n    procedure\n    in the set is being executed, then a \n    process \n    that attempts to execute any\n    procedure\n    in the set will be forced to wait\n    until the first execution has finished.\n  ","3.4.2#p4":"\n    We can use serialization to control access to shared variables.\n    For example, if we want to update a shared variable based on the\n    previous value of that variable, we put the access to the previous\n    value of the variable and the assignment of the new value to the\n    variable in the same\n    procedure.\n    We then ensure that no other\n    procedure\n    that assigns to the variable can run concurrently with this\n    procedure\n    by serializing all of these\n    procedures\n    with the same serializer.  This guarantees that the value of the\n    variable cannot be changed between an access and the corresponding\n    assignment.\n  ","3.4.2#h2":"Serializers in Scheme","3.4.2#p5":"\n    To make the above mechanism more concrete, suppose that we have\n    extended \n    Scheme\n    to include a\n    procedure\n    called \n    parallel-execute:\n(parallel-execute p$_{1}$ p$_{2}$ $\\ldots$ p$_{k}$)\n      \n        Each p must be a procedure of no arguments.  \n        Parallel-execute\n        creates a separate process for each\n        p, which applies\n        p (to no arguments).\n      \n    These \n    processes \n    all run concurrently.","3.4.2#footnote-link-1":"1","3.4.2#p6":"\n    As an example of how this is used, consider\n    (define x 10)\n\n(parallel-execute (lambda () (set! x (* x x)))\n  (lambda () (set! x (+ x 1)))) \n        This creates two concurrent\n        processes—$P_1$, which\n        sets x to\n        x times x,\n        and $P_2$, which\n\tincrements x.  After execution is\n\tcomplete, x will be left with one of five\n\tpossible values, depending on the interleaving of the events of\n        $P_1$ and $P_2$:\n        101:$P_1$\n            sets x to 100 and then\n            $P_2$ increments\n            x to 101.121:$P_2$ increments\n\t    x to 11 and then\n            $P_1$ sets x\n\t    to x times\n\t    x.110:$P_2$ changes\n            x from 10 to 11 between the two\n            times that $P_1$\n\t      accesses the value of\n            x during the evaluation of\n            (* x x).\n\t    11:$P_2$ accesses\n\t    x, then\n            $P_1$ sets x\n\t    to 100,\n            then $P_2$ sets\n\t    x.100:$P_1$ accesses\n\t    x (twice),\n            then $P_2$ sets\n\t    x to 11,\n            then $P_1$ sets\n\t    x.","3.4.2#p7":"\n    We can constrain the concurrency by using serialized\n    procedures,\n    which are created by serializers. Serializers are constructed by\n    make-serializer,\n      \n    whose implementation is given below.  A serializer takes a\n    procedure\n    as argument and returns a serialized\n    procedure\n    that behaves like the original\n    procedure.\n    All calls to a given serializer return serialized\n    procedures\n    in the same set.\n  ","3.4.2#p8":"\n    Thus, in contrast to the example above, executing\n    (define x 10)\n\n(define s (make-serializer))\n\n(parallel-execute (s (lambda () (set! x (* x x))))\n                  (s (lambda () (set! x (+ x 1))))) \n    can produce only two possible values for\n    x, 101 or 121.\n    The other possibilities are eliminated, because the execution of\n    $P_1$ and\n\t$P_2$ cannot be interleaved.\n      ","3.4.2#p9":"\n    Here is a version of the\n    make-accountprocedure\n    from section 3.1.1,\n    where the deposits and withdrawals have been\n    \n    serialized:\n    (define (make-account balance)\n  (define (withdraw amount)\n    (if (>= balance amount)\n      (begin (set! balance (- balance amount))\n             balance)\n      \"Insufficient funds\"))\n  (define (deposit amount)\n    (set! balance (+ balance amount))\n    balance)\n  (let ((protected (make-serializer)))\n    (define (dispatch m)\n      (cond ((eq? m 'withdraw) (protected withdraw))\n            ((eq? m 'deposit) (protected deposit))\n            ((eq? m 'balance) balance)\n            (else (error \"Unknown request - - MAKE-ACCOUNT\"\n                         m))))\n    dispatch)) \n    With this implementation, two \n    processes \n    cannot be withdrawing from or\n    depositing into a single account concurrently.  This eliminates the source\n    of the error illustrated in figure 3.29,\n    where Peter changes the account balance between the times when Paul accesses\n    the balance to compute the new value and when Paul actually performs the\n    assignment.  On the other hand, each account has its own serializer,\n    so that deposits and withdrawals for different accounts can proceed\n    concurrently.\n  ","3.4.2#ex-3.39":"\n    Which of the five possibilities in the \n    parallel \n    execution shown above remain if we instead serialize execution as follows:\n    (define x 10)\n\n(define s (make-serializer))\n\n(parallel-execute (lambda () (set! x ((s (lambda () (* x x))))))\n                  (s (lambda () (set! x (+ x 1))))) ","3.4.2#ex-3.40":"\n    Give all possible values of x\n    that can result from executing\n    (define x 10)\n\n(parallel-execute (lambda () (set! x (* x x)))\n                  (lambda () (set! x (* x x x))))  Which of these possibilities remain if we instead use serialized\n    procedures:(define x 10)\n\n(define s (make-serializer))\n\n(parallel-execute (s (lambda () (set! x (* x x))))\n                  (s (lambda () (set! x (* x x x))))) ","3.4.2#ex-3.41":"\n    Ben Bitdiddle worries that it would be better to implement the bank\n    account as follows (where the commented line has been changed):\n    (define (make-account balance)\n  (define (withdraw amount)\n    (if (>= balance amount)\n        (begin (set! balance (- balance amount))\n          balance)\n        \"Insufficient funds\"))\n  (define (deposit amount)\n    (set! balance (+ balance amount))\n    balance)\n  (let ((protected (make-serializer)))\n    (define (dispatch m)\n      (cond ((eq? m 'withdraw) (protected withdraw))\n            ((eq? m 'deposit) (protected deposit))\n            ((eq? m 'balance)\n            ((protected (lambda () balance)))) \n            (else (error \"Unknown request - - MAKE-ACCOUNT\"\n                         m))))\n    dispatch)) \n    because allowing unserialized access to the bank balance can\n    result in anomalous behavior.  Do you agree?  Is there any\n    scenario that demonstrates Ben's concern?\n  ","3.4.2#ex-3.42":"\n    Ben Bitdiddle suggests that it's a waste of time to\n    create a new serialized\n    procedure\n    in response to every withdraw and\n    deposit message.  He says that\n    make-account\n    could be changed so that the calls to\n    protected\n    are done outside the dispatchprocedure.\n    That is, an account would return the same serialized\n    procedure\n    (which was created at the same time as the account) each time\n    it is asked for a withdrawal\n    procedure.(define (make-account balance)\n  (define (withdraw amount)\n    (if (>= balance amount)\n        (begin (set! balance (- balance amount))\n          balance)\n        \"Insufficient funds\"))\n  (define (deposit amount)\n    (set! balance (+ balance amount))\n    balance)\n  (let ((protected (make-serializer)))\n        (let ((protected-withdraw (protected withdraw))\n              (protected-deposit (protected deposit)))\n          (define (dispatch m)\n            (cond ((eq? m 'withdraw) protected-withdraw)\n                  ((eq? m 'deposit) protected-deposit)\n                  ((eq? m 'balance) balance)\n                  (else (error \"Unknown request - - MAKE-ACCOUNT\"\n                               m))))\n  dispatch))) \n    Is this a safe change to make?  In particular, is there any difference\n    in what concurrency is allowed by these two versions of\n    make-account?\n    ","3.4.2#h3":"Complexity of using multiple shared resources","3.4.2#p10":"\n    Serializers provide a powerful abstraction that helps isolate the\n    complexities of concurrent programs so that they can be dealt with\n    carefully and (hopefully) correctly.  However, while using serializers\n    is relatively straightforward when there is only a single shared\n    resource (such as a single bank account), concurrent programming can\n    be treacherously difficult when there are multiple shared resources.\n  ","3.4.2#p11":"\n    To illustrate one of the difficulties that can arise, suppose we wish to\n    \n    swap the balances in two bank accounts.  We access each account to find\n    the balance, compute the difference between the balances, withdraw this\n    difference from one account, and deposit it in the other account.\n    We could implement this as\n    follows:(define (exchange account1 account2)\n  (let ((difference (- (account1 'balance)\n                       (account2 'balance))))\n       ((account1 'withdraw) difference)\n    ((account2 'deposit) difference))) ","3.4.2#footnote-link-2":"2","3.4.2#p12":"\n    This\n    procedure\n    works well when only a single \n    process \n    is trying to do the exchange.  Suppose, however, that Peter and Paul both\n    have access to accounts $a_1$,\n    $a_2$, and $a_3$, and\n    that Peter exchanges $a_1$ and\n    $a_2$ while Paul concurrently exchanges\n    $a_1$ and $a_3$.\n    Even with account deposits and withdrawals\n    serialized for individual accounts (as in the\n    make-accountprocedure\n    shown above in this section), exchange can\n    still produce incorrect results.  For example, Peter might compute the\n    difference in the balances for $a_1$ and\n    $a_2$, but then Paul might change the balance in\n    $a_1$ before Peter is able to complete the\n    exchange.\n    For correct behavior, we must arrange for the\n    exchangeprocedure\n    to lock out any other concurrent accesses to the accounts during the\n    entire time of the exchange.\n  ","3.4.2#footnote-link-3":"3","3.4.2#p13":"\n    One way we can accomplish this is by using both accounts' serializers\n    to serialize the entire exchangeprocedure.\n    To do this, we will arrange for access to an account's serializer.\n    Note that we are deliberately breaking the modularity of the bank-account\n    object by exposing the serializer.  The following version of\n    make-account\n    is identical to the original version given in\n    section 3.1.1, except that a\n    serializer is provided to protect the balance variable, and the serializer\n    is exported via message passing:\n    (define (make-account-and-serializer balance)\n  (define (withdraw amount)\n    (if (>= balance amount)\n        (begin (set! balance (- balance amount))\n          balance)\n        \"Insufficient funds\"))\n  (define (deposit amount)\n    (set! balance (+ balance amount))\n    balance)\n  (let ((balance-serializer (make-serializer)))\n    (define (dispatch m)\n      (cond ((eq? m 'withdraw) withdraw)\n            ((eq? m 'deposit) deposit)\n            ((eq? m 'balance) balance)\n            ((eq? m 'serializer) balance-serializer)\n            (else (error \"Unknown request - - MAKE-ACCOUNT\"\n                         m))))\n    dispatch)) ","3.4.2#p14":"\n    We can use this to do serialized deposits and withdrawals.  However,\n    unlike our earlier serialized account, it is now the responsibility of\n    each user of bank-account objects to explicitly manage the\n    serialization, for example as\n    follows:(define (deposit account amount)\n  (let ((s (account 'serializer))\n        (d (account 'deposit)))\n    ((s d) amount))) ","3.4.2#footnote-link-4":"4","3.4.2#p15":"\n    Exporting the serializer in this way gives us enough flexibility to\n    implement a serialized exchange program.  We simply serialize the original\n    exchangeprocedure\n    with the serializers for both accounts:\n    (define (serialized-exchange account1 account2)\n  (let ((serializer1 (account1 'serializer))\n        (serializer2 (account2 'serializer)))\n    ((serializer1 (serializer2 exchange))\n     account1\n     account2))) ","3.4.2#ex-3.43":"\n    Suppose that the balances in three accounts start out as $10,\n    $20, and $30, and that multiple \n    processes \n    run, exchanging the balances in the accounts.  Argue that if the \n    processes \n    are run sequentially,\n    after any number of concurrent exchanges, the account balances should be \n    $10, $20, and $30 in some order.\n    Draw a timing diagram like the one in\n    figure 3.29 to\n    show how this condition can be violated if the exchanges are\n    implemented using the first version of the account-exchange program in\n    this section.  On the other hand, argue that even with this\n    exchange program, the sum of the balances\n    in the accounts will be preserved.  Draw a timing diagram to show how\n    even this condition would be violated if we did not serialize the\n    transactions on individual accounts.\n    ","3.4.2#ex-3.44":"\n    Consider the problem of\n    \n    transferring an amount from one account to\n    another.  Ben Bitdiddle claims that this can be accomplished with the\n    following\n    procedure,\n    even if there are multiple people concurrently\n    transferring money among multiple accounts, using any account\n    mechanism that serializes deposit and withdrawal transactions, for\n    example, the version of\n    make-account\n    in the text above.\n    (define (transfer from-account to-account amount)\n  ((from-account 'withdraw) amount)\n  ((to-account 'deposit) amount)) Louis Reasoner claims that there is a problem here, and that we need\n    to use a more sophisticated method, such as the one required for\n    dealing with the exchange problem.  Is Louis right?  If not, what is\n    the essential difference between the transfer problem and the exchange\n    problem?  (You should assume that the balance in\n    from-account\n    is at least amount.)\n    ","3.4.2#ex-3.45":"\n    Louis Reasoner thinks our bank-account system is unnecessarily complex\n    and error-prone now that deposits and withdrawals aren't\n    automatically serialized. He suggests that\n    make-account-and-serializer\n    should have exported the serializer \n    \n        (for use by such procedures as\n        serialized-exchange)\n      \n    in addition to (rather than instead of) using it to serialize accounts and\n    deposits as\n    make-account\n    did. He proposes to redefine accounts as follows:\n    (define (make-account-and-serializer balance)\n  (define (withdraw amount)\n    (if (>= balance amount)\n      (begin (set! balance (- balance amount))\n        balance)\n      \"Insufficient funds\"))\n  (define (deposit amount)\n    (set! balance (+ balance amount))\n    balance)\n  (let ((balance-serializer (make-serializer)))\n    (define (dispatch m)\n      (cond ((eq? m 'withdraw) (balance-serializer withdraw))\n            ((eq? m 'deposit) (balance-serializer deposit))\n            ((eq? m 'balance) balance)\n            ((eq? m 'serializer) balance-serializer)\n            (else (error \"Unknown request - - MAKE-ACCOUNT\"\n                         m))))\n    dispatch)) \n    Then deposits are handled as with the original\n    make-account:(define (deposit account amount)\n  ((account 'deposit) amount)) \n    Explain what is wrong with Louis's reasoning.  In particular,\n    consider what happens when\n    serialized-exchange\n    is called.\n    ","3.4.2#h4":"Implementing serializers","3.4.2#p16":"\n    We implement serializers in terms of a more primitive synchronization\n    mechanism called a \n    mutex.  A mutex is an object that supports two\n    operations—the mutex can be \n    acquired, and the mutex can be\n    released.  Once a mutex has been acquired, no other acquire\n    operations on that mutex may proceed until the mutex is\n    released.\n    In our implementation, each serializer has an associated mutex.  Given a\n    \n\tprocedure\n\tp, \t\n      \n    the serializer returns a\n    procedure\n    that acquires the mutex, runs\n    p,\n      \n    and then releases the mutex.  This ensures that only one of the\n    procedures\n    produced by the serializer can be running at once, which is\n    precisely the serialization property that we need to guarantee.\n    (define (make-serializer)\n  (let ((mutex (make-mutex)))\n    (lambda (p)\n      (define (serialized-p . args)\n        (mutex 'acquire)\n        (let ((val (apply p args)))\n          (mutex 'release)\n          val))\n      serialized-p))) ","3.4.2#footnote-link-5":"5","3.4.2#p17":"\n    The mutex is a mutable object (here we'll use a one-element list,\n    which we'll refer to as a \n    cell) that can hold the value true or false.  When the value is\n    false, the mutex is available to be acquired.  When the value is true, the\n    mutex is unavailable, and any\n    process \n    that attempts to acquire the mutex must wait.\n  ","3.4.2#p18":"\n    Our mutex constructor\n    make-mutex\n    begins by initializing the cell contents to false.  To acquire the mutex,\n    we test the cell.  If the mutex is available, we set the cell contents to\n    true and proceed. Otherwise, we wait in a loop, attempting to acquire over\n    and over again, until we find that the mutex is available.\n    To release the mutex, we set the cell contents to false.\n    (define (make-mutex)\n  (let ((cell (list false)))            \n    (define (the-mutex m)\n      (cond ((eq? m 'acquire)\n             (if (test-and-set! cell)\n                 (the-mutex 'acquire))) \n                 ((eq? m 'release) (clear! cell))))\n    the-mutex))\n\n(define (clear! cell)\n  (set-car! cell false)) ","3.4.2#footnote-link-6":"6","3.4.2#p19":"Test-and-set!\n    tests the cell and returns the result of the test.  In addition, if the\n    test was false,\n    test-and-set!\n    sets the cell contents to true before returning false.  We can express this\n    behavior as the following\n    procedure:(define (test-and-set! cell)\n  (if (car cell)\n      true\n      (begin (set-car! cell true)\n        false))) ","3.4.2#p20":"\n    However, this implementation of\n    test-and-set!\n    does not suffice as it stands.  There is a crucial subtlety here, which is\n    the essential place where concurrency control enters the system: The\n    test-and-set!\n    operation must be performed \n    atomically. That is, we must guarantee that, once a \n    process \n    has tested the cell and found it to be false, the cell contents will\n    actually be set to true before any other \n    process \n    can test the cell.  If we do not make this guarantee, then the mutex can\n    fail in a way similar to the bank-account failure in\n    figure 3.29. (See\n    exercise 3.46.)\n  ","3.4.2#p21":"\n    The actual implementation of\n    test-and-set!\n    depends on the details of how our system runs concurrent \n    processes.\n    For example, we might be executing concurrent \n    processes \n    on a sequential processor using a \n    \n    time-slicing mechanism that cycles through the \n    processes,\n    permitting each \n    process \n    to run for a short time before interrupting it\n    and moving on to the next \n    process.\n    In that case,\n    test-and-set!\n    can work by disabling time slicing during the testing and\n    setting.\n    Alternatively, multiprocessing computers provide instructions that\n    support atomic operations directly in hardware.","3.4.2#footnote-link-7":"7","3.4.2#footnote-link-8":"8","3.4.2#ex-3.46":"\n    Suppose that we implement\n    test-and-set!\n    using an ordinary\n    procedure\n    as shown in the text, without attempting to make the operation atomic.\n    Draw a timing diagram like the one in\n    figure 3.29 to demonstrate how the mutex\n    implementation can fail by allowing two \n    processes \n    to acquire the mutex at the same time.\n    ","3.4.2#ex-3.47":"\n    A semaphore\n    \n    (of size $n$) is a generalization of\n    a mutex. Like a mutex, a semaphore supports acquire and release operations,\n    but it is more general in that up to $n$processes \n    can acquire it\n    concurrently.  Additional \n    processes \n    that attempt to acquire the semaphore must wait for release operations.  \n    Give implementations of semaphores\n    in terms of mutexes\n      \n        in terms of atomic\n        test-and-set!\n        operations.\n      ","3.4.2#h5":"Deadlock","3.4.2#p22":"\n    Now that we have seen how to implement serializers, we can see\n    that account exchanging still has a problem, even with the\n    serialized-exchangeprocedure\n    above.\n    Imagine that Peter attempts to exchange $a_1$\n    with $a_2$ while Paul concurrently attempts to\n    exchange $a_2$ with\n    $a_1$. Suppose that Peter's \n    process \n    reaches the point where it has entered a serialized\n    procedure\n    protecting $a_1$ and, just after that,\n    Paul's \n    process \n    enters a serialized\n    procedure\n    protecting $a_2$.  Now Peter cannot proceed (to\n    enter a serialized\n    procedure\n    protecting $a_2$) until Paul exits the serialized\n    procedure\n    protecting $a_2$.  Similarly, Paul cannot proceed\n    until Peter exits the serialized\n    procedure\n    protecting $a_1$.  Each\n    process \n    is stalled forever, waiting for the other.  This situation is called a\n    deadlock.  Deadlock is always a danger in systems that provide\n    concurrent access to multiple shared resources.\n  ","3.4.2#p23":"\n    One way to avoid the\n    \n    deadlock in this situation is to give each account a\n    unique identification number and rewrite\n    serialized-exchange\n    so that a \n    process \n    will always attempt to enter a\n    procedure\n    protecting the lowest-numbered account first.  Although this method works\n    well for the exchange problem, there are other situations that require more\n    sophisticated deadlock-avoidance techniques, or where deadlock cannot\n    be avoided at all.  (See exercises 3.48\n    and 3.49.)","3.4.2#footnote-link-9":"9","3.4.2#ex-3.48":"\n    Explain in detail why the\n    \n    deadlock-avoidance method described above,\n    (i.e., the accounts are numbered, and each \n    process \n    attempts to acquire the smaller-numbered account first) avoids \n    deadlock in the exchange problem.  Rewrite\n    serialized-exchange\n    to incorporate this idea. (You will also need to modify\n    make-account\n    so that each account is created with a number, which can be accessed by\n    sending an appropriate message.)\n    ","3.4.2#ex-3.49":"\n    Give a scenario where the deadlock-avoidance mechanism described\n    above does not work.  (Hint: In the exchange problem, each \n    process\n    knows in advance which accounts it will need to get access to.  Consider a\n    situation where a \n    process \n    must get access to some shared resources before it can know which additional\n    shared resources it will require.)\n    ","3.4.2#h6":"Concurrency, time, and communication","3.4.2#p24":"\n    We've seen how programming concurrent systems requires controlling\n    the ordering of events when different \n    processes \n    access shared state, and we've seen how to achieve this control \n    through judicious use of serializers.  But the problems of concurrency\n    lie deeper than this, because, from a fundamental point of view, it's\n    not always clear what is meant by \"shared state.\"","3.4.2#p25":"\n    Mechanisms such as\n    test-and-set!\n    require \n    processes \n    to examine a global shared flag at arbitrary times.  This is problematic \n    and inefficient to implement in modern high-speed processors, where\n    due to optimization techniques such as pipelining and cached memory,\n    the contents of memory may not be in a consistent state at every instant.\n    In\n    contemporary\n    multiprocessing systems, therefore, the serializer paradigm\n    is being supplanted by\n    new\n    approaches to concurrency\n    control.","3.4.2#footnote-link-10":"10","3.4.2#p26":"\n    The problematic aspects of shared state also arise in large, distributed\n    systems.  For instance, imagine a distributed banking system where\n    individual branch banks maintain local values for bank balances and\n    periodically compare these with values maintained by other branches.  In\n    such a system the value of \"the account balance\" would be\n    undetermined, except right after synchronization.  If Peter deposits money\n    in an account he holds jointly with Paul, when should we say that the\n    account balance has changed—when the balance in the local branch\n    changes, or not until after the synchronization?  And if Paul accesses the\n    account from a different branch, what are the reasonable constraints to\n    place on the banking system such that the behavior is\n    \"correct\"?  The only thing that might matter for correctness\n    is the behavior observed by Peter and Paul individually and the\n    \"state\" of the account immediately after synchronization.\n    Questions about the \"real\" account balance or the order of\n    events between synchronizations may be irrelevant or\n    meaningless.","3.4.2#footnote-link-11":"11","3.4.2#p27":"\n    The basic phenomenon here is that synchronizing different \n    processes,\n    establishing shared state, or imposing an order on events requires\n    communication among the \n    processes.\n    In essence, any notion of time in concurrency control must be intimately\n    tied to communication. It is intriguing that a similar\n    connection between time and communication also arises in the \n    \n    Theory of Relativity, where the speed of light (the fastest signal that can\n    be used to synchronize events) is a fundamental constant relating time and\n    space.  The complexities we encounter in dealing with time and state in our\n    computational models may in fact mirror a fundamental complexity of\n    the physical universe.\n\n    ","3.4.2#footnote-link-12":"12","3.4.2#footnote-1":"Parallel-execute is not part of standard\n\tScheme, but it can be implemented in MIT Scheme. In our implementation,\n\tthe new concurrent processes also run concurrently with the original\n\tScheme process. Also, in our implementation, the value returned by\n        parallel-execute is a special control\n\tobject that can be used to halt the newly created processes.\n      ","3.4.2#footnote-2":"We have simplified exchange\n    by exploiting the fact that our deposit\n    message accepts negative amounts.  (This is a serious bug in our banking\n    system!)","3.4.2#footnote-3":"If the account balances start out as $10,\n    $20, and $30, then after any number of concurrent exchanges,\n    the balances should still be $10, $20, and $30 in\n    some order.  Serializing the deposits to individual accounts is not\n    sufficient to guarantee this. See\n    exercise 3.43.","3.4.2#footnote-4":"Exercise 3.45\n    investigates why deposits and withdrawals are no longer automatically\n    serialized by the account.","3.4.2#footnote-5":"The term \"mutex\" is an abbreviation for \n    mutual exclusion.  The general problem of arranging a mechanism\n    that permits concurrent \n    processes \n    to safely share resources is called the mutual exclusion problem.  Our\n    mutex is a simple variant of the \n    semaphore mechanism (see\n    exercise 3.47), which was introduced in the \n    \"THE\" Multiprogramming System developed at the\n    \n    Technological University of Eindhoven and named for the university's\n    initials in Dutch\n    \n    (Dijkstra 1968a).  The acquire and\n    release operations were originally called \n    \n    P and V, from the Dutch\n    words passeren (to pass) and vrijgeven (to release), in\n    reference to the semaphores used on railroad systems.  Dijkstra's\n    classic exposition (1968b) was one of the first to clearly present the\n    issues of concurrency control, and showed how to use semaphores to\n    handle a variety of concurrency problems.","3.4.2#footnote-6":"In most\n    time-shared operating systems,\n    processes \n    that are\n    \n    blocked by a mutex do\n    not waste time\n    \"busy-waiting\" as above.  Instead, the system\n    schedules another \n    process \n    to run while the first is waiting, and the blocked\n    process \n    is awakened when the mutex becomes available.","3.4.2#footnote-7":"In MIT Scheme\n    for a single processor, which uses a time-slicing\n    model, test-and-set!\n    can be implemented as follows:\n    (define (test-and-set! cell)\n  (without-interrupts\n    (lambda ()\n      (if (car cell)\n          true\n          (begin (set-car! cell true)\n            false)))))\n    Without-interrupts disables time-slicing interrupts while its procedure\n    argument is being executed.","3.4.2#footnote-8":"There are many\n    variants of such\n    \n    instructions—including test-and-set, test-and-clear, swap,\n    compare-and-exchange, load-reserve, and store-conditional—whose\n    design must be carefully matched to the machine's\n    processor–memory interface. One issue that arises here is to\n    determine what happens if two \n    processes \n    attempt to acquire the same resource at exactly the same time by using such\n    an instruction. This requires some mechanism for making a decision about\n    which \n    process \n    gets control. Such a mechanism is called an \n    arbiter.  Arbiters usually boil down to some sort of hardware\n    device. Unfortunately, it is possible to prove that one cannot physically\n    construct a fair arbiter that works 100% of the time unless one\n    allows the arbiter an arbitrarily long time to make its decision.\n    The fundamental phenomenon here was originally observed by the\n    fourteenth-century French philosopher \n    \n    Jean Buridan in his commentary on\n    \n    Aristotle's De caelo. Buridan argued that a perfectly rational\n    \n    dog placed between two equally attractive sources of food will starve to\n    death, because it is incapable of deciding which to go to first.","3.4.2#footnote-9":"The general\n    technique for avoiding\n    \n    deadlock by numbering the\n    shared resources and acquiring them in order is due to \n    Havender (1968).  Situations where deadlock cannot be\n    avoided require deadlock-recovery methods, which entail having \n    processes\"back out\" of the deadlocked state and try again.\n    Deadlock-recovery mechanisms are widely used in\n    data-base-management systems, a topic that is treated in detail in\n    Gray and Reuter 1993.","3.4.2#footnote-10":"One such alternative to serialization is called \n    barrier synchronization.  The programmer permits concurrent \n    processes\n    to execute as they please, but establishes certain synchronization points\n    (\"barriers\") through which no \n    process \n    can proceed until all the\n    processes \n    have reached the barrier.\n    Modern\n    processors provide machine instructions\n    that permit programmers to establish synchronization points at places where\n    consistency is required.  The \n    \n    PowerPC$^{\\textrm{TM}}$, for example, includes\n    for this purpose two instructions called\n    \n    SYNC and \n    \n    EIEIO (Enforced In-order Execution of Input/Output).","3.4.2#footnote-11":"This may seem like a strange point of view, but there\n    are\n    systems that work this way.\n    \n    International charges to credit-card accounts,\n    for example, are normally cleared on a per-country basis, and the charges\n    made in different countries are periodically reconciled.  Thus the account\n    balance may be different in different countries.","3.4.2#footnote-12":"For distributed systems, this perspective\n    was pursued by \n    \n    Lamport (1978), who showed how to use communication to establish\n    \"global clocks\" that can be used to establish orderings on\n    events in distributed systems.","3.5":"3.5  Streams","3.5#p1":"\n    We've gained a good understanding of assignment as a tool in modeling,\n    as well as an appreciation of the complex problems that assignment\n    raises. It is time to ask whether we could have gone about things in a\n    different way, so as to avoid some of these problems.  In this\n    section, we explore an alternative approach to modeling state, based\n    on data structures called streams.  As we shall see, streams can\n    mitigate some of the complexity of modeling state.\n  ","3.5#p2":"\n    Let's step back and review where this complexity comes from.  In an\n    attempt to model real-world phenomena, we made some apparently\n    reasonable decisions: We modeled real-world objects with local state\n    by computational objects with local variables.  We identified time\n    variation in the real world with time variation in the computer.  We\n    implemented the time variation of the states of the model objects in\n    the computer with assignments to the local variables of the model\n    objects.\n  ","3.5#p3":"\n    Is there another approach?  Can we avoid identifying time in the\n    computer with time in the modeled world?  Must we make the model\n    change with time in order to model phenomena in a changing world?\n    Think about the issue in terms of mathematical functions.  We can\n    describe the time-varying behavior of a quantity\n    $x$ as a function of time\n    $x(t)$.  \n    If we concentrate on $x$ instant by instant,\n    we think of it as a changing quantity.  Yet if we concentrate on the entire\n    time history of values, we do not emphasize change—the function\n    itself does not change.","3.5#footnote-link-1":"1","3.5#p4":"\n    If time is measured in discrete steps, then we can model a time function as\n    a (possibly infinite) sequence.  In this section, we will see how to\n    model change in terms of sequences that represent the time histories\n    of the systems being modeled.  To accomplish this, we introduce new\n    data structures called streams.  From an abstract point of view,\n    a stream is simply a sequence.  However, we will find that the\n    straightforward implementation of streams as lists (as in\n    section 2.2.1) doesn't fully reveal\n    the power of stream processing.  As an alternative, we introduce the\n    technique of \n    delayed evaluation, which enables us to represent\n    very large (even infinite) sequences as streams.\n  ","3.5#p5":"\n    Stream processing lets us model systems that have state without ever\n    using assignment or mutable data.  This has important implications,\n    both theoretical and practical, because we can build models that avoid\n    the drawbacks inherent in introducing assignment.  On the other hand,\n    the stream framework raises difficulties of its own, and the question\n    of which modeling technique leads to more modular and more easily\n    maintained systems remains open.\n  ","3.5#footnote-1":"Physicists sometimes adopt this view by\n    introducing the\n    \"world lines\" of particles as a device for reasoning about\n    motion. We've also already mentioned\n    (section 2.2.3) that\n    this is the natural way to think about signal-processing systems.  We will\n    explore applications of streams to signal processing in\n    section 3.5.3.","3.5.1":"3.5.1  \n    Streams Are Delayed Lists","3.5.1#p1":"\n    As we saw in\n    section 2.2.3,\n    sequences can serve as standard interfaces for combining program\n    modules.  We formulated powerful abstractions for manipulating\n    sequences, such as map,\n    filter, and\n    accumulate, that capture a wide variety of\n    operations in a manner that is both succinct and elegant.\n  ","3.5.1#p2":"\n    Unfortunately, if we represent sequences as lists, this elegance is\n    bought at the price of severe inefficiency with respect to both the\n    time and space required by our computations.\n    When we represent manipulations on sequences as transformations\n    of lists, our programs must construct and copy data structures (which\n    may be huge) at every step of a process.\n  ","3.5.1#p3":"\n    To see why this is true, let us compare two programs for computing the\n    sum of all the prime numbers in an interval.  The first program is\n    written in standard iterative style:(define (sum-primes a b)\n  (define (iter count accum)\n    (cond ((> count b) accum)\n          ((prime? count) (iter (+ count 1) (+ count accum)))\n          (else (iter (+ count 1) accum))))\n  (iter a 0)) \n    The second program performs the same computation using the sequence\n    operations of\n    section 2.2.3:\n    (define (sum-primes a b)\n  (accumulate +\n              0\n              (filter prime? (enumerate-interval a b)))) ","3.5.1#footnote-link-1":"1","3.5.1#p4":"\n    In carrying out the computation, the first program needs to store only\n    the sum being accumulated.  In contrast, the filter in the second\n    program cannot do any testing until \n    enumerate-interval\n    has constructed a complete list of the numbers in the interval.\n    The filter generates another list, which in turn is passed to \n    accumulate before being collapsed to form\n    a sum.  Such large intermediate storage is not needed by the first program,\n    which we can think of as enumerating the interval incrementally, adding\n    each prime to the sum as it is generated.\n  ","3.5.1#p5":"\n    The inefficiency in using lists becomes painfully apparent if we use\n    the sequence paradigm to compute the second prime in the interval from\n    10,000 to 1,000,000 by evaluating the expression\n    (car (cdr (filter prime?\n                  (enumerate-interval 10000 1000000))))\n    This expression does find the second prime, but the computational overhead\n    is outrageous. We construct a list of almost a million integers, filter\n    this list by testing each element for primality, and then ignore almost\n    all of the result.  In a more traditional programming style, we would\n    interleave the enumeration and the filtering, and stop when we reached\n    the second prime.\n  ","3.5.1#p6":"\n    Streams are a clever idea that allows one to use sequence\n    manipulations without incurring the costs of manipulating sequences as\n    lists.  With streams we can achieve the best of both worlds: We can\n    formulate programs elegantly as sequence manipulations, while attaining\n    the efficiency of incremental computation.  The basic idea is to arrange\n    to construct a stream only partially, and to pass the partial\n    construction to the program that consumes the stream.  If the consumer\n    attempts to access a part of the stream that has not yet been\n    \n    constructed, the stream will automatically construct just enough more\n    of itself to produce the required part, thus preserving the illusion\n    that the entire stream exists.  In other words, although we will write\n    programs as if we were processing complete sequences, we design our\n    stream implementation to automatically and transparently interleave\n    the construction of the stream with its use.\n  ","3.5.1#p7":"\n        On the surface, streams are just lists with different names for the\n\tprocedures that manipulate them.  There is a constructor,\n        cons-stream, and two selectors, \n        stream-car and \n        stream-cdr, which satisfy the constraints\n        \\begin{eqnarray*}\n        \\mbox{(stream-car (cons-stream x y))} &=& \\mbox{x} \\\\\n        \\mbox{(stream-cdr (cons-stream x y))} &=& \\mbox{y}\n        \\end{eqnarray*}\n        There is a distinguishable object, \n        the-empty-stream, which\n        cannot be the result of any cons-stream\n\toperation, and which can be identified with the predicate \n        stream-null?.","3.5.1#footnote-link-2":"2","3.5.1#p8":"\n        Thus we can make and use streams, in just the same way as we can make\n        and use lists, to represent aggregate data arranged in a sequence.  In\n        particular, we can build stream analogs of the list operations from\n        chapter 2, such as list-ref,\n\tmap, and\n\tfor-each:(define (stream-ref s n)\n  (if (= n 0)\n      (stream-car s)\n      (stream-ref (stream-cdr s) (- n 1))))\n\n(define (stream-map proc s)\n  (if (stream-null? s)\n      the-empty-stream\n      (cons-stream (proc (stream-car s))\n                   (stream-map proc (stream-cdr s)))))\n\n(define (stream-for-each proc s)\n  (if (stream-null? s)\n      'done\n      (begin (proc (stream-car s))\n             (stream-for-each proc (stream-cdr s)))))","3.5.1#footnote-link-3":"3","3.5.1#p9":"Stream-for-each is useful for viewing streams:\n        (define (display-stream s)\n  (stream-for-each display-line s))\n\n(define (display-line x)\n  (newline)\n  (display x)) ","3.5.1#p10":"\n        To make the stream implementation automatically and transparently\n        interleave the construction of a stream with its use, we will arrange\n        for the cdr of a stream to be evaluated\n\twhen it is accessed by the stream-cdr\n\tprocedure rather than when the stream is constructed by\n\tcons-stream.  This implementation choice\n\tis reminiscent of our discussion of rational numbers in\n        section 2.1.2, where we saw\n\tthat we can choose to implement rational numbers so that the reduction\n\tof numerator and denominator to lowest terms is performed either at\n        construction time or at selection time.  The two rational-number\n        implementations produce the same data abstraction, but the choice has\n        an effect on efficiency.  There is a similar relationship between\n        streams and ordinary lists.  As a data abstraction, streams are the\n        same as lists.  The difference is the time at which the elements are\n        evaluated.  With ordinary lists, both the\n\tcar and the cdr\n        are evaluated at construction time.  With streams, the\n\tcdr is evaluated at selection time.\n      ","3.5.1#p11":"\n        Our implementation of streams will be based on a special form called\n        delay.  Evaluating\n\t(delay exp)\n\tdoes not evaluate the expression exp, but\n\trather returns a so-called\n        delayed object, which we can think of as a\n\t\"promise\" to evaluate exp at\n\tsome future time. As a companion to delay,\n\tthere is a procedure called \n        force that takes a delayed object as\n        argument and performs the evaluation—in effect, forcing the\n        delay to fulfill its promise.  We will see\n\tbelow how delay and\n\tforce can be implemented, but first let us\n\tuse these to construct streams.\n      ","3.5.1#p12":"Cons-stream\n\tis a special form defined so that\n        (cons-stream a b)\n        is equivalent to\n        (cons a (delay b))","3.5.1#p13":"\n        What this means is that we will construct streams using pairs.  However,\n        rather than placing the value of the rest of the stream\n        into the cdr of the\n        pair we will put there a promise to compute the rest if it is ever\n        requested.  Stream-car and\n\tstream-cdr can now be defined as procedures:\n        (define (stream-car stream) (car stream))\n\n(define (stream-cdr stream) (force (cdr stream))) ","3.5.1#p14":"Stream-car selects the\n\tcar of the pair;\n\tstream-cdr selects the\n\tcdr of the pair and evaluates the delayed\n\texpression found there to obtain the rest of the\n\tstream.","3.5.1#footnote-link-4":"4","3.5.1#h1":"The stream implementation in action","3.5.1#p15":"\n        To see how this implementation behaves, let us analyze the\n        \"outrageous\" prime computation we saw above, reformulated\n\tin terms of streams:\n        (stream-car\n (stream-cdr\n  (stream-filter prime?\n                 (stream-enumerate-interval 10000 1000000)))) \n        We will see that it does indeed work efficiently.\n      ","3.5.1#p16":"\n        We begin by calling\n\tstream-enumerate-interval with the\n\targuments 10,000 and 1,000,000.\n\tStream-enumerate-interval is the stream\n\tanalog of enumerate-interval\n        (section 2.2.3):\n        (define (stream-enumerate-interval low high)\n  (if (> low high)\n      the-empty-stream\n      (cons-stream\n       low\n       (stream-enumerate-interval (+ low 1) high)))) \n        and thus the result returned by\n\tstream-enumerate-interval, formed by the\n\tcons-stream, is(cons 10000\n      (delay (stream-enumerate-interval 10001 1000000))) \n        That is, stream-enumerate-interval returns\n\ta stream represented as a pair whose car\n        is 10,000 and whose cdr is a promise to\n\tenumerate more of the interval if so requested.  This stream is now\n\tfiltered for primes, using the stream analog of the\n\tfilter procedure\n        (section 2.2.3):\n        (define (stream-filter pred stream)\n  (cond ((stream-null? stream) the-empty-stream)\n        ((pred (stream-car stream))\n         (cons-stream (stream-car stream)\n                      (stream-filter pred\n                                     (stream-cdr stream))))\n        (else (stream-filter pred (stream-cdr stream))))) Stream-filter tests the\n\tstream-car of the stream (the\n\tcar of the pair, which is 10,000).\n\tSince this is not prime,\n        stream-filter examines the\n\tstream-cdr of its input\n        stream.  The call to stream-cdr forces\n\tevaluation of the delayed\n\tstream-enumerate-interval, which now returns\n        (cons 10001\n      (delay (stream-enumerate-interval 10002 1000000))) Stream-filter now looks at the\n\tstream-car of this stream,\n        10,001, sees that this is not prime either, forces another\n\tstream-cdr, and so on, until\n\tstream-enumerate-interval yields\n        the prime 10,007, whereupon stream-filter,\n\taccording to its definition, returns\n        (cons-stream (stream-car stream)\n             (stream-filter pred (stream-cdr stream))) \n        which in this case is\n        (cons 10007\n      (delay\n       (stream-filter\n        prime?\n        (cons 10008\n              (delay\n               (stream-enumerate-interval 10009\n                                          1000000)))))) \n        This result is now passed to stream-cdr in\n\tour original expression.  This forces the delayed\n\tstream-filter, which in turn keeps forcing\n\tthe delayed stream-enumerate-interval until\n\tit finds the next prime, which is 10,009.  Finally, the result passed to\n\tstream-car\tin our original expression is\n        (cons 10009\n      (delay\n       (stream-filter\n        prime?\n        (cons 10010\n              (delay\n               (stream-enumerate-interval 10011\n                                          1000000)))))) Stream-car returns 10,009, and the\n\tcomputation is complete. Only as many integers were tested for primality\n\tas were necessary to find the second prime, and the interval was\n\tenumerated only as far as was necessary to feed the prime filter.\n      ","3.5.1#footnote-link-5":"5","3.5.1#p17":"\n        In general, we can think of delayed evaluation as \n        \"demand-driven\"\n        programming, whereby each stage in the stream process is activated\n        only enough to satisfy the next stage.  What we have done is to\n        \n        decouple the actual order of events in the computation from the apparent\n\tstructure of our procedures. We write procedures as if the streams\n\texisted \"all at once\" when, in reality, the computation is\n        performed incrementally, as in traditional programming styles.\n      ","3.5.1#h2":"Implementing delay and\n\tforce","3.5.1#p18":"\n        Although\n        delay and\n\tforce may seem like mysterious operations,\n\ttheir implementation is really quite straightforward.\n        Delay must package an expression so that it\n\tcan be evaluated later on demand, and we can accomplish this simply by\n\ttreating the expression as the body of a procedure.\n\tDelay can be a special form such that\n        (delay exp)\n        is syntactic sugar for\n        (lambda () exp)","3.5.1#p19":"Force simply calls the procedure\n        (of no arguments) produced by delay,\n\tso we can implement force as\n        a procedure:\n        (define (force delayed-object)\n  (delayed-object)) ","3.5.1#p20":"\n        This implementation suffices for delay and\n\tforce to work as advertised, but there is\n\tan important optimization that we can include.  In many applications, we\n\tend up forcing the same delayed object many times.  This can lead to\n\tserious inefficiency in recursive programs involving streams.  (See\n        exercise 3.57.)  The solution\n\tis to build delayed objects so that the first time they are forced, they\n\tstore the value that is computed.  Subsequent forcings will simply\n\treturn the stored value without repeating the computation.  In other\n\twords, we implement delay as a\n\tspecial-purpose\n        \n\tmemoized procedure similar to the one described in\n\texercise 3.27. One way to accomplish this\n\tis to use the following procedure, which takes as argument a procedure\n        (of no arguments) and returns a memoized version of the procedure. The\n\tfirst time the memoized procedure is run, it saves the computed result.\n\tOn subsequent evaluations, it simply returns the result.\n        (define (memo-proc proc)\n  (let ((already-run? false) (result false))\n    (lambda ()\n      (if (not already-run?)\n          (begin (set! result (proc))\n            (set! already-run? true)\n            result)\n          result)))) ","3.5.1#p21":"Delay is then defined so that\n\t(delay exp) is equivalent to\n        (memo-proc (lambda () exp))\n        and force is as defined\n\tpreviously.","3.5.1#footnote-link-6":"6","3.5.1#ex-3.50":"\n\tComplete the following definition, which\n\tgeneralizes stream-map to allow procedures\n\tthat take multiple arguments, analogous to\n\tmap in\n\tsection 2.2.3,\n\tfootnote .\n\t(define (stream-map proc . argstreams)\n  (if (?? (car argstreams))\n      the-empty-stream\n      (??\n       (apply proc (map ?? argstreams))\n       (apply stream-map\n              (cons proc (map ?? argstreams))))))","3.5.1#ex-3.51":"\n\tIn order to take a closer look at\n\t\n\tdelayed evaluation, we will use the\n\tfollowing procedure, which simply returns its argument after printing it:\n\t(define (show x)\n  (display-line x)\n  x) \n\tWhat does the interpreter print in response to evaluating each\n\texpression in the following sequence?(define x (stream-map show (stream-enumerate-interval 0 10)))\n\n(stream-ref x 5)\n\n(stream-ref x 7) ","3.5.1#footnote-link-7":"7","3.5.1#ex-3.52":"\n        Consider the sequence of expressions\n        (define sum 0)\n\n(define (accum x)\n  (set! sum (+ x sum))\n  sum)\n\n(define seq (stream-map accum (stream-enumerate-interval 1 20)))\n(define y (stream-filter even? seq))\n(define z (stream-filter (lambda (x) (= (remainder x 5) 0))\n                         seq))\n\n(stream-ref y 7)\n\n(display-stream z) \n        What is the value of sum after each of the\n\tabove expressions is evaluated?\n        \n\tWhat is the printed response to\n\tevaluating the stream-ref and\n\tdisplay-stream expressions?  Would these\n\tresponses differ if we had implemented\n\t(delay exp) simply as\n        (lambda () exp) without using the\n\toptimization provided by\n        memo-proc$\\,$?\n\tExplain.\n        ","3.5.1#footnote-1":"Assume that we have a\n    predicate\n    prime?\n    (e.g., as in section 1.2.6) that\n    tests for primality.","3.5.1#footnote-2":"In the MIT\n\timplementation, \n        the-empty-stream is the\n        same as the empty list '(), and\n\tstream-null? is the same\n        as null?.","3.5.1#footnote-3":"This should bother you.\n\tThe fact that we are defining such similar procedures\n        for streams and lists indicates that we are missing some\n        underlying abstraction.  Unfortunately, in order to exploit this\n        abstraction, we will need to exert finer control over the process of\n        evaluation than we can at present.  We will discuss this point further\n        at the end of\n\tsection 3.5.4.\n        In section 4.2, we'll\n\tdevelop a framework that unifies lists and streams.","3.5.1#footnote-4":"Although stream-car and\n         stream-cdr can be defined as procedures,\n\tcons-stream must be a special form.  If\n\tcons-stream were a procedure, then,\n        according to our model of evaluation, evaluating\n\t(cons-stream a b)\n\twould automatically cause b to be\n\tevaluated, which is precisely what we do not want to happen.  For the\n\tsame reason, delay must be a special form,\n\tthough force can be an ordinary\n\tprocedure.","3.5.1#footnote-5":"The numbers shown\n\there do not really appear in the delayed expression.  What actually\n\tappears is the original expression, in an environment in which the\n\tvariables are bound to the appropriate numbers.  For example,\n\t(+ low 1) with\n        low bound to 10,000 actually appears where\n\t10001 is shown.","3.5.1#footnote-6":"There are many possible implementations of streams\n\tother than the one described in this section.  Delayed evaluation, which\n\tis the key to making streams practical, was inherent in \n        \n        Algol 60's call-by-name parameter-passing method.  The\n\tuse of this mechanism to implement streams was first described by \n        \n        Landin (1965). Delayed evaluation for streams was introduced into Lisp by\n        \n        Friedman and Wise (1976). In their implementation,\n\tcons\n\talways delays evaluating its\n\targuments, so that lists automatically behave as streams.  The\n\tmemoizing optimization is also known as \n        call-by-need.  The Algol community would refer to our original\n\tdelayed objects as call-by-name thunks and to the optimized\n\tversions as call-by-need thunks.","3.5.1#footnote-7":"Exercises such\n\tas 3.51\n\tand 3.52\n\tare valuable for testing our understanding of how\n\tdelay works. On the other hand, intermixing\n\tdelayed evaluation with printing—and, even worse, with\n\tassignment—is extremely confusing, and instructors of courses\n\ton computer languages have traditionally tormented their students with\n\texamination questions such as the ones in this section.\tNeedless to say,\n\twriting programs that depend on such subtleties is\n\t\n\todious programming style.  Part of the power of stream processing is\n\tthat it lets us ignore the order in which events actually happen in\n\tour programs.  Unfortunately, this is precisely what we cannot afford\n\tto do in the presence of assignment, which forces us to be concerned\n\twith time and change.","3.5.2":"3.5.2  \n    Infinite Streams","3.5.2#p1":"\n    We have seen how to support the illusion of manipulating streams\n    as complete entities even though, in actuality, we compute only\n    as much of the stream as we need to access.  We can exploit this\n    technique to represent sequences efficiently as streams, even if the\n    sequences are very long.  What is more striking, we can use streams to\n    represent sequences that are infinitely long.  For instance, consider\n    the following definition of the stream of positive integers:\n    (define (integers-starting-from n)\n  (cons-stream n (integers-starting-from (+ n 1)))) (define integers (integers-starting-from 1)) \n    This makes sense because integers will be a\n    pair whose\n    car\n    is 1 and whose\n    cdr\n    is a promise to produce the integers beginning with 2. This is an infinitely\n    long stream, but in any given time we can examine only a finite portion of\n    it.  Thus, our programs will never know that the entire infinite stream is\n    not there.\n  ","3.5.2#p2":"\n    Using integers we can define other infinite\n    streams, such as the stream of integers that are not divisible by 7:\n    (define (divisible? x y) (= (remainder x y) 0)) (define no-sevens\n  (stream-filter (lambda (x) (not (divisible? x 7)))\n                 integers)) \n    Then we can find integers not divisible by 7 simply by accessing\n    elements of this stream:\n    (stream-ref no-sevens 100) ","3.5.2#p3":"\n    In analogy with integers, we can define the\n    infinite stream of Fibonacci numbers:\n    (define (fibgen a b)\n  (cons-stream a (fibgen b (+ a b))))\n\n(define fibs (fibgen 0 1)) Fibs\n    is a pair whose\n    car\n    is 0 and whose\n    cdr\n    is a promise to evaluate\n    (fibgen 1 1).\n    When we evaluate this delayed\n    (fibgen 1 1),\n    it will produce a pair whose\n    car\n    is 1 and whose\n    cdr\n    is a promise to evaluate\n    (fibgen 1 2),\n    and so on.\n  ","3.5.2#p4":"\n    For a look at a more exciting infinite stream, we can generalize the\n    no-sevens\n    example to construct the infinite stream of prime\n    numbers, using a\n    sieving method.\n    We start with the integers beginning with 2, which is the first prime.\n    To get the rest of the primes, we start by filtering the multiples of\n    2 from the rest of the integers.  This leaves a stream beginning with\n    3, which is the next prime.  Now we filter the multiples of 3 from the\n    rest of this stream.  This leaves a stream beginning with 5, which is\n    the next prime, and so on.  In other words, we construct the primes by\n    a sieving process, described as follows: To sieve a stream\n    S,\n    form a stream whose first element is the first element of\n    S and\n    the rest of which is obtained by filtering all multiples of the\n    first element of S out of the rest\n    of S and sieving the result. This\n    process is readily described in terms of stream operations:\n\n    (define (sieve stream)\n  (cons-stream\n   (stream-car stream)\n   (sieve (stream-filter\n           (lambda (x)\n             (not (divisible? x (stream-car stream))))\n           (stream-cdr stream)))))\n\n(define primes (sieve (integers-starting-from 2))) \n    Now to find a particular prime we need only ask for it:\n    (stream-ref primes 50) ","3.5.2#footnote-link-1":"1","3.5.2#p5":"\n    It is interesting to contemplate the signal-processing system set up\n    by sieve, shown in the\n    \"Henderson diagram\" in\n    figure . The input stream feeds into an\n    \"unconser\"\n    that separates the first element of the stream from the rest of the stream.\n    The first element is used to construct a divisibility filter, through\n    which the rest is passed, and the output of the filter is fed to\n    another sieve box.  Then the original first element is\n    consed\n\tonto the output of the internal sieve to form the output stream.\n      \n    Thus, not only is the stream infinite, but the signal processor is also\n    infinite, because the sieve contains a sieve within it.\n    ","3.5.2#footnote-link-2":"2","3.5.2#fig-":"","3.5.2#h1":"Defining streams implicitly","3.5.2#p6":"\n    The integers and\n    fibs streams above were defined by specifying\n    \"generating\"procedures\n    that explicitly compute the stream elements one by one. An alternative way\n    to specify streams is to take advantage of delayed evaluation to define\n    streams implicitly. For example, the following\n    expression\n    defines the\n    stream ones to be an infinite stream of ones:\n    (define ones (cons-stream 1 ones)) \n    This works much like the declaration of a recursive\n    procedure:ones is a pair whose\n    car\n    is 1 and whose\n    cdr\n    is a promise to evaluate ones.  Evaluating the\n    cdr\n    gives us again a 1 and a promise to evaluate\n    ones, and so on.\n  ","3.5.2#p7":"\n    We can do more interesting things by manipulating streams with\n    operations such as\n    add-streams,\n    which produces the elementwise sum of two given streams:(define (add-streams s1 s2)\n  (stream-map + s1 s2)) \n    Now we can define the integers as follows:\n    (define integers (cons-stream 1 (add-streams ones integers))) \n    This defines integers to be a stream whose\n    first element is 1 and the rest of which is the sum of\n    ones and integers.\n    Thus, the second element of integers is 1 plus\n    the first element of integers, or 2; the third\n    element of integers is 1 plus the second\n    element of integers, or 3; and so on.  This\n    definition works because, at any point, enough of the\n    integers stream has been generated so that we\n    can feed it back into the definition to produce the next integer.\n  ","3.5.2#footnote-link-3":"3","3.5.2#p8":"\n    We can define the Fibonacci numbers in the same style:\n    (define fibs\n  (cons-stream 0\n               (cons-stream 1\n                            (add-streams (stream-cdr fibs)\n                                         fibs)))) \n    This definition says that fibs is a stream\n    beginning with 0 and 1, such that the rest of the stream can be generated\n    by adding fibs to itself shifted by one place:\n    \n\\[\n\\begin{array}{ccccccccccccl}\n  &   & 1 & 1 & 2 & 3 & 5 &  8 & 13 & 21 & \\ldots & = & \\texttt{(stream-cdr fibs)} \\\\\n  &   & 0 & 1 & 1 & 2 & 3 &  5 &  8 & 13 & \\ldots & = & \\texttt{fibs} \\\\ \\hline\n0 & 1 & 1 & 2 & 3 & 5 & 8 & 13 & 21 & 34 & \\ldots & = & \\texttt{fibs}\n\\end{array}\n\\]\n\t","3.5.2#p9":"Scale-stream\n\tis another useful procedure\n      \n    in formulating such stream definitions. This multiplies each item in a\n    stream by a given constant:\n    (define (scale-stream stream factor)\n  (stream-map (lambda (x) (* x factor)) stream)) \n    For example,\n    (define double (cons-stream 1 (scale-stream double 2))) \n    produces the stream of powers of 2:\n    $1, 2, 4, 8, 16, 32,$….\n  ","3.5.2#p10":"\n    An alternate definition of the stream of primes can be given by\n    starting with the integers and filtering them by testing for\n    primality.  We will need the first prime, 2, to get started:\n    (define primes\n  (cons-stream\n   2\n   (stream-filter prime? (integers-starting-from 3)))) \n    This definition is not so straightforward as it appears, because we will\n    test whether a number $n$ is prime by checking\n    whether $n$ is divisible by a prime (not by just\n    any integer) less than or equal to $\\sqrt{n}$:\n    (define (prime? n)\n  (define (iter ps)\n    (cond ((> (square (stream-car ps)) n) true)\n          ((divisible? n (stream-car ps)) false)\n          (else (iter (stream-cdr ps)))))\n  (iter primes)) \n    This is a recursive definition, since primes\n    is defined in terms of the\n    prime?\n    predicate, which itself uses the primes stream.\n    The reason this\n    procedure\n    works is that, at any point, enough of the\n    primes stream has been generated to test the\n    primality of the numbers we need to check next.  That is, for every\n    $n$ we test for primality, either\n    $n$ is not prime (in which case there is a prime\n    already generated that divides it) or $n$ is\n    prime (in which case there is a prime already generated—i.e., a\n    prime less than $n$—that is greater than\n    $\\sqrt{n}$).","3.5.2#footnote-link-4":"4","3.5.2#ex-3.53":"\n    Without running the program, describe the elements of the\n    stream defined by\n    (define s (cons-stream 1 (add-streams s s))) ","3.5.2#ex-3.54":"\n    Define a\n    proceduremul-streams,\n    analogous to\n    add-streams,\n    that produces the elementwise product of its two input streams. Use this\n    together with the stream of integers to\n    complete the following definition of the stream whose\n    $n$th element (counting from 0) is\n    $n+1$ factorial:\n    \n(define factorials (cons-stream 1 (mul-streams ?? ??)))\n      ","3.5.2#ex-3.55":"\n    Define a\n    procedurepartial-sums\n    that takes as argument a stream $S$ and returns\n    the stream whose elements are\n    $S_0, S_0+S_1, S_0+S_1+S_2,$….\n    For example,\n    (partial-sums integers)\n    should be the stream $1, 3, 6, 10, 15,\\ldots$.\n\n    ","3.5.2#ex-3.56":"\n    A famous problem, first raised by\n    \n    R. Hamming, is to enumerate, in ascending order with no repetitions, all\n    positive integers with no prime factors other than 2, 3, or 5.  One obvious\n    way to do this is to simply test each integer in turn to see whether it has\n    any factors other than 2, 3, and 5.  But this is very inefficient, since, as\n    the integers get larger, fewer and fewer of them fit the requirement.  As\n    an alternative, let us call the required stream of numbers\n    S and notice the following facts about it.\n    S begins with 1.\n      \n        The elements of\n\t(scale-stream S 2)\n\tare also elements of S.\n      \n        The same is true for\n\t(scale-stream S 3)\n        and\n\t(scale-stream S 5).\n        These are all the elements of S.\n      \n    Now all we have to do is combine elements from these sources. For this we\n    define a\n    proceduremerge that combines two ordered\n    streams into one ordered result stream, eliminating repetitions:\n    (define (merge s1 s2)\n  (cond ((stream-null? s1) s2)\n        ((stream-null? s2) s1)\n        (else\n         (let ((s1car (stream-car s1))\n               (s2car (stream-car s2)))\n           (cond ((< s1car s2car)\n                  (cons-stream s1car (merge (stream-cdr s1) s2)))\n                 ((> s1car s2car)\n                  (cons-stream s2car (merge s1 (stream-cdr s2))))\n                 (else\n                  (cons-stream s1car\n                               (merge (stream-cdr s1)\n                                      (stream-cdr s2))))))))) \n    Then the required stream may be constructed with\n    merge, as follows:\n    \n(define S (cons-stream 1 (merge ?? ??)))\n      \n    Fill in the missing expressions in the places marked\n    〈??〉 above.\n    ","3.5.2#ex-3.57":"\n\tHow many additions are performed when we compute the\n\t$n$th Fibonacci number using the definition of\n\tfibs based on the\n      add-streams procedure?\n\tShow that the number of additions would be exponentially greater if we\n\thad implemented (delay exp) simply as\n\t(lambda () exp), without using the\n\toptimization provided by the memo-proc\n\tprocedure described in\n\tsection 3.5.1.","3.5.2#footnote-link-5":"5","3.5.2#ex-3.58":"\n    Give an interpretation of the stream computed by the \n    following procedure:(define (expand num den radix)\n  (cons-stream\n   (quotient (* num radix) den)\n   (expand (remainder (* num radix) den) den radix)))\n\t(Quotient\n\tis a primitive that returns the\n\tinteger quotient of two integers.)\n      \n    What are the successive elements produced by\n    (expand 1 7 10)?\n      \n    What is produced by\n    (expand 3 8 10)?\n      ","3.5.2#ex-3.59":"\n  In section 2.5.3 we saw how to implement a\n  polynomial arithmetic system representing polynomials as lists of\n  terms.  In a similar way, we can work with\n  power series, such as\n  \n    \\[\n    \\begin{array}{rll}\n    e^{x} &=&\n    1+x+\\dfrac{x^{2}}{2}+\\dfrac{x^{3}}{3\\cdot2}\n                       +\\dfrac{x^{4}}{4\\cdot 3\\cdot 2}+\\cdots, \\\\[9pt]\n    \\cos x &=& 1-\\dfrac{x^{2}}{2}+\\dfrac{x^{4}}{4\\cdot 3\\cdot 2}-\\cdots, \\\\[9pt]\n    \\sin x &=& x-\\dfrac{x^{3}}{3\\cdot 2}\n                          +\\dfrac{x^{5}}{5\\cdot 4\\cdot 3\\cdot 2}- \\cdots,\n    \\end{array}\n    \\]\n  \n  represented as infinite streams.\n  We will represent the series\n  $a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \\cdots$\n  as the stream whose elements are the coefficients\n  $a_0, a_1, a_2, a_3,$….\n  \n      The\n      \n      integral of the series\n      $a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \\cdots$\n      is the series\n      \n\t\\[\n\t\\begin{array}{l}\n         c + a_0 x + \\frac{1}{2}a_1 x^2 + \\frac{1}{3}a_2 x^3 + \\frac{1}{4}a_3\n         x^4 + \\cdots\n\t \\end{array}\n\t \\]\n      \n      where $c$ is any constant.\n      Define a\n      procedureintegrate-series\n      that takes as input a stream\n      $a_0, a_1, a_2,\\ldots$ representing\n      a power series and returns the stream\n      $a_0, \\frac{1}{2}a_1, \\frac{1}{3}a_2,\\ldots$\n      of coefficients of the nonconstant terms of the integral of the series.\n      (Since the result has no constant term, it doesn't represent a power\n      series; when we use\n      integrate-series,\n      we will\n      cons on the appropriate constant.)\n\t\n      The function $x\\mapsto e^x$ is its own\n      derivative.  This implies that $e^x$ and the\n      integral of $e^x$ are the\n      same series, except for the constant term, which is\n      $e^0 = 1$. Accordingly, we can generate the\n      series for $e^x$ as\n      (define exp-series\n  (cons-stream 1 (integrate-series exp-series)))\n      Show how to generate the series for sine and cosine, starting from the\n      facts that the derivative of sine is cosine and the derivative of cosine\n      is the negative of sine:\n      \n(define cosine-series\n  (cons-stream 1 ??))\n\n(define sine-series\n  (cons-stream 0 ??))\n        ","3.5.2#ex-3.60":"\n    With\n     \n    power series represented as streams of coefficients as in\n    exercise 3.59, adding series is implemented\n    by\n    add-streams.\n      \n    Complete the declaration of\n    the following\n    procedure\n    for multiplying series:\n    \n(define (mul-series s1 s2)\n  (cons-stream ?? (add-streams ?? ??)))\n      \n\n    You can test your\n    procedure\n    by verifying that $sin^2 x + cos^2 x = 1$,\n    using the series from exercise 3.59.\n  ","3.5.2#ex-3.61":"\n    Let $S$ be a power series\n    (exercise 3.59)\n    whose constant term is 1.  Suppose we want to find the power series\n    $1/S$, that is, the series\n    $X$ such that\n    $S\\cdot X= 1$.\n    Write $S=1+S_R$ where\n    $S_R$ is the part of\n    $S$ after the constant term. Then we can solve\n    for $X$ as follows:\n    \\[\n    \\begin{array}{rll}\n    S \\cdot X &=& 1 \\\\\n    (1+S_R)\\cdot X &=& 1 \\\\\n    X + S_R \\cdot X &=& 1 \\\\\n    X &=& 1 - S_R \\cdot X\n    \\end{array}\n    \\]\n    In other words, $X$ is the power series whose\n    constant term is 1 and whose higher-order terms are given by the negative of\n    $S_R$ times $X$.\n    Use this idea to write a\n    procedureinvert-unit-series\n    that computes $1/S$ for a power series\n    $S$ with constant term 1. You will need to use\n    mul-series\n    from exercise 3.60.\n    ","3.5.2#ex-3.62":"\n    Use the results of exercises 3.60\n    and 3.61\n    to define a\n    procedurediv-series\n    that divides two power series.\n    Div-series\n    should work for any two series, provided that the denominator series begins\n    with a nonzero constant term.  (If the denominator has a zero constant term,\n    then\n    div-series\n    should signal an error.) Show how to use\n    div-series\n    together with the result of exercise 3.59\n    to generate the power series for\n    \n    tangent.\n  ","3.5.2#footnote-1":"This method is reminiscent of the ancient\n    sieve of Eratosthenes, named for\n    \n    Eratosthenes, a third-century BCE Alexandrian Greek mathematician.\n    His sieve finds the primes by repeatedly crossing off the multiples of each\n    prime in turn; although ancient, it has formed the basis\n    for special-purpose hardware \"sieves\" that, until the 1970s,\n    were the\n    most powerful tools in existence for locating large primes.  Since then,\n    however, these methods have been superseded by outgrowths of the\n    \n    probabilistic techniques discussed in\n    section 1.2.6. But the stream-based method\n    given here is not actually Eratosthenes's sieve: as Melissa E.\n    O'Neill shows in O'Neill 2009, it tests\n    each candidate for divisibility by the primes found so far—a form of\n    trial division—rather than crossing off multiples, and is\n    substantially less efficient than the sieve of Eratosthenes.","3.5.2#footnote-2":"We have named these\n    figures after\n    \n    Peter Henderson, who was the first person to show us diagrams of this sort\n    as a way of thinking about stream processing.","3.5.2#footnote-3":"\n\tThis uses the generalized version of\n\tstream-map\n\tfrom exercise 3.50.\n      ","3.5.2#footnote-4":"This last point is very\n    subtle and relies on the fact that $p_{n+1} \\leq p_{n}^2$. (Here, $p_{k}$ denotes the\n    $k$th prime.)  Estimates such as these are very\n    difficult to establish.  The ancient proof by\n    \n    Euclid that there are an infinite number of primes shows that\n    $p_{n+1}\\leq p_{1} p_{2}\\,\\cdots\\,\\, p_{n} +1$,\n    and no substantially better result was proved until 1851, when the Russian\n    mathematician\n    \n    P. L. Chebyshev established\n    that $p_{n+1}\\leq 2p_{n}$ for all\n    $n$.  This result, originally conjectured in\n    1845, is known as\n    Bertrand's hypothesis.  A proof can be\n    found in section 22.3 of\n    Hardy and Wright 1960.","3.5.2#footnote-5":"This\n\texercise shows how call-by-need is closely related to\n\t\n\tordinary memoization as described in\n\texercise 3.27. In that exercise, we used\n\tassignment to explicitly construct a local table.  Our call-by-need stream\n\toptimization effectively constructs such a table automatically, storing\n\tvalues in the previously forced parts of the stream.","3.5.3":"3.5.3  \n    Exploiting the Stream Paradigm","3.5.3#p1":"\n    Streams with delayed evaluation can be a powerful modeling tool,\n    providing many of the benefits of local state and assignment.\n    Moreover, they avoid some of the theoretical tangles that accompany\n    the introduction of assignment into a programming language.\n  ","3.5.3#p2":"\n    The stream approach can be illuminating because it allows us to build\n    systems with different\n    \n    module boundaries than systems organized around\n    assignment to state variables.  For example, we can think of an entire\n    time series (or signal) as a focus of interest, rather than the values\n    of the state variables at individual moments.  This makes it\n    convenient to combine and compare components of state from different\n    moments.\n  ","3.5.3#h1":"Formulating iterations as stream processes","3.5.3#p3":"\n    In section 1.2.1, we introduced\n    iterative processes, which proceed by updating state variables.  We know now\n    that we can represent state as a \"timeless\" stream of values\n    rather than as a set of variables to be updated.  Let's adopt this\n    perspective in revisiting the square-root\n    procedure\n    from section 1.1.7.  Recall that the idea is to\n    generate a sequence of better and better guesses for the square root of\n    $x$ by applying over and over again the\n    procedure\n    that improves guesses:\n    (define (sqrt-improve guess x)\n  (average guess (/ x guess))) ","3.5.3#p4":"\n    In our original\n    sqrtprocedure,\n    we made these guesses be the successive values of a state variable. Instead\n    we can generate the infinite stream of guesses, starting with an initial\n    \n\tguess of 1:(define (sqrt-stream x)\n  (define guesses\n    (cons-stream 1.0\n                 (stream-map (lambda (guess)\n                               (sqrt-improve guess x))\n                             guesses)))\n  guesses) (display-stream (sqrt-stream 2)) \n    We can generate more and more terms of the stream to get better and\n    better guesses.  If we like, we can write a\n    procedure\n    that keeps generating terms until the answer is good enough.\n    (See exercise 3.64.)\n  ","3.5.3#footnote-link-1":"1","3.5.3#p5":"\n    Another iteration that we can treat in the same way is to generate an\n    approximation to\n    $\\pi$, based upon the\n    alternating series that we saw in\n    section 1.3.1:\n    \n      \\[\n      \\begin{array}{lll}\n      \\dfrac {\\pi}{4} &=& 1-\\dfrac{1}{3}+\\dfrac{1}{5}-\\dfrac{1}{7}+\\cdots\n      \\end{array}\n      \\]\n    \n    We first generate the stream of summands of the series (the reciprocals\n    of the odd integers, with alternating signs).  Then we take the stream\n    of sums of more and more terms (using the \n    partial-sums procedure\n    of exercise 3.55) and scale the result by 4:\n    (define (pi-summands n)\n  (cons-stream (/ 1.0 n)\n               (stream-map - (pi-summands (+ n 2)))))\n\n(define pi-stream\n  (scale-stream (partial-sums (pi-summands 1)) 4)) (display-stream pi-stream) \n    This gives us a stream of better and better approximations to\n    $\\pi$, although the approximations converge\n    rather slowly.  Eight terms of the sequence bound the value of\n    $\\pi$ between 3.284 and 3.017.\n  ","3.5.3#p6":"\n    So far, our use of the stream of states approach is not much different\n    from updating state variables.  But streams give us an opportunity to do\n    some interesting tricks.  For example, we can transform a stream with a \n    sequence accelerator that converts a sequence of approximations to a\n    new sequence that converges to the same value as the original, only faster.\n  ","3.5.3#p7":"\n    One such accelerator, due to the eighteenth-century Swiss mathematician\n    \n    Leonhard Euler, works well with sequences that are partial sums of\n    alternating series (series of terms with alternating signs). In\n    Euler's technique, if $S_n$ is the\n    $n$th term of the original sum sequence, then\n    the accelerated sequence has terms\n    \n      \\[\n      \\begin{array}{l}\n      S_{n+1} - \\dfrac{(S_{n+1}-S_n)^2}{S_{n-1}-2S_n+S_{n+1}}\n      \\end{array}\n      \\]\n     \n    Thus, if the original sequence is represented as a stream of values,\n    the transformed sequence is given by\n    \n(define (euler-transform s)\n  (let ((s0 (stream-ref s 0))           ; $S_{n-1}$\n        (s1 (stream-ref s 1))           ; $S_{n}$\n        (s2 (stream-ref s 2)))          ; $S_{n+1}$\n    (cons-stream (- s2 (/ (square (- s2 s1))\n                          (+ s0 (* -2 s1) s2)))\n                 (euler-transform (stream-cdr s)))))\n      ","3.5.3#p8":"\n    We can demonstrate Euler acceleration with our sequence of\n    approximations to $\\pi$:\n    (display-stream (euler-transform pi-stream)) ","3.5.3#p9":"\n    Even better, we can accelerate the accelerated sequence, and recursively\n    accelerate that, and so on.  Namely, we create a stream of streams (a\n    structure we'll call a \n    tableau) in which each stream is the transform of the preceding one:\n    (define (make-tableau transform s)\n  (cons-stream s\n               (make-tableau transform\n                             (transform s)))) \n    The tableau has the form\n    \n      \\[\n      \\begin{array}{llllll}\n      s_{00} & s_{01} & s_{02} & s_{03} & s_{04} & \\ldots\\\\\n      & s_{10} & s_{11} & s_{12} & s_{13} & \\ldots\\\\\n      &        & s_{20} & s_{21} & s_{22} & \\ldots\\\\\n      &        &        &        & \\ldots &\n      \\end{array}\n      \\]\n    \n    Finally, we form a sequence by taking the first term in each row of\n    the tableau:\n    (define (accelerated-sequence transform s)\n  (stream-map stream-car\n              (make-tableau transform s))) ","3.5.3#p10":"\n    We can demonstrate this kind of \"super-acceleration\" of the\n    $\\pi$ sequence:\n    (display-stream (accelerated-sequence euler-transform\n                                      pi-stream)) \n    The result is impressive.  Taking eight terms of the sequence yields the\n    correct value of $\\pi$ to 14 decimal places.\n    If we had used only the original $\\pi$ sequence,\n    we would need to compute on the order of $10^{13}$\n    terms (i.e., expanding the series far enough so that the individual terms\n    are less then $10^{-13}$) to get that much\n    accuracy!\n    ","3.5.3#p11":"\n    We could have implemented these acceleration techniques without using\n    streams.  But the stream formulation is particularly elegant and convenient\n    because the entire sequence of states is available to us as a data structure\n    that can be manipulated with a uniform set of operations.\n  ","3.5.3#ex-3.63":"\n\tLouis Reasoner asks why the \n\tsqrt-stream procedure\n\twas not written in the following more straightforward way, without\n\tthe local variable guesses:\n\t(define (sqrt-stream x)\n  (cons-stream 1.0\n               (stream-map (lambda (guess)\n                             (sqrt-improve guess x))\n                           (sqrt-stream x))))\n\tAlyssa P. Hacker replies that this version of the procedure\n\tis considerably less efficient because it performs redundant computation.\n\tExplain Alyssa's answer.  Would the two versions still differ in\n\tefficiency if our implementation of delay\n\tused only (lambda () exp)\n\twithout using the optimization provided by\n\tmemo-proc\n\t(section 3.5.1)?\n      ","3.5.3#ex-3.64":"\n  Write a\n  procedure stream-limit\n  that takes as arguments a stream\n  and a number (the tolerance).  It should examine the stream until it\n  finds two successive elements that differ in absolute value by less\n  than the tolerance, and return the second of the two elements.  Using\n  this, we could compute square roots up to a given tolerance by\n  (define (sqrt x tolerance)\n  (stream-limit (sqrt-stream x) tolerance))","3.5.3#ex-3.65":"\n    Use the series \n    \n      \\[\n      \\begin{array}{lll}\n      \\ln 2 &=& 1-\\dfrac{1}{2}+\\dfrac{1}{3}-\\dfrac{1}{4}+\\cdots\n      \\end{array}\n      \\]\n    \n    to compute three sequences of approximations to the natural logarithm of 2,\n  \n    in the same way we did above for $\\pi$.\n    How rapidly do these sequences converge?\n    ","3.5.3#h2":"Infinite streams of pairs","3.5.3#p12":"\n    In section 2.2.3, we saw how the\n    sequence paradigm handles traditional nested loops as processes defined\n    on sequences of pairs. If we generalize this technique to infinite streams,\n    then we can write programs that are not easily represented as loops, because\n    the \"looping\" must range over an infinite set.\n  ","3.5.3#p13":"\n    For example, suppose we want to generalize the \n    prime-sum-pairs procedure\n    of section 2.2.3 to produce the stream\n    of pairs of all integers $(i,j)$ with\n    $i \\leq j$ such that\n    $i+j$\n    is prime.  If\n    int-pairs\n    is the sequence of all pairs of integers $(i,j)$\n    with $i \\leq j$, then our required stream is\n    simply(stream-filter (lambda (pair)\n                 (prime? (+ (car pair) (cadr pair))))\n               int-pairs) ","3.5.3#footnote-link-2":"2","3.5.3#p14":"\n    Our problem, then, is to produce the stream\n    int-pairs.\n    More generally, suppose we have two streams\n    $S = (S_i)$ and\n    $T = (T_j)$,\n    and imagine the infinite rectangular array\n    \n      \\[\n      \\begin{array}{cccc}\n      (S_0,T_0) & (S_0,T_1) & (S_0, T_2) & \\ldots\\\\\n      (S_1,T_0) & (S_1,T_1) & (S_1, T_2) & \\ldots\\\\\n      (S_2,T_0) & (S_2,T_1) & (S_2, T_2) & \\ldots\\\\\n      \\ldots\n      \\end{array}\n      \\]\n    \n    We wish to generate a stream that contains all the pairs in the array\n    that lie on or above the diagonal, i.e., the pairs\n    \n      \\[\n      \\begin{array}{cccc}\n      (S_0,T_0) & (S_0,T_1) & (S_0, T_2) & \\ldots\\\\\n      & (S_1,T_1) & (S_1, T_2) & \\ldots\\\\\n      &           & (S_2, T_2) & \\ldots\\\\\n      &           &            & \\ldots\n      \\end{array}\n      \\]\n    \n    (If we take both $S$ and\n    $T$ to be the stream of integers, then this\n    will be our desired stream\n    int-pairs.)","3.5.3#p15":"\n    Call the general stream of pairs\n    (pairs S T),\n    and consider it to be composed of three parts: the pair\n    $(S_0,T_0)$, the rest of the pairs in the first\n    row, and the remaining pairs:\n      \\[\n      \\begin{array}{c|ccc}\n      (S_0,T_0) & (S_0,T_1) & (S_0, T_2) & \\ldots\\\\\n      \\hline{} %--------------------------------------------------- \\\\\n      & (S_1,T_1) & (S_1, T_2) & \\ldots\\\\\n      &           & (S_2, T_2) & \\ldots\\\\\n      &           &            & \\ldots\n      \\end{array}\n      \\]\n    \n    Observe that the third piece in this decomposition (pairs that are not in\n    the first row) is (recursively) the pairs formed from \n    (stream-cdr S)\n    and\n    (stream-cdr T).\n    Also note that the second piece (the rest of the first row) is\n    (stream-map (lambda (x) (list (stream-car s) x))\n            (stream-cdr t))\n    Thus we can form our stream of pairs as follows:\n    \n(define (pairs s t)\n  (cons-stream\n   (list (stream-car s) (stream-car t))\n   ($\\langle combine-in-some-way \\rangle$\n      (stream-map (lambda (x) (list (stream-car s) x))\n                  (stream-cdr t))\n      (pairs (stream-cdr s) (stream-cdr t)))))\n      ","3.5.3#footnote-link-3":"3","3.5.3#p16":"\n    In order to complete the\n    procedure,\n    we must choose some way to\n    \n    combine the two inner streams.  One idea is to\n    use the stream analog of the appendprocedure\n    from section 2.2.1:\n    (define (stream-append s1 s2)\n  (if (stream-null? s1)\n      s2\n      (cons-stream (stream-car s1)\n                   (stream-append (stream-cdr s1) s2)))) \n    This is unsuitable for infinite streams, however, because it takes all the\n    elements from the first stream before incorporating the second stream. In\n    particular, if we try to generate all pairs of positive integers using\n    (pairs integers integers) \n    our stream of results will first try to run through all pairs with the\n    first integer equal to 1, and hence will never produce pairs with any\n    other value of the first integer.\n  ","3.5.3#p17":"\n    To handle infinite streams, we need to devise an order of combination\n    that ensures that every element will eventually be reached if we let\n    our program run long enough.  An elegant way to accomplish this is\n    with the following interleaveprocedure:(define (interleave s1 s2)\n  (if (stream-null? s1)\n      s2\n      (cons-stream (stream-car s1)\n                   (interleave s2 (stream-cdr s1))))) \n    Since interleave takes elements alternately\n    from the two streams, every element of the second stream will eventually\n    find its way into the interleaved stream, even if the first stream is\n    infinite.\n  ","3.5.3#footnote-link-4":"4","3.5.3#p18":"\n    We can thus generate the required stream of pairs as\n    (define (pairs s t)\n  (cons-stream\n   (list (stream-car s) (stream-car t))\n   (interleave\n    (stream-map (lambda (x) (list (stream-car s) x))\n                (stream-cdr t))\n    (pairs (stream-cdr s) (stream-cdr t))))) ","3.5.3#ex-3.66":"\n    Examine the stream\n    (pairs integers integers).\n    Can you make any general comments about the order in which the pairs are\n    placed into the stream? For example, approximately how many pairs precede\n    the pair (1,100)? the pair (99,100)? the pair (100,100)? (If you can make\n    precise mathematical statements here, all the better. But feel free to give\n    more qualitative answers if you find yourself getting bogged down.)\n    ","3.5.3#ex-3.67":"\n    Modify the pairsprocedure\n    so that\n    (pairs integers integers)\n    will produce the stream of all pairs of integers\n    $(i,j)$ (without the condition\n    $i \\leq j$).  Hint: You will need to\n    mix in an additional stream.\n    ","3.5.3#ex-3.68":"\n    Louis Reasoner thinks that building a stream of pairs from three parts is\n    unnecessarily complicated.  Instead of separating the pair\n    $(S_0,T_0)$ from the rest of the pairs in the\n    first row, he proposes to work with the whole first row, as follows:\n    (define (pairs s t)\n  (interleave\n   (stream-map (lambda (x) (list (stream-car s) x))\n               t)\n   (pairs (stream-cdr s) (stream-cdr t))))\n    Does this work?  Consider what happens if we evaluate\n    (pairs integers integers)\n    using Louis's definition of pairs.\n    ","3.5.3#ex-3.69":"\n    Write a\n    proceduretriples that takes three infinite streams,\n    $S$, $T$, and\n    $U$, and produces the stream of triples\n    $(S_i,T_j,U_k)$ such that\n    $i \\leq j \\leq k$. Use\n    triples to generate the stream of all \n    \n    Pythagorean triples of positive integers, i.e., the triples\n    $(i,j,k)$ such that\n    $i \\leq j$ and\n    $i^2 + j^2 =k^2$.\n    ","3.5.3#ex-3.70":"\n    It would be nice to be able to generate\n    \n    streams in which the pairs\n    appear in some useful order, rather than in the order that results\n    from an ad hoc interleaving process.  We can use a technique\n    similar to the mergeprocedure\n    of exercise 3.56, if we define a way to say that\n    one pair of integers is \"less than\" another.  One way to do\n    this is to define a\n    \"weighting function\"$W(i,j)$ and stipulate that\n    $(i_1,j_1)$ is less than\n    $(i_2,j_2)$ if\n    $W(i_1,j_1) < W(i_2,j_2)$.  Write a\n    procedure merge-weighted\n    that is like merge, except that\n    merge-weighted\n    takes an additional argument weight, which is a\n    procedure\n    that computes the weight of a pair, and is used to determine the order in\n    which elements should appear in the resulting merged stream. Using this, generalize pairs\n    to a\n    procedure weighted-pairs\n    that takes two streams, together with a\n    procedure\n    that computes a weighting function, and generates the stream of pairs,\n    ordered according to weight.  Use your\n    procedure\n    to generate\n    \n\tthe stream of all pairs of positive integers\n\t$(i,j)$ with $i \\leq         j$ ordered according to the sum\n\t$i + j$\n\tthe stream of all pairs of positive integers\n\t$(i,j)$ with $i \\leq         j$, where neither $i$ nor\n\t$j$ is divisible by 2, 3, or 5, and the\n\tpairs are ordered according to the sum\n\t$2 i + 3 j + 5 i j$.\n      ","3.5.3#footnote-link-5":"5","3.5.3#ex-3.71":"\n    Numbers that can be expressed as the sum of two cubes in more than one\n    way are sometimes called\n    Ramanujan numbers, in honor of the\n    mathematician Srinivasa Ramanujan. Ordered streams of pairs provide an elegant solution\n    to the problem of computing these numbers.  To find a number that can be\n    written as the sum of two cubes in two different ways, we need only generate\n    the stream of pairs of integers $(i,j)$ weighted\n    according to the sum $i^3 + j^3$ (see\n    exercise 3.70), then search the stream for\n    two consecutive pairs with the same weight.  Write a\n    procedure\n    to generate the Ramanujan numbers.  The first\n    such number is 1,729.  What are the next five?\n    ","3.5.3#footnote-link-6":"6","3.5.3#ex-3.72":"\n    In a similar way to exercise 3.71 generate\n    a stream of all numbers that can be written as the sum of two squares in\n    three different ways (showing how they can be so written).\n    ","3.5.3#h3":"Streams as signals","3.5.3#p19":"\n    We began our discussion of streams by describing them as computational\n    analogs of the \"signals\" in signal-processing systems.\n    In fact, we can use streams to model signal-processing systems in a very\n    direct way, representing the values of a signal at successive time\n    intervals as consecutive elements of a stream.  For instance, we can\n    implement an \n    integrator or \n    summer that, for an input stream\n    $x=(x_{i})$, an initial value $C$, and a small increment $dt$,\n    accumulates the sum\n    \n      \\[\n      \\begin{array}{lll}\n      S_i &=& C +\\sum_{j=1}^{i} x_{j} \\, dt\n      \\end{array}\n      \\]\n    \n    and returns the stream of values $S=(S_{i})$.\n    The following integralprocedure\n    is reminiscent of the \"implicit style\" definition of the\n    stream of integers (section 3.5.2):\n    (define (integral integrand initial-value dt)\n  (define int\n          (cons-stream initial-value\n                       (add-streams (scale-stream integrand dt)\n                                    int)))\n  int) ","3.5.3#fig-":"","3.5.3#p20":"\n\tFigure \n    is a picture of a signal-processing\n    system that corresponds to the integralprocedure.\n    The input stream is scaled by $dt$ and passed\n    through an adder, whose output is passed back through the same adder.\n    The self-reference in the definition of\n    int\n    is reflected in the figure by the feedback loop that\n    connects the output of the adder to one of the inputs.\n  ","3.5.3#ex-3.73":"\n      We can model electrical circuits using streams to represent the values\n      of currents or voltages at a sequence of times.  For instance, suppose\n      we have an\n      RC circuit consisting of a resistor of resistance\n      $R$ and a capacitor of capacitance\n      $C$ in series.  The voltage response\n      $v$ of the circuit to an injected current\n      $i$ is determined by the formula in\n      figure 3.33, whose structure is shown by the\n      accompanying signal-flow diagram.\n  \n      Write a\n      procedureRC that models this circuit.\n      RC should take as inputs the values of\n      $R$, $C$, and\n      $dt$ and should return a\n      procedure\n      that takes as inputs a stream representing the current\n      $i$ and an initial value for the capacitor\n      voltage $v_{0}$ and produces as output the\n      stream of voltages $v$.  For example, you\n      should be able to use RC to model an RC\n      circuit with $R = 5$ ohms,\n      $C = 1$ farad, and a 0.5-second time step by\n      evaluating\n      (define RC1 (RC 5 1 0.5)).\n\t\n      This defines RC1 as a\n      procedure\n      that takes a stream representing the time sequence of currents and an\n      initial capacitor voltage and produces the output stream of voltages.\n    ","3.5.3#fig-3.33":"","3.5.3#ex-3.74":"\n    Alyssa P. Hacker is designing a system to process signals coming from\n    physical sensors.  One important feature she wishes to produce is a signal\n    that describes the\n    zero crossings of the input signal. That is,\n    the resulting signal should be $+1$ whenever the\n    input signal changes from negative to positive,\n    $-1$ whenever the input signal changes from\n    positive to negative, and 0 otherwise.  (Assume that the sign of a 0 input\n    is positive.)  For example, a typical input signal with its associated\n    zero-crossing signal would be\n    \n$\\ldots$ 1  2  1.5  1  0.5  -0.1  -2  -3  -2  -0.5  0.2  3  4 $\\ldots$\n$\\ldots$  0  0    0  0    0     -1  0   0   0     0    1  0  0 $\\ldots$\n      \n\n    In Alyssa's system, the signal from the sensor is represented as a\n    stream\n    sense-data \n    and the stream\n    zero-crossings\n    is the corresponding stream of zero crossings.  Alyssa first writes a\n    proceduresign-change-detector\n    that takes two values as arguments and compares the signs of the values to\n    produce an appropriate $0$,\n    $1$, or $-1$.  She\n    then constructs her zero-crossing stream as follows:\n\n    (define (make-zero-crossings input-stream last-value)\n  (cons-stream\n   (sign-change-detector (stream-car input-stream) last-value)\n   (make-zero-crossings (stream-cdr input-stream)\n                        (stream-car input-stream))))\n\n(define zero-crossings (make-zero-crossings sense-data 0))\n\n    Alyssa's boss, Eva Lu Ator, walks by and suggests that this program is\n    approximately equivalent to the following one, which uses\n    \n\tthe generalized version of\n\tstream-map\n\tfrom exercise 3.50:\n      \n(define zero-crossings\n  (stream-map sign-change-detector sense-data expression))\n      \n    Complete the program by supplying the indicated\n    expression.\n    ","3.5.3#ex-3.75":"\n    Unfortunately, Alyssa's\n    \n    zero-crossing detector in\n    exercise 3.74 proves to be insufficient,\n    because the noisy signal from the sensor leads to spurious zero crossings.\n    Lem E. Tweakit, a hardware specialist, suggests that Alyssa smooth the\n    signal to filter out the noise before extracting the zero crossings.\n    Alyssa takes his advice and decides to extract the zero crossings from\n    the signal constructed by averaging each value of the sense data with\n    the previous value.  She explains the problem to her assistant, Louis\n    Reasoner, who attempts to implement the idea, altering Alyssa's\n    program as follows:\n    (define (make-zero-crossings input-stream last-value)\n  (let ((avpt (/ (+ (stream-car input-stream) last-value) 2)))\n    (cons-stream (sign-change-detector avpt last-value)\n                 (make-zero-crossings (stream-cdr input-stream)\n                                      avpt))))\n    This does not correctly implement Alyssa's plan.\n    Find the bug that Louis has installed\n    and fix it without changing the structure of the program.  (Hint: You\n    will need to increase the number of arguments to\n    make-zero-crossings.)\n      ","3.5.3#ex-3.76":"\n    Eva Lu Ator has a criticism of Louis's approach in\n    exercise 3.75.\n    \n    The program he wrote is\n    not modular, because it intermixes the operation of smoothing with the\n    zero-crossing extraction.  For example, the extractor should not have\n    to be changed if Alyssa finds a better way to condition her input\n    signal.  Help Louis by writing a\n    proceduresmooth that takes a stream as input and\n    produces a stream in which each element is the average of two successive\n    input stream elements.  Then use smooth as a\n    component to implement the zero-crossing detector in a more modular style.\n    ","3.5.3#footnote-1":"We can't use\n\tlet to bind the local variable\n\tguesses because the value of\n\tguesses depends on\n\tguesses itself.\n\tExercise 3.63\n\taddresses why we want\n\ta local name here.","3.5.3#footnote-2":"As in\n    section 2.2.3, we\n    represent a pair of integers as a list rather than a\n    Lisp\n    pair.","3.5.3#footnote-3":"See\n    exercise 3.68 for some insight into why we\n    chose this decomposition.","3.5.3#footnote-4":"The\n    precise statement of the required property on the order of combination is\n    as follows: There should be a function $f$ of\n    two arguments such that the pair corresponding to\n    element $i$ of the first stream and\n    element $j$ of the second stream will\n    appear as element number $f(i,j)$ of the output\n    stream.  The trick of using interleave\n    to accomplish this was shown to us by \n    \n    David Turner, who employed it in the language \n    \n    KRC (Turner 1981).","3.5.3#footnote-5":"We\n    will require that the weighting function be such that the weight of a pair\n    increases as we move out along a row or down along a column of the array of\n    pairs.","3.5.3#footnote-6":"To quote from G. H.\n    Hardy's obituary of\n    \n    Ramanujan (Hardy 1921): \"It was\n    Mr. Littlewood\n    (I believe) who remarked that 'every positive integer was one of his\n    friends.'  I remember once going to see him when he was lying ill\n    at Putney.  I had ridden in taxi-cab No. 1729, and remarked that the number\n    seemed to me a rather dull one, and that I hoped it was not an unfavorable\n    omen. 'No,' he replied, 'it is a very interesting number;\n    it is the smallest number expressible as the sum of two cubes in two\n    different ways.'\" The trick of using weighted pairs to\n    generate the Ramanujan numbers was shown to us by\n    \n    Charles\n    Leiserson.","3.5.4":"3.5.4  \n    Streams and Delayed Evaluation","3.5.4#p1":"\n    The integralprocedure\n    at the end of the preceding section shows how we can use streams to model\n    signal-processing systems that contain\n    \n    feedback loops.  The feedback loop for the adder shown in\n    figure 3.32 is modeled by the fact that\n    integral's\n    internal stream\n    int\n    is defined in terms of itself:\n\n    (define int\n  (cons-stream initial-value\n               (add-streams (scale-stream integrand dt)\n  int)))\n\tThe interpreter's ability to deal with such an implicit definition\n\tdepends on the delay that is incorporated\n\tinto cons-stream.  Without this\n\tdelay, the interpreter could not construct\n\tint before evaluating both arguments to\n\tcons-stream, which would require that\n\tint already be defined.\n\tIn general, delay is crucial for using\n\tstreams to model signal-processing systems that contain loops.  Without\n\tdelay, our models would have to be\n\tformulated so that the inputs to any signal-processing component would\n\tbe fully evaluated before the output could be produced.  This would\n\toutlaw loops.\n      ","3.5.4#p2":"\n\tUnfortunately, stream models of systems with loops may require uses of\n\tdelay beyond the \"hidden\"delay supplied by\n\tcons-stream.  For instance,\n\tfigure  shows a\n\tsignal-processing system for solving the \n\t\n\tdifferential equation $dy/dt=f(y)$ where\n\t$f$ is a given function.  The figure shows a\n\tmapping component, which applies $f$ to its\n\tinput signal, linked in a feedback loop to an integrator in a manner\n\tvery similar to that of the analog computer circuits that are actually\n\tused to solve such equations.\n\t","3.5.4#fig-":"","3.5.4#p3":"\n    Assuming we are given an initial value $y_0$ for\n    $y$, we could try to model this system using the\n    procedure(define (solve f y0 dt)\n  (define y (integral dy y0 dt))\n  (define dy (stream-map f y))\n  y)\n    This\n    procedure\n    does not work, because in the first line of\n    solve the call to\n    integral requires that the input\n    dy be defined, which does not happen until the\n    second line of solve.\n  ","3.5.4#p4":"\n    On the other hand, the intent of our definition does make sense, because we\n    can, in principle, begin to generate the y\n    stream without knowing dy.\n    \n\tIndeed, integral and many other stream\n\toperations have properties similar to those of\n\tcons-stream, in that we can generate part\n\tof the answer given only partial information about the arguments.\n      \n    For integral, the first element of the output\n    stream is the specified initial_value.  Thus,\n    we can generate the first element of the output stream without evaluating\n    the integrand dy.  Once we know the first\n    element of y, the\n    stream-map\n    in the second line of solve can begin working\n    to generate the first element of dy, which will\n    produce the next element of y, and so on.\n  ","3.5.4#p5":"\n    To take advantage of this idea, we will redefine\n    integral to expect the integrand stream to be a \n    delayed argument.\n    Integral will\n\tforce\n    the integrand to be evaluated only when it is required to generate more than\n    the first element of the output stream:\n    (define (integral delayed-integrand initial-value dt)\n  (define int\n          (cons-stream initial-value\n                       (let ((integrand (force delayed-integrand)))\n                         (add-streams (scale-stream integrand dt)\n                                      int))))\n  int) \n    Now we can implement our solveprocedure\n    by delaying the evaluation of dy in the\n    \n\tdefinition of\n      y:(define (solve f y0 dt)\n  (define y (integral (delay dy) y0 dt))\n  (define dy (stream-map f y))\n  y) \n    In general, every caller of integral must now\n    delay\n    the integrand argument.  We can demonstrate that the\n    solveprocedure\n    works by approximating \n    $e\\approx 2.718$ by computing the value at\n    $y=1$ of the solution to the differential\n    equation $dy/dt=y$ with initial condition\n    $y(0)=1$:(stream-ref (solve (lambda (y) y) 1 0.001) 1000) ","3.5.4#footnote-link-1":"1","3.5.4#ex-3.77":"\n    The integralprocedure\n    used above was analogous to the \"implicit\" definition of the\n    infinite stream of integers in\n    section 3.5.2.  Alternatively, we can\n    give a definition of integral that is more\n    like integers-starting-from (also in\n    section 3.5.2):\n    (define (integral integrand initial-value dt)\n  (cons-stream initial-value\n               (if (stream-null? integrand)\n                   the-empty-stream\n                   (integral (stream-cdr integrand)\n                             (+ (* dt (stream-car integrand))\n                                initial-value)\n                             dt)))) \n    When used in systems with loops, this\n    procedure\n    has the same problem\n    as does our original version of integral.\n    Modify the\n    procedure\n    so that it expects the integrand as a\n    delayed argument and hence can be used in the\n    solveprocedure\n    shown above.\n    ","3.5.4#ex-3.78":"\n    Consider the problem of designing a signal-processing system to study\n    the homogeneous\n    \n    second-order linear differential equation\n    \n      \\[\\begin{array}{lll}\n      \\dfrac {d^{2} y}{dt^{2}}-a\\dfrac{dy}{dt}-by &=& 0\n      \\end{array}\\]\n    \n    The output stream, modeling $y$, is generated by\n    a network that contains a loop. This is because the value of\n    $d^{2}y/dt^{2}$ depends upon the values of\n    $y$ and $dy/dt$ and\n    both of these are determined by integrating\n    $d^{2}y/dt^{2}$.  The diagram we would like to\n    encode is shown in figure 3.35.  Write a\n    procedure solve-2nd\n    that takes as arguments the constants $a$,\n    $b$, and $dt$ and the\n    initial values $y_{0}$ and\n    $dy_{0}$ for $y$ and\n    $dy/dt$ and generates the stream of successive\n    values of $y$.\n    ","3.5.4#fig-3.35":"","3.5.4#ex-3.79":"\n    Generalize the \n    solve-2nd procedure\n    of exercise 3.78 so that it can be used to\n    solve general second-order differential equations\n    $d^{2} y/dt^{2}=f(dy/dt,\\, y)$.\n    ","3.5.4#fig-3.36":"","3.5.4#ex-3.80":"\n    A series RLC circuit\n    consists of a resistor, a capacitor, and an\n    inductor connected in series, as shown in\n    figure 3.36. If\n    $R$, $L$, and\n    $C$ are the resistance, inductance, and\n    capacitance, then the relations between voltage\n    ($v$) and current\n    ($i$) for the three components are described\n    by the equations\n    \n      \\[\\begin{array}{lll}\n      v_{R} &=& i_{R} R\\\\[9pt]\n      v_{L} &=& L\\dfrac{di_{L}}{dt}\\\\[11pt]\n      i_{C} &=& C\\dfrac{dv_{C}}{dt}\n      \\end{array}\\]\n    \n      and the circuit connections dictate the relations\n    \n      \\[\\begin{array}{lll}\n      i_{R} &=& i_{L}=-i_{C}\\\\[3pt]\n      v_{C} &=& v_{L}+v_{R}\n      \\end{array}\\]\n    \n    Combining these equations shows that the state of the circuit (summarized by\n    $v_{C}$, the voltage across the capacitor, and\n    $i_{L}$, the current in the inductor) is\n    described by the pair of differential equations\n    \n    \\[\\begin{array}{lll}\n    \\dfrac{dv_{C}}{dt}  &=& -\\dfrac{i_{L}}{C}\\\\[11pt]\n    \\dfrac {di_{L}}{dt} &=& \\dfrac{1}{L}v_{C}-\\dfrac{R}{L}i_{L}\n    \\end{array}\\]\n    \n    The signal-flow diagram representing this system of differential equations\n    is shown in figure 3.37.\n  ","3.5.4#fig-3.37":"","3.5.4#p6":"\n    Write a\n    procedureRLC that takes as arguments the parameters\n    $R$, $L$, and\n    $C$ of the circuit and the time increment\n    $dt$.  In a manner similar to that of the\n    RCprocedure\n    of exercise 3.73,\n    RLC should produce a\n    procedure\n    that takes the initial values of the state variables,\n    $v_{C_{0}}$ and\n    $i_{L_{0}}$, and produces a pair \n    (using cons)\n    of the streams of states $v_{C}$ and\n    $i_{L}$.  Using RLC,\n    generate the pair of streams that models the behavior of a series RLC\n    circuit with $R = 1$ ohm,\n    $C= 0.2$ farad,\n    $L = 1$ henry,\n    $dt = 0.1$ second, and initial values\n    $i_{L_{0}} = 0$ amps and\n    $v_{C_{0}} = 10$ volts.\n    ","3.5.4#h1":"Normal-order evaluation","3.5.4#p7":"\n    The examples in this section illustrate how\n    the explicit use of\n      delay and force\n    provides great programming flexibility, but the same examples also show how\n    this can make our programs more complex. Our new\n    integralprocedure,\n    for instance, gives us the power to model systems with loops, but we must\n    now remember that integral should be called\n    with a delayed integrand, and every\n    procedure\n    that  uses integral must be aware of this.\n    In effect, we have created two classes of\n    procedures:\n    ordinary\n    procedures\n    and\n    procedures\n    that take delayed arguments.  In general, creating separate classes of\n    procedures\n    forces us to create separate classes of higher-order\n    procedures\n    as well.","3.5.4#footnote-link-2":"2","3.5.4#p8":"\n    One way to avoid the need for two different classes of\n    procedures\n    is to make all\n    procedures\n    take delayed arguments.  We could adopt a model of evaluation in which all\n    arguments to\n    procedures\n    are automatically delayed and arguments are forced only when they are\n    actually needed (for example, when they are required by a primitive\n    operation).  This would transform our language to use normal-order\n    evaluation, which we first described when we introduced the substitution\n    model for evaluation in section 1.1.5.\n    Converting to normal-order evaluation provides a uniform and elegant way to\n    simplify the use of delayed evaluation, and this would be a natural strategy\n    to adopt if we were concerned only with stream processing.  In\n    section 4.2, after we have studied the\n    evaluator, we will see how to transform our language in just this way.\n    Unfortunately, including delays in\n    procedure\n    calls wreaks havoc with our ability to design programs that depend on the\n    order of events, such as programs that use assignment, mutate data, or\n    perform input or output.\n    \n\tEven the single delay in\n\tcons-stream can cause great confusion, as\n\tillustrated by exercises \n\tand .\n      \n    As far as anyone knows, mutability and delayed evaluation do not mix well\n    in programming\n    \n\tlanguages, and devising ways to deal with both of these at\n\tonce is an active area of research.\n      ","3.5.4#footnote-1":"This\n    procedure is not guaranteed to work in all Scheme implementations, although\n    for any implementation there is a simple variation that will work.  The\n    problem has to do with subtle differences in the ways that Scheme\n    implementations handle internal definitions. (See\n    section 4.1.6.)","3.5.4#footnote-2":"This is a small reflection, in\n    Lisp,\n    of the difficulties that\n    \n\tconventional strongly\n      \n    typed languages such as Pascal\n    have\n    in coping with higher-order\n    procedures.\n    In\n    such\n    languages, the programmer\n    must\n    specify the data types of the\n    arguments and the result of each\n    procedure:\n    number, logical value, sequence, and so on. Consequently, we could not\n    express an abstraction such as \"map a given\n    procedureproc\n    over all the elements in a sequence\" by a single higher-order\n    procedure\n    such as\n    stream-map.\n    Rather, we would need a different mapping\n    procedure\n    for each different combination of argument and result data types that might\n    be specified for a\n    proc.\n    Maintaining a practical notion of \"data type\" in the presence\n    of higher-order\n    procedures\n    raises many difficult issues. One way of dealing with this problem is\n    illustrated by the language\n    \n    ML\n    \n    (Gordon, Milner, and Wadsworth 1979), \n    whose\n    \"polymorphic data types\"\n    include templates for\n    higher-order transformations between data types. Moreover, data types for\n    most\n    procedures\n    in ML are never explicitly declared by the programmer.  Instead, ML\n    includes a \n    type-inferencing mechanism that uses information in the environment\n    to deduce the data types for newly defined\n    procedures.","3.5.5":"3.5.5  \n    Modularity of Functional Programs and Modularity of Objects","3.5.5#p1":"\n    As we saw in section 3.1.2, one of\n    the major benefits of introducing assignment is that we can increase the\n    modularity of our systems by encapsulating, or \"hiding,\" parts\n    of the state of a large system within local variables.  Stream models can\n    provide an equivalent modularity without the use of assignment.  As an\n    illustration, we can reimplement the\n    \n    Monte Carlo estimation\n    of $\\pi$, which we examined in\n    section 3.1.2, from a\n    stream-processing point of view.\n  ","3.5.5#p2":"\n    The key modularity issue was that we wished to hide the internal state\n    of a random-number generator from programs that used random numbers.\n    We began with a\n    procedure rand-update,\n    whose successive values furnished our supply of random numbers, and used\n    this to produce a random-number generator:\n    (define rand\n  (let ((x random-init))\n    (lambda ()\n      (set! x (rand-update x))\n    x))) ","3.5.5#p3":"\n    In the stream formulation there is no random-number generator per\n    se, just a stream of random numbers produced by successive calls to\n    rand-update:\n      (define random-numbers\n  (cons-stream random-init\n               (stream-map rand-update random-numbers))) \n    We use this to construct the stream of outcomes of the Cesàro\n    experiment performed on consecutive pairs in the\n    random-numbers\n    stream:\n    (define cesaro-stream\n  (map-successive-pairs (lambda (r1 r2) (= (gcd r1 r2) 1))\n                        random-numbers))\n\n(define (map-successive-pairs f s)\n  (cons-stream\n   (f (stream-car s) (stream-car (stream-cdr s)))\n      (map-successive-pairs f (stream-cdr (stream-cdr s))))) \n    The\n    cesaro-stream\n    is now fed to a\n    monte-carloprocedure,\n    which produces a stream of estimates of probabilities.  The results are then\n    converted into a stream of estimates of $\\pi$.\n    This version of the program doesn't need a parameter telling how many\n    trials to perform.  Better estimates of $\\pi$\n    (from performing more experiments) are obtained by looking farther into the\n    pi stream:\n    (define (monte-carlo experiment-stream passed failed)\n  (define (next passed failed)\n    (cons-stream\n     (/ passed (+ passed failed))\n     (monte-carlo\n      (stream-cdr experiment-stream) passed failed)))\n  (if (stream-car experiment-stream)\n      (next (+ passed 1) failed)\n      (next passed (+ failed 1))))\n\n(define pi\n  (stream-map \n   (lambda (p) (sqrt (/ 6 p)))\n   (monte-carlo cesaro-stream 0 0))) \n    There is considerable\n    \n    modularity in this approach, because we still\n    can formulate a general \n    monte-carlo procedure\n    that can deal with arbitrary experiments.  Yet there is no assignment or\n    local state.\n  ","3.5.5#ex-3.81":"\n    Exercise 3.6 discussed generalizing\n    the random-number generator to allow one to\n    \n    reset the random-number sequence\n    so as to produce repeatable sequences of \"random\" numbers.\n    Produce a stream formulation of this same generator that operates on an\n    input stream of requests to\n    generate\n    a new\n    random number or to\n    reset\n    the sequence to a\n    specified value and that produces the desired stream of random numbers.\n    Don't use assignment in your solution.\n    ","3.5.5#ex-3.82":"\n    Redo exercise 3.5 on\n    \n    Monte Carlo integration in terms of streams.  The stream version of\n    estimate-integral\n    will not have an argument telling how many trials to perform.  Instead, it\n    will produce a stream of estimates based on successively more trials.\n    ","3.5.5#h1":"A functional-programming view of time","3.5.5#p4":"\n    Let us now return to the issues of objects and state that were raised\n    at the beginning of this chapter and examine them in a new light.  We\n    introduced assignment and mutable objects to provide a mechanism for\n    modular construction of programs that model systems with state.\n    We constructed computational objects with local state variables and used\n    assignment to modify these variables.  We modeled the temporal behavior of\n    the objects in the world by the temporal behavior of the corresponding\n    computational objects.\n  ","3.5.5#p5":"\n    Now we have seen that streams provide an alternative way to model\n    objects with local state.  We can model a changing quantity, such as\n    the local state of some object, using a stream that represents the\n    time history of successive states.  In essence, we represent time\n    explicitly, using streams, so that we decouple time in our simulated\n    world from the sequence of events that take place during evaluation.\n    Indeed, because of the presence of\n    delay\n    there may be little relation between simulated time in the model and the\n    order of events during the evaluation.\n  ","3.5.5#p6":"\n    In order to contrast these two approaches to modeling, let us\n    reconsider the implementation of a \"withdrawal processor\" that\n    monitors the balance in a\n    \n    bank account.  In\n    section 3.1.3 we implemented a\n    simplified version of such a processor:\n    (define (make-simplified-withdraw balance)\n  (lambda (amount)\n    (set! balance (- balance amount))\n    balance))\n    Calls to\n    make-simplified-withdraw\n    produce computational objects, each with a local state variable\n    balance that is decremented by successive calls\n    to the object.  The object takes an amount as\n    an argument and returns the new balance.  We can imagine the user of a bank\n    account typing a sequence of inputs to such an object and observing the\n    sequence of returned values shown on a display screen.\n  ","3.5.5#p7":"\n    Alternatively, we can model a withdrawal processor as a\n    procedure\n    that takes as input a balance and a stream of amounts to withdraw and\n    produces the stream of successive balances in the account:\n    (define (stream-withdraw balance amount-stream)\n  (cons-stream\n   balance\n   (stream-withdraw (- balance (stream-car amount-stream))\n                    (stream-cdr amount-stream)))) Stream-withdraw\n    implements a well-defined mathematical function whose output is fully\n    determined by its input.  Suppose, however, that the input\n    amount-stream\n    is the stream of successive values typed by the user and that the resulting\n    stream of balances is displayed. Then, from the perspective of the user who\n    is typing values and watching results, the stream process has the same\n    behavior as the object created by\n    make-simplified-withdraw.\n    However, with the stream version, there is no assignment, no local state\n    variable, and consequently none of the theoretical difficulties that we\n    encountered\n    \n    in section 3.1.3.  Yet the system\n    has state!\n  ","3.5.5#p8":"\n    This is really remarkable.  Even though\n    stream-withdraw\n    implements a well-defined mathematical function whose behavior does not\n    change, the user's perception here is one of interacting with a system\n    that has a changing state.  One way to resolve this paradox is to realize\n    that it is the user's temporal existence that imposes state on the\n    system.  If the user could step back from the interaction and think in terms\n    of streams of balances rather than individual transactions, the system\n    would appear stateless.","3.5.5#footnote-link-1":"1","3.5.5#p9":"\n    From the point of view of one part of a complex process, the other parts\n    appear to change with time.  They have hidden time-varying local state.  If\n    we wish to write programs that model this kind of natural decomposition in\n    our world (as we see it from our viewpoint as a part of that world) with\n    structures in our computer, we make computational objects that are not\n    functional—they must change with time.  We model state with local\n    state variables, and we model the changes of state with assignments to\n    those variables.  By doing this we make the time of execution of a\n    computation model time in the world that we are part of, and thus we\n    get \"objects\" in our computer.\n  ","3.5.5#p10":"\n    Modeling with objects is powerful and intuitive, largely because this\n    matches the perception of interacting with a world of which we are\n    part.  However, as we've seen repeatedly throughout this chapter,\n    these models raise thorny problems of constraining the order of events\n    and of synchronizing multiple processes.  The possibility of avoiding\n    these problems has stimulated the development of \n    functional programming languages, which do not include any\n    provision for assignment or mutable data.  In such a language, all\n    procedures\n    implement well-defined mathematical functions of their arguments,\n    whose behavior does not change.  The functional approach is extremely\n    attractive for dealing with\n    \n    concurrent systems.","3.5.5#footnote-link-2":"2","3.5.5#p11":"\n    On the other hand, if we look closely, we can see time-related problems\n    creeping into functional models as well.  One particularly troublesome area\n    arises when we wish to design interactive systems, especially ones that\n    model interactions between independent entities. For instance, consider once\n    more the implementation of a banking system that permits joint bank accounts.\n    In a conventional system using assignment and objects, we would model the\n    fact that Peter and Paul share an account by having both Peter and Paul send\n    their transaction requests to the same bank-account object, as we saw in\n    section 3.1.3. From the stream point\n    of view, where there are no \"objects\"per se, we have\n    already indicated that a bank account can be modeled as a process that\n    operates on a stream of transaction requests to produce a stream of\n    responses.  Accordingly, we could model the fact that Peter and Paul have a\n    joint bank account by merging Peter's stream of transaction requests\n    with Paul's stream of requests and feeding the result to the\n    bank-account stream process, as shown in\n    figure 3.38.\n    ","3.5.5#fig-3.38":"","3.5.5#p12":"\n    The trouble with this formulation is in the notion of merge.  It\n    will not do to merge the two streams by simply taking alternately one\n    request from Peter and one request from Paul. Suppose Paul accesses\n    the account only very rarely.  We could hardly force Peter to wait for\n    Paul to access the account before he could issue a second transaction.\n    \n    However such a merge is implemented, it must interleave the two\n    transaction streams in some way that is constrained by \"real\n    time\" as perceived by Peter and Paul, in the sense that, if Peter and\n    Paul meet, they can agree that certain transactions were processed\n    before the meeting, and other transactions were processed after the\n    meeting.\n    This is precisely the same constraint that we had to deal with in\n    section 3.4.1, where we found the need to\n    introduce explicit synchronization to ensure a \"correct\" order\n    of events in concurrent processing of objects with state.  Thus, in an\n    attempt to support the functional style, the need to merge inputs from\n    different agents reintroduces the same problems that the functional style\n    was meant to eliminate.\n  ","3.5.5#footnote-link-3":"3","3.5.5#p13":"\n    We began this chapter with the goal of building computational models\n    whose structure matches our perception of the real world we are trying\n    to model.  We can model the world as a collection of separate,\n    time-bound, interacting objects with state, or we can model the world\n    as a single, timeless, stateless unity.  Each view has powerful\n    advantages, but neither view alone is completely satisfactory.  A\n    grand unification has yet to emerge.","3.5.5#footnote-link-4":"4","3.5.5#footnote-1":"Similarly in physics, when we observe a\n    moving particle, we say that the position (state) of the particle is\n    changing.  However, from the perspective of the particle's \n    \n    world line in space-time there is no change involved.","3.5.5#footnote-2":"John Backus, the\n    \n    inventor of Fortran, gave high\n    visibility to functional programming when he was awarded the ACM Turing\n    award in 1978.  His acceptance speech\n    \n    (Backus 1978)\n    strongly advocated the functional approach.  A good overview of functional\n    programming is given in \n    Henderson 1980 and in \n    Darlington, Henderson, and Turner 1982.","3.5.5#footnote-3":"Observe that, for any two streams, there is in general\n    more than one\n    acceptable order of interleaving.  Thus, technically, \"merge\"\n    is a\n    \n    relation rather than a function—the answer is not a\n    deterministic function of the inputs.  We already mentioned\n    (footnote 5) that nondeterminism\n    is essential when dealing with concurrency.  The merge relation illustrates\n    the same essential nondeterminism, from the functional perspective.\n    In section 4.3, we\n    will look at nondeterminism from yet another point of view.","3.5.5#footnote-4":"The object model approximates\n    the world by dividing it into separate pieces.  The functional model does\n    not\n    \n    modularize along object boundaries.  The object model is useful when\n    the unshared state of the \"objects\" is much larger than the\n    state that they share.  An example of a place where the object viewpoint\n    fails is \n    \n    quantum mechanics, where thinking of things as individual particles leads\n    to paradoxes and confusions.  Unifying the object view with the\n    functional view may have little to do with programming, but rather\n    with fundamental epistemological issues.","4#p1":"\n    In our study of program design, we have seen that expert programmers\n    control the complexity of their designs with the same general\n    techniques used by designers of all complex systems.  They combine\n    primitive elements to form compound objects, they abstract compound\n    objects to form higher-level building blocks, and they preserve\n    modularity by adopting appropriate large-scale views of system\n    structure.  In illustrating these techniques, we have used \n    Lisp\n    as a language for describing processes and for constructing computational\n    data objects and processes to model complex phenomena in the real world.\n    However, as we confront increasingly complex problems, we will find that\n    Lisp,\n    or indeed any fixed programming language, is not sufficient for our needs.\n    We must constantly turn to new languages in order to express our ideas more\n    effectively.  Establishing new languages is a powerful strategy for\n    controlling complexity in engineering design; we can often enhance our\n    ability to deal with a complex problem by adopting a new language that\n    enables us to describe (and hence to think about) the problem in a different\n    way, using primitives, means of combination, and means of abstraction that\n    are particularly well suited to the problem at hand.","4#footnote-link-1":"1","4#p2":"\n    Programming is endowed with a multitude of languages.  There are\n    physical languages, such as the\n    \n    machine languages for particular\n    computers.  These languages are concerned with the representation of\n    data and control in terms of individual bits of storage and primitive\n    machine instructions.  The machine-language programmer is concerned\n    with using the given hardware to erect systems and utilities for the\n    efficient implementation of resource-limited computations.  High-level\n    languages, erected on a machine-language substrate, hide concerns\n    about the representation of data as collections of bits and the\n    representation of programs as sequences of primitive instructions.\n    These languages have means of combination and abstraction, such as\n    procedure definition,\n    that are appropriate to the larger-scale organization of systems.\n  ","4#p3":"Metalinguistic abstraction—establishing\n    \n    new languages—plays an important role in all branches of engineering\n    design.  It is particularly important to computer programming, because\n    in programming not only can we formulate new languages but we can also\n    implement these languages by constructing evaluators.  An \n    evaluator (or interpreter) for a programming language is a\n      procedure\n      that, when applied to\n      an expression\n      of the language, performs the actions required to evaluate that\n      \n      expression.\n    It is no exaggeration to regard this as the most fundamental idea in\n    programming:\n    \n      The evaluator, which determines the meaning of\n      \n      expressions in a programming language, is just another program.\n    \n    To appreciate this point is to change our images of ourselves as\n    programmers.  We come to see ourselves as designers of languages,\n    rather than only users of languages designed by others.\n  ","4#p5":"\n    We now embark on a tour of the technology by which languages are\n    established in terms of other languages.  In this chapter we shall use\n    Lisp \n    as a base, implementing evaluators as \n    Lisp procedures.\n\tLisp is particularly well suited to this task, because of its ability\n\tto represent and manipulate symbolic expressions.\n      \n    We will take the first step in understanding how languages are implemented\n    by building\tan evaluator for\n    Lisp\n    itself.\n    The language implemented by our evaluator will be a subset of\n    the Scheme dialect of Lisp that we use in this\n      book.\n    Although the evaluator described in this chapter is written for a\n    particular \n    dialect of Lisp,\n    it contains the essential structure of an evaluator for any\n    expression-oriented language\n    designed for writing programs for a sequential machine.  (In fact, most\n    language processors contain, deep within them, a little \n    \"Lisp\" evaluator.)\n    The evaluator has been simplified for the purposes of illustration and\n    discussion, and some features have been left out that would be\n    important to include in a production-quality \n    Lisp\n    system. Nevertheless, this simple evaluator is adequate to execute most of\n    the programs in this book.","4#footnote-link-2":"2","4#p6":"\n    An important advantage of making the evaluator accessible as a \n    Lisp\n    program is that we can implement alternative evaluation rules by describing\n    these as modifications to the evaluator program.  One place where we can use\n    this power to good effect is to gain extra control over the ways in which\n    computational models embody the notion of time, which was so central to the\n    discussion in chapter 3.  There, we mitigated some of the complexities\n    of state and assignment by using streams to decouple the representation of\n    time in the world from time in the computer.  Our stream programs, however,\n    were sometimes cumbersome, because they were constrained by the\n    applicative-order evaluation of \n    Scheme.\n    In section 4.2, we'll change\n    the underlying language to provide for a more elegant approach, by modifying\n    the evaluator to provide for normal-order evaluation.\n  ","4#p7":"\n    Section 4.3 implements a\n    more ambitious linguistic change, whereby statements and expressions\n    have many values, rather than just a single value.  In this language of\n    nondeterministic computing, it is natural to express processes that\n    generate all possible values for statements and expressions and then search\n    for those values that satisfy certain constraints.  In terms of models of\n    computation and time, this is like having time branch into a set of\n    \"possible futures\" and then searching for appropriate time\n    lines. With our nondeterministic evaluator, keeping track of multiple values\n    and performing searches are handled automatically by the underlying\n    mechanism of the language.\n  ","4#p8":"\n    In section 4.4 we implement a\n    logic-programming language in which knowledge is expressed in terms\n    of relations, rather than in terms of computations with inputs and outputs.\n    Even though this makes the language drastically different from\n    Lisp,\n    or indeed from any conventional language, we will see that\n    the logic-programming evaluator shares the essential structure of the\n    Lisp\n    evaluator.\n  ","4#footnote-1":"The same idea\n    is pervasive throughout all of engineering.  For example, electrical\n    engineers use many different languages for describing circuits.  Two\n    of these are the language of electrical networks and the\n    language of electrical systems.  The network language emphasizes\n    the physical modeling of devices in terms of discrete electrical\n    elements.  The primitive objects of the network language are primitive\n    electrical components such as resistors, capacitors, inductors, and\n    transistors, which are characterized in terms of physical variables\n    called voltage and current.  When describing circuits in the network\n    language, the engineer is concerned with the physical characteristics\n    of a design.  In contrast, the primitive objects of the system\n    language are signal-processing modules such as filters and amplifiers.\n    Only the functional behavior of the modules is relevant, and signals\n    are manipulated without concern for their physical realization as\n    voltages and currents.  The system language is erected on the network\n    language, in the sense that the elements of signal-processing systems\n    are constructed from electrical networks.  Here, however, the concerns\n    are with the large-scale organization of electrical devices to solve a\n    given application problem; the physical feasibility of the parts is\n    assumed.  This layered collection of languages is another example of\n    the stratified design technique illustrated by the picture\n    language of section 2.2.4.","4#footnote-2":"The most important features that our\n    evaluator leaves out are mechanisms for handling errors and supporting\n    debugging.  For a more extensive discussion of evaluators, see \n    \n    Friedman, Wand, and Haynes\n    1992, which gives an exposition of programming languages that proceeds\n    via a sequence of evaluators written in \n    Scheme.","4.1":"4.1  The Metacircular Evaluator","4.1#p1":"\n    Our evaluator for \n    Lisp \n    will be implemented as a \n    Lisp \n    program.  It may\n    seem circular to think about evaluating \n    Lisp \n    programs using an evaluator that is itself implemented in \n    Lisp.\n    However, evaluation is a process, so it is appropriate to describe the\n    evaluation process using\n    Lisp,\n    which, after all, is our tool for describing processes.  \n    An evaluator that is written in the same language\n    that it evaluates is said to be\n    metacircular.\n  ","4.1#footnote-link-1":"1","4.1#p2":"\n    The metacircular evaluator is essentially a \n    Scheme\n    formulation of the\n    \n    environment model of evaluation described in\n    section 3.2.\n    Recall that the model has two basic parts:","4.1#p3":"\n            To evaluate a combination (a compound expression other than a \n            special form), evaluate the subexpressions and then apply the value\n\t    of the operator subexpression to the values of the operand\n            subexpressions.\n\t  \n        To apply a compound\n\tprocedure\n\tto a set of arguments, evaluate the\n        body of the\n\tprocedure\n\tin a new environment.  To construct this\n        environment, extend the environment part of the\n\tprocedure\n\tobject by a\n        frame in which the\n\tformal\n\tparameters of the\n\tprocedure\n\tare bound to\n\tthe arguments to which the\n\tprocedure\n\tis applied.\n      ","4.1#p4":"\n    These two rules describe the essence of the evaluation process, a basic\n    \n    cycle in which \n    \n    expressions to be evaluated in environments are reduced to\n    procedures\n    to be applied to arguments, which in turn are reduced to new \n    \n    expressions to be evaluated in new environments, and so on, until we get\n    down to\n    \n\tsymbols,\n      \n    whose values are looked up in the environment, and to \n    primitive procedures,\n    which are applied directly (see\n    figure 4.1).\n    This evaluation cycle will be embodied by the interplay between the two\n    critical\n    procedures\n    in the evaluator,\n    eval\n    and apply, which are described in\n    section 4.1.1\n    (see figure 4.1).\n  ","4.1#footnote-link-2":"2","4.1#p5":"\n\tThe implementation of the evaluator will depend upon procedures\n\tthat define the syntax of the expressions to be evaluated.\n\tWe will use\n\t\n\tdata abstraction to make the evaluator independent of the representation of\n\tthe language.  For example, rather than committing to a choice that an\n\tassignment is to be represented by a list beginning with the \n\tsymbol set! we use an abstract predicate\n\tassignment?\n\tto test for an assignment, and we use abstract selectors\n\tassignment-variable and \n\tassignment-value\n\tto access the parts of an assignment.\n\tImplementation of expressions will be described in detail in\n\tsection 4.1.2. \t\n      \n    There are also\n    operations, described in\n    section 4.1.3, that specify the\n    representation of\n    procedures\n    and environments.  For example,\n    make-procedure\n    constructs compound\n    procedures,lookup-variable-value\n    accesses the values of\n    variables,\n    and\n    apply-primitive-procedure\n    applies a primitive\n    procedure\n    to a given list of arguments.\n  ","4.1#footnote-1":"Even so,\n    there will remain important aspects of the evaluation process that are not\n    elucidated by our evaluator.  The most important of these are the detailed\n    mechanisms by which\n    procedures\n    call other\n    procedures\n    and return values to their callers. We will address these issues in\n    chapter 5, where we take a closer look at the evaluation process by\n    implementing the evaluator as a simple register machine.","4.1#footnote-2":"If we grant ourselves\n    the ability to apply primitives,\n    then what remains for us to implement in the evaluator?  The\n    \n    job of the\n    evaluator is not to specify the primitives of the language, but rather to\n    provide the connective tissue—the means of combination and the means\n    of abstraction—that binds a collection of primitives to form a\n    language.  Specifically:\n    \n\tThe evaluator enables us to deal with nested expressions. For example,\n\talthough simply applying primitives would suffice for evaluating the\n\texpression\n\t(+ 1 6),\n\tit is not adequate for handling \n\t(+ 1 (* 2 3)).\n\tAs far as the \n\t\n\t    primitive procedure\n\t    +\n\tis concerned, its arguments must be numbers, and it would choke if we\n\tpassed it the expression\n\t(* 2 3)\n\tas an argument.\tOne important role of the evaluator is to choreograph\n\tprocedure\n\tcomposition so that\n\t(* 2 3)\n\tis reduced to 6 before being passed as an \n\t\n\t    argument to +.\n\t  \n\tThe evaluator allows us to use\n\tvariables.\n\tFor example, the \n\tprimitive procedure for addition\n\thas no way to deal with expressions such as\n\t(+ x 1).\n\tWe need an evaluator to keep track of\n\tvariables\n\tand obtain their values before invoking the \n\tprimitive procedures.\n\tThe evaluator allows us to define compound\n\tprocedures.\n\t    This involves keeping track of procedure definitions, knowing\n\t    how to use these definitions in evaluating\n\t    expressions, and providing a mechanism that enables\n\t    procedures to accept arguments.\n\t  The evaluator provides the special forms, which must be\n\t  evaluated differently from procedure calls.","4.1.1":"4.1.1  \n    The Core of the Evaluator","4.1.1#p1":"\n    The evaluation\n    \n    process can be described as the interplay between two\n    procedures:eval\n    and apply.\n  ","4.1.1#fig-":"","4.1.1#h1":"Eval","4.1.1#p2":"","4.1.1#p3":"Eval\n    takes as arguments\n    \n\tand an environment.\n      \n    It  classifies the \n    expression \n    and directs its evaluation.  \n    Eval\n    is structured as a case analysis of the syntactic type of the\n    expression \n    to be evaluated.  In order to keep the\n    procedure\n    general, we express\n    the determination of the type of \n    an expression\n    abstractly, making no\n    commitment to any particular \n    \n    representation for the various types of\n    expressions.\n    Each type of \n    expression\n    has a\n    predicate\n    that tests for it and an abstract means for selecting its parts.  This \n    abstract syntax\n    makes it easy to see how we can change the syntax of the language by\n    using the same evaluator, but with a different collection of syntax\n    \n\tprocedures.\n      ","4.1.1#h2":"Primitive expressions","4.1.1#p4":"\n\t\tFor\n\t\t\n\t\tself-evaluating expressions, such as numbers,\n\t\teval\n\t\treturns the expression itself.\n\t      Eval\n\t    must look up\n\t    variables\n\t    in the environment to find their values.\n\t  \n\t      For quoted expressions, \n\t      eval\n\t      returns the expression that was\n\t      quoted.\n\t    \n\t      An assignment to (or a definition of) a variable\n\t      must recursively call      \n\t      eval\n\t      to compute the new value to be associated with the variable.\n\t      The environment must be modified to change (or create) the\n\t      binding of the variable.\n\t    \n\t      An if expression requires special processing of its parts, so as to\n\t      evaluate the consequent if the predicate is true, and otherwise to\n\t      evaluate the alternative.\n\t    \n\t      A lambda expression\n\t      must be transformed into an applicable\n\t      procedure by packaging together the parameters and body\n\t      specified by the\n\t      lambda\n\t      expression with the environment of the evaluation.\n\t    \n\t      A begin expression\n\t      requires evaluating its sequence of\n\t      expressions in the order in which they appear.\n\t    \n\t      A case analysis (cond) is transformed\n\t      into a nest of if\n\t      expressions and then evaluated.\n\t    \n\t  For a procedure application, eval\n\t  must recursively evaluate the operator part and the operands of\n\t  the combination.  The resulting procedure\n\t  and arguments are passed to apply,\n\t  which handles the actual procedure application.\n\t\n    Here is the definition of eval:\n    (define (eval exp env)\n  (cond ((self-evaluating? exp) exp)\n        ((variable? exp) (lookup-variable-value exp env))\n        ((quoted? exp) (text-of-quotation exp))\n        ((assignment? exp) (eval-assignment exp env))\n        ((definition? exp) (eval-definition exp env))\n        ((if? exp) (eval-if exp env))\n        ((lambda? exp)\n         (make-procedure (lambda-parameters exp)\n                         (lambda-body exp)\n                         env))\n        ((begin? exp) \n         (eval-sequence (begin-actions exp) env))\n        ((cond? exp) (eval (cond->if exp) env))\n        ((application? exp)\n         (apply (eval (operator exp) env)\n                (list-of-values (operands exp) env)))\n        (else\n         (error \"Unknown expression type - - EVAL\" exp)))) ","4.1.1#h3":"Special forms","4.1.1#h4":"Combinations","4.1.1#p5":"\n    For clarity, \n    eval\n    has been implemented as a\n    \n    case analysis using\n    cond.\n    The disadvantage of this is that our\n    procedure\n    handles only a few distinguishable types of \n    \n    expressions, and no new ones can be defined without editing the\n    \n\tdefinition of \n\teval.\n      \n    In most\n    Lisp\n    implementations, dispatching on the type of\n    an expression\n    is done in a data-directed style.  This allows a user to add new types of    \n    expressions that eval\n    can distinguish, without modifying the \n    \n\tdefinition of eval\n    itself. (See exercise .)\n  ","4.1.1#p6":"\n    The representation of names is handled by the syntax abstractions. Internally,\n    the evaluator uses strings to represent names, and we refer to such strings as\n    symbols. The function\n    symbol_of_name used in\n    evaluate extracts from a\n    name the symbol by which it is represented.\n  ","4.1.1#h5":"Apply","4.1.1#p7":"Apply\n    takes two arguments, a\n    procedure\n    and a list of arguments  to which the\n    procedure\n    should be applied.  \n    Apply\n    classifies\n    procedures\n    into two kinds: It calls \n    apply-primitive-procedure\n    to apply primitives; it applies compound\n    procedures\n\tby sequentially evaluating the \t\n\texpressions that make up the body of the procedure.\n      \n    The environment for the evaluation of the body of a compound\n    procedure\n    is constructed by extending the base environment carried by the\n    procedure\n    to include a frame that binds the parameters of the\n    procedure\n    to the arguments to which the\n    procedure\n    is to be applied. \n    Here is the\n    \n\tdefinition\n      \n    of apply:\n    (define (apply procedure arguments)\n  (cond ((primitive-procedure? procedure)\n         (apply-primitive-procedure procedure arguments))\n        ((compound-procedure? procedure)\n         (eval-sequence\n           (procedure-body procedure)\n           (extend-environment\n             (procedure-parameters procedure)\n             arguments\n             (procedure-environment procedure))))\n        (else\n         (error\n          \"Unknown procedure type - - APPLY\" procedure)))) ","4.1.1#h6":"Procedure\n      arguments\n    ","4.1.1#p8":"\n    When\n    eval\n    processes a\n    procedure\n    application, it uses\n    list-of-values\n    to produce the list of arguments to which the\n    procedure\n    is to be applied.\n    List-of-values\n    takes as an argument the\n    operands of the combination.\n    It evaluates each\n    operand\n    and returns a\n    list of the corresponding values:(define (list-of-values exps env)\n  (if (no-operands? exps)\n      '()\n      (cons (eval (first-operand exps) env)\n            (list-of-values (rest-operands exps) env)))) ","4.1.1#footnote-link-1":"1","4.1.1#h7":"\n      Conditionals\n    ","4.1.1#p9":"Eval-if\n    evaluates the predicate part of \n    an if expression\n    in the given environment.  If the result is true, \n    eval-if\n      evaluates the consequent, otherwise it evaluates the alternative:\n      (define (eval-if exp env)\n  (if (true? (eval (if-predicate exp) env))\n      (eval (if-consequent exp) env)\n      (eval (if-alternative exp) env))) ","4.1.1#p10":"\n    The use of\n    true?\n    in \n    eval-if\n    highlights the issue of the connection between an implemented language and\n    an implementation language.  The \n    if-predicate\n    is evaluated in the language being implemented and thus yields a value in\n    that language.  The interpreter predicate \n    true? \n    translates that value into a value that can be tested by the\n    if\n    in the implementation language: The metacircular representation of truth\n    might not be the same as that of the underlying \n    Scheme.","4.1.1#footnote-link-2":"2","4.1.1#h8":"    \n\tSequences\n    ","4.1.1#p11":"Eval-sequence is used by\n\tapply to evaluate the sequence of\n\texpressions in a procedure body and by eval\n\tto evaluate the sequence of expressions in a\n\tbegin expression.  It takes as arguments a\n\tsequence of expressions and an environment, and evaluates the\n\texpressions in the order in which they occur.  The value returned is the\n\tvalue of the final expression.\n\t(define (eval-sequence exps env)\n  (cond ((last-exp? exps) (eval (first-exp exps) env))\n        (else (eval (first-exp exps) env)\n              (eval-sequence (rest-exps exps) env)))) ","4.1.1#h9":"Assignments and\n    definitions","4.1.1#p12":"\n    The \n    following procedure\n    handles assignments to\n    variables.\n\tIt calls\n\teval\n\tto find the value to be assigned and transmits the variable\n\tand the resulting value to\n\tset-variable-value!\n\tto be installed  in the designated environment.\n      (define (eval-assignment exp env)\n  (set-variable-value! (assignment-variable exp)\n                       (eval (assignment-value exp) env)\n                       env)\n  'ok) ","4.1.1#p13":"\n\tDefinitions of variables are handled in a similar manner.(define (eval-definition exp env)\n  (define-variable! (definition-variable exp)\n                    (eval (definition-value exp) env)\n                    env)\n  'ok) ","4.1.1#footnote-link-3":"3","4.1.1#p14":"\n\tWe have chosen here to return the symbol ok\n\tas the value of an assignment or a definition.","4.1.1#footnote-link-4":"4","4.1.1#ex-4.1":"\n\tNotice that we cannot tell whether the metacircular evaluator\n\t\n\tevaluates operands from left to right or from right to left.  Its\n\tevaluation order is inherited from the underlying Lisp:\tIf the\n\targuments to cons in\n\tlist-of-values are evaluated from left to\n\tright, then list-of-values\twill evaluate\n\toperands from left to right; and if the arguments to\n\tcons are evaluated from right to left, then\n\tlist-of-values will evaluate operands\n\tfrom right to left.\n\t\n\tWrite a version of list-of-values that\n\tevaluates operands from left to right regardless of the\n\torder of evaluation in the underlying Lisp. Also write a version of\n\tlist-of-values that evaluates\n\toperands from right to left.\n\t","4.1.1#footnote-1":"We could\n    have simplified the \n    application?\n    clause in\n    eval\n    by using\n    map (and stipulating that\n    operands\n    returns a list) rather than writing\n    an explicit\n    list-of-values procedure.\n    We chose not to use map here to emphasize the\n    fact that the\n    \n    evaluator can be implemented without any use of higher-order\n    procedures\n    (and thus could be written in a language that doesn't\n    have higher-order\n    procedures),\n    even though the language that it supports\n    will include higher-order\n    procedures.\n    ","4.1.1#footnote-2":"In this case, the language being implemented and the\n    implementation language are the same. Contemplation of the meaning of\n    true?\n    here yields\n    \n    expansion of consciousness without the abuse of\n    substance.","4.1.1#footnote-3":"This\n\timplementation of define ignores a subtle\n\tissue in the handling of internal definitions, although it works\n\tcorrectly in most cases.  We will see what the problem is and how to\n\tsolve it in\n\tsection 4.1.6.","4.1.1#footnote-4":"As we said when\n\twe introduced define and \n        set!, these values\n        are implementation-dependent in Scheme—that is, the implementor\n        can choose what value to return.","4.1.2":"4.1.2  \n    Representing","4.1.2#p1":"\n    The evaluator is reminiscent of the\n    \n    symbolic differentiation program\n    discussed in section 2.3.2.\n    Both programs operate on symbolic\n    expressions.\n    In both programs, the\n    result of operating on \n    a compound expression\n    is determined by\n    operating recursively on the pieces of the\n    expression\n    and combining\n    the results in a way that depends on the type of the\n    expression.\n    In both programs we used \n    \n    data abstraction to decouple the general rules\n    of operation from the details of how\n    expressions\n    are represented.  In\n    the differentiation program this meant that the same differentiation\n    procedure\n    could deal with algebraic expressions in prefix form, in\n    infix form, or in some other form.  For the evaluator, this means that\n    \n\tthe syntax of the language being evaluated is determined solely by the\n      procedures that classify and extract pieces of expressions.\n      ","4.1.2#p2":"\n\tHere is the specification of the syntax of our language:\n\t\n\t    The only self-evaluating items are numbers and\n\t    strings:\n\t    (define (self-evaluating? exp)\n  (cond ((number? exp) true)\n        ((string? exp) true)\n        ((null? exp) true)\n        (else false))) \n\t    Variables are represented by symbols:\n\t    (define (variable? exp) (symbol? exp)) \n\t    Quotations have the form\n\t    (quote text-of-quotation) \t    :(define (quoted? exp)\n  (tagged-list? exp 'quote))\n\n(define (text-of-quotation exp) (cadr exp)) Quoted? is defined in terms of the\n\t    procedure tagged-list?, which\n\t    identifies lists beginning with a designated symbol:\n\t    (define (tagged-list? exp tag)\n  (if (pair? exp)\n      (eq? (car exp) tag)\n      false)) \n\t    Assignments have the form\n\t    (set!var value):\n\t    (define (assignment? exp)\n  (tagged-list? exp 'set!))\n\n(define (assignment-variable exp) (cadr exp))\n\n(define (assignment-value exp) (caddr exp)) \n\t    Definitions have the form\n\t    (define var value)\n\t    or the form\n\t    (define (var parameter$_{1}$ $\\ldots$ parameter$_{n}$)\n  body)\n\t    The latter form (standard\n\t    procedure definition) is\n\t    \n\t    syntactic sugar for\n\t    (define var\n  (lambda (parameter$_{1}$ $\\ldots$ parameter$_{n}$)\n    body))\n\t    The corresponding syntax\n\t    procedures\n\t    are the following:\n\t    (define (definition? exp)\n  (tagged-list? exp 'define))\n\n(define (definition-variable exp)\n  (if (symbol? (cadr exp))\n      (cadr exp)\n      (caadr exp)))\n\n(define (definition-value exp)\n  (if (symbol? (cadr exp))\n      (caddr exp)\n      (make-lambda (cdadr exp)\n                   (cddr exp)))) Lambda expressions are lists that begin\n\t    with the symbol lambda:\n\t    (define (lambda? exp) (tagged-list? exp 'lambda))\n\n(define (lambda-parameters exp) (cadr exp))\n(define (lambda-body exp) (cddr exp))\n\n(define (make-lambda parameters body)\n  (cons 'lambda (cons parameters body))) \n\t    We also provide a constructor for\n\t    lambda expressions, which is used by\n\t    definition-value, above:\n\t    (define (make-lambda parameters body)\n  (cons 'lambda (cons parameters body))) \n\t    Conditionals begin with if and have a\n\t    predicate, a consequent, and an (optional) alternative.  If the\n\t    expression has no alternative part, we provide\n\t    false as the alternative.(define (if? exp) (tagged-list? exp 'if))\n\n(define (if-predicate exp) (cadr exp))\n\n(define (if-consequent exp) (caddr exp))\n\n(define (if-alternative exp)\n  (if (not (null? (cdddr exp)))\n      (cadddr exp)\n      'false))\n\n(define (make-if predicate consequent alternative)\n  (list 'if predicate consequent alternative)) \n\t  We also provide a constructor for if expressions,\n\t  to be used by cond->if to transform cond expressions\n\t  into if expressions:\n\t  (define (make-if predicate consequent alternative)\n  (list 'if predicate consequent alternative)) Begin packages a sequence of\n\t    expressions into a single expression.  We include syntax operations\n\t    on begin expressions to extract the\n\t    actual sequence from the begin\n\t    expression, as well as selectors that return the first expression\n\t    and the rest of the expressions in the sequence.(define (begin? exp) (tagged-list? exp 'begin))\n\n(define (begin-actions exp) (cdr exp))\n\n(define (last-exp? seq) (null? (cdr seq)))\n(define (first-exp seq) (car seq))\n(define (rest-exps seq) (cdr seq)) \n\t    We also include a constructor \n\t    sequence->exp (for use by\n\t    cond->if) \n\t    that transforms a sequence into a single expression,\n\t    using begin if necessary:\n\t    (define (sequence->exp seq)\n  (cond ((null? seq) seq)\n        ((last-exp? seq) (first-exp seq))\n        (else (make-begin seq))))\n\n(define (make-begin seq) (cons 'begin seq)) \n\t    A procedure application is any compound expression that is not one\n\t    of the above expression types. The car\n\t    of the expression is the operator, and the \n\t    cdr is the list of operands:\n\t    (define (application? exp) (pair? exp))\n(define (operator exp) (car exp))\n(define (operands exp) (cdr exp))\n\n(define (no-operands? ops) (null? ops))\n(define (first-operand ops) (car ops))\n(define (rest-operands ops) (cdr ops)) ","4.1.2#footnote-link-1":"1","4.1.2#footnote-link-2":"2","4.1.2#footnote-link-3":"3","4.1.2#h1":"Derived expressions","4.1.2#p3":"\n\tSome special forms in our language can be defined in terms of\n\texpressions involving\n\t\n\tother special forms, rather than being\n\timplemented directly.  One example is cond,\n\twhich can be implemented as a nest of if\n\texpressions.  For example, we can reduce the problem of evaluating the\n\texpression\n\t(cond ((> x 0) x)\n      ((= x 0) (display 'zero) 0)\n      (else (- x))) \n\tto the problem of evaluating the following expression involving\n\tif and begin\n\texpressions:\n\t(if (> x 0)\n    x\n    (if (= x 0)\n        (begin (display 'zero)\n               0)\n        (- x))) \n\tImplementing the evaluation of cond in this\n\tway simplifies the evaluator because it reduces the number of special\n\tforms for which the evaluation process must be explicitly specified.\n      ","4.1.2#p4":"\n\tWe include syntax procedures that extract the parts of a\n\tcondexpression, and a procedure\n\tcond->if that transforms \n\tcond expressions into\n\tif expressions. A case analysis begins with\n\tcond and has a list of predicate-action\n\tclauses.  A clause is an else clause if its\n\tpredicate is the symbol\telse.(define (cond? exp) (tagged-list? exp 'cond))\n\n(define (cond-clauses exp) (cdr exp))\n\n(define (cond-else-clause? clause)\n  (eq? (cond-predicate clause) 'else))\n\n(define (cond-predicate clause) (car clause))\n\n(define (cond-actions clause) (cdr clause))\n\n(define (cond->if exp)\n  (expand-clauses (cond-clauses exp)))\n\n(define (expand-clauses clauses)\n  (if (null? clauses)\n      'false                          ; no else clause\n      (let ((first (car clauses))\n            (rest (cdr clauses)))\n        (if (cond-else-clause? first)\n            (if (null? rest)\n                (sequence->exp (cond-actions first))\n                (error \"ELSE clause isn't last - - COND->IF\"\n                       clauses))\n            (make-if (cond-predicate first)\n                     (sequence->exp (cond-actions first))\n                     (expand-clauses rest)))))) ","4.1.2#footnote-link-4":"4","4.1.2#p5":"\n\tExpressions (such as cond) that we choose\n\tto implement as syntactic transformations are called derived\n\texpressions. Let expressions are also\n\tderived expressions (see\n\texercise 4.6).","4.1.2#footnote-link-5":"5","4.1.2#ex-4.2":"\n\tLouis Reasoner plans to reorder the cond\n\tclauses in eval so that the clause for\n\t\tprocedure applications appears before the clause for assignments.  He\n\targues that this will make the interpreter more efficient:  Since\n\tprograms usually contain more applications than assignments,\n\tdefinitions, and so on, his modified eval\n\twill usually check fewer clauses than the original\n\teval before identifying the type of an\n\texpression.\n\t\n\t    What is wrong with Louis's plan?  (Hint: What will\n\t    Louis's evaluator do with the expression \n\t    (define x 3)?)\n\t  \n\t    Louis is upset that his plan didn't work. He is willing to go\n\t    to any lengths to make his evaluator recognize procedure\n\t    applications before it checks for most other kinds of expressions.\n\t    Help him by changing the\n\t    \n\t    syntax of the evaluated language so that\n\t    procedure applications start with call.\n\t    For example, instead of (factorial 3)\n\t    we will now have to write\n\t    (call factorial 3) and instead of \n\t    (+ 1 2) we will have to write \n\t    (call + 1 2).\n\t  ","4.1.2#ex-4.3":"\n\tRewrite\n\teval so that the dispatch is done in\n\tdata-directed style.  Compare this with the data-directed differentiation\n\tprocedure of\n\texercise 2.73.\t(You may\n\tuse the car of a compound expression as the\n\ttype of the expression, as is appropriate for the syntax implemented in\n\tthis section.)\n\t","4.1.2#ex-4.4":"\n\tRecall\n\t\n\tspecial forms\n\tand\n\tand\n\tor\n\tfrom section 1.1.6:\n      and: The expressions are evaluated from\n\t    left to right.  If any expression evaluates to false, false is\n\t    returned; any remaining expressions are not evaluated.  If all the\n\t    expressions evaluate to true values, the value of the last\n\t    expression is returned.  If there are no expressions then true is\n\t    returned.\n\t  or: The expressions are evaluated from\n\t    left to right.  If any expression evaluates to a true value, that\n\t    value is returned; any remaining expressions are not evaluated.  If\n\t    all expressions evaluate to false, or if there are no expressions,\n\t    then false is returned.\n\t  \n\tInstall and and\n\tor as new special forms for the evaluator\n\tby defining appropriate syntax procedures and evaluation procedures\n\teval-and and \n\teval-or. Alternatively, show how to\n\timplement and and\n\tor\tas derived expressions.\n      ","4.1.2#ex-4.5":"\n\tScheme allows an additional syntax for \n\tcond clauses, \n\t(test=>recipient)\n\tIf test evaluates to a true value, then\n\trecipient is evaluated. Its value must be a\n\tprocedure of one argument; this\tprocedure is then invoked on the value\n\tof the test, and the result is returned as\n\tthe value of the cond expression.  For\n\texample\n\t(cond ((assoc 'b '((a 1) (b 2))) => cadr)\n      (else false)) \n\treturns 2. Modify the handling of cond \n\tso that it supports this extended syntax.\n\t","4.1.2#ex-4.6":"Let expressions are derived expressions,\n\t\n\tbecause\n\t(let ((var$_{1}$ exp$_{1}$) $\\ldots$ (var$_{n}$ exp$_{n}$))\n  body)\n\tis equivalent to\n\t((lambda (var$_{1}$ $\\ldots$ var$_{n}$)\n   body)\n exp$_{1}$\n $\\vdots$\n exp$_{n}$)\n\tImplement a syntactic transformation\n\tlet->combination that reduces evaluating\n\tlet expressions to evaluating combinations\n\tof the type shown above, and add the appropriate clause to\n\teval to handle\n\tlet expressions.\n\t","4.1.2#ex-4.7":"Let* is similar to\n\tlet, except that the bindings of the\n\tlet* variables are performed sequentially\n\tfrom left to right, and each binding is made in an environment in which\n\tall of the preceding bindings are visible.  For example\n\t(let* ((x 3)\n       (y (+ x 2))\n       (z (+ x y 5)))\n  (* x z)) \n\treturns 39.  Explain how a let* expression\n\tcan be rewritten as a set of nested let\n\texpressions, and write a procedure\n\tlet*->nested-lets that performs this\n\ttransformation. If we have already implemented\n\tlet\n\t(exercise 4.6)\n\tand we want to extend the evaluator to handle\n\tlet*, is it sufficient to add a clause to\n\teval whose action is\n\t(eval (let*->nested-lets exp) env) \n\tor must we\n\texplicitly expand let* in terms of\n\tnon-derived expressions?\n\t","4.1.2#ex-4.8":"\"Named let\" is a variant of\n\tlet that has the form\n\t(let var bindings body) \n\tThe bindings and\n\tbody are just as in ordinary\n\tlet, except that\n\tvar is bound within\n\tbody to a procedure whose body is\n\tbody and whose parameters are the variables\n\tin the bindings.  Thus, one can repeatedly\n\texecute the body by invoking the procedure\n\tnamed var.  For example, the iterative\n\tFibonacci procedure (section 1.2.2)\n\tcan be rewritten using named let as follows:\n\t(define (fib n)\n  (let fib-iter ((a 1)\n                 (b 0)\n                 (count n))\n    (if (= count 0)\n        b\n        (fib-iter (+ a b) a (- count 1))))) \n\tModify let->combination of\n\texercise 4.6 to also support named\n\tlet.\n\t","4.1.2#ex-4.9":"\n\tMany languages support a variety of iteration constructs, such as\n  \tdo, for,\n\twhile, and\n\tuntil.  In Scheme,\titerative processes can\n\tbe expressed in terms of ordinary procedure calls, so special iteration\n\tconstructs provide no essential gain in\tcomputational power.  On the\n\tother hand, such constructs are often convenient.  Design some iteration\n\tconstructs, give examples of their use, and show how to implement them\n\tas derived expressions.\n      ","4.1.2#ex-4.10":"\n\tBy using data abstraction, we were able to write an\n\teval procedure that is independent of the\n\tparticular syntax of the language to be evaluated.  To illustrate this,\n\tdesign and implement a new\n\t\n\tsyntax for Scheme by modifying the procedures\n\tin this section, without changing eval or\n\tapply.\n      ","4.1.2#footnote-1":"As mentioned in\n\t    section , the evaluator sees a\n\t    quoted expression as a list beginning with\n\t    quote, even if the expression is typed\n\t    with the quotation mark.  For example, the expression\n\t    'a would be seen by the evaluator as\n\t    (quote a).\n\t    See exercise 2.55.","4.1.2#footnote-2":"The\n\t    value of an if expression when the\n\t    predicate is false and there is no alternative is unspecified in\n\t    Scheme; we have chosen here to make it false. We will support the\n\t    use of the variables true and\n\t    false in expressions to be evaluated\n\t    by binding them in the global environment.  See\n\t    section 4.1.4.\n\t  ","4.1.2#footnote-3":"These\n\t    selectors for a list of expressions—and the corresponding\n\t    ones for a list of operands—are not intended as a data\n\t    abstraction. They are introduced as mnemonic names for the basic list\n\t    operations in order to make it easier to understand the\n\t    explicit-control evaluator in\n\t    section 5.4.\n\t    ","4.1.2#footnote-4":"The\n\tvalue of a cond expression when all the\n\tpredicates are false and there is no else\n\tclause is unspecified in Scheme; we have chosen here to make it\n\tfalse.","4.1.2#footnote-5":"Practical Lisp\n\tsystems provide a mechanism that allows a user to add new derived\n\texpressions and specify their implementation as syntactic\n\ttransformations without\tmodifying the evaluator.  Such a user-defined\n\ttransformation is called a\n\tmacro.\tAlthough it is easy to add an elementary mechanism for\n\tdefining macros, the resulting language has subtle name-conflict\n\tproblems. There has been much research on mechanisms for macro definition\n\tthat do not cause these difficulties.  See,\n\t\n\tfor example, Kohlbecker 1986, \n\tClinger and Rees 1991, and \n\tHanson 1991.","4.1.3":"4.1.3  \n    Evaluator Data Structures","4.1.3#p1":"\n    In addition to defining the \n    external syntax of expressions,\n    the evaluator implementation must also define the data structures that the\n    evaluator manipulates internally, as part of the execution of a\n    program, such as the representation of\n    procedures\n    and environments and the representation of true and false.\n  ","4.1.3#h1":"Testing of predicates","4.1.3#p2":"\n\tFor conditionals, we accept anything to be true that is not the explicit\n\tfalse object.\n      (define (true? x)\n  (not (eq? x false)))\n\n(define (false? x)\n  (eq? x false)) ","4.1.3#h2":"Representing\n    procedures","4.1.3#p3":"\n    To handle primitives, we assume that we have available the following\n    procedures:(apply-primitive-procedureproc args)\n\n\tapplies the given primitive\n\tprocedure\n\tto the argument values in the list args and returns the result of\n\tthe application.\n      (primitive-procedure?proc)\n\n\ttests whether\n\tproc\n        is a primitive\n        procedure.\n\n    These mechanisms for handling primitives are further described in\n    section 4.1.4.\n  ","4.1.3#p4":"\n    Compound\n    procedures\n    are constructed from parameters,\n    procedure\n    bodies, and environments using the constructor \n    make-procedure:(define (make-procedure parameters body env)\n  (list 'procedure parameters body env))\n\n(define (compound-procedure? p)\n  (tagged-list? p 'procedure))\n\n(define (procedure-parameters p) (cadr p))\n(define (procedure-body p) (caddr p))\n(define (procedure-environment p) (cadddr p)) ","4.1.3#h3":"Operations on Environments","4.1.3#p5":"\n    The evaluator needs operations for\n    \n    manipulating environments.  As explained\n    in section 3.2, an environment is a\n    sequence of frames, where each frame is a table of bindings that associate\n    variables\n    with their corresponding values.  We use the following operations for\n    manipulating environments:\n    (lookup-variable-value var env)\n\t    returns the value that is bound to the symbol\n\t    var in the environment\n\t    env, or signals an error if the variable\n\t    is unbound.\n\t  (extend-environment variables values base-env)\n\t      returns a new environment, consisting of a new frame in which the\n\t      symbols in the list variables are bound\n\t      to the corresponding elements in the list\n\t      values, where the enclosing environment\n\t      is the environment base-env.\n\t    (define-variable! var value env)\n\t      adds to the first frame in the environment\n\t      env a new binding that associates the\n\t      variable var with the value\n\t      value.\n\t    (set-variable-value! var value env)\n\t      changes the binding of the variable var\n\t      in the environment env so that the\n\t      variable is now bound to the value\n\t      value, or signals an error if the\n\t      variable is unbound.\n\t    ","4.1.3#p6":"\n    To implement these operations we\n    \n    represent an environment as a list of\n    frames.  The enclosing environment of an environment is the\n    cdr\n    of the list.  The empty environment is simply the empty list.\n    (define (enclosing-environment env) (cdr env))\n\n(define (first-frame env) (car env))\n\n(define the-empty-environment '()) \n    Each frame of an environment is represented as a pair of lists: a list\n    of the\n    variables\n    bound in that frame and a list of the associated\n    values.(define (make-frame variables values)\n  (cons variables values))\n\n(define (frame-variables frame) (car frame))\n(define (frame-values frame) (cdr frame))\n\n(define (add-binding-to-frame! var val frame)\n  (set-car! frame (cons var (car frame)))\n  (set-cdr! frame (cons val (cdr frame)))) ","4.1.3#footnote-link-1":"1","4.1.3#p7":"\n    To extend an environment by a new frame that associates\n    variables\n    with values, we make a frame consisting of the list of\n    variables\n    and the list of values, and we adjoin this to the environment.  We signal\n    an error if the number of\n    variables\n    does not match the number of values.\n    (define (extend-environment vars vals base-env)\n  (if (= (length vars) (length vals))\n      (cons (make-frame vars vals) base-env)\n      (if (< (length vars) (length vals))\n          (error \"Too many arguments supplied\" vars vals)\n          (error \"Too few arguments supplied\" vars vals)))) ","4.1.3#p8":"\n    To look up a\n    variable\n    in an environment, we scan the list of\n    variables\n    in the first frame.  If we find the desired\n    variable,\n    we return the corresponding element in the list of values. If we do not\n    find the\n    variable\n    in the current frame, we search the enclosing environment, and so on.\n    If we reach the empty environment, we signal an\n    \"unbound\n\tvariable\"  \n    error.\n    (define (lookup-variable-value var env)\n  (define (env-loop env)\n    (define (scan vars vals)\n      (cond ((null? vars)\n             (env-loop (enclosing-environment env)))\n            ((eq? var (car vars))\n             (car vals))\n            (else (scan (cdr vars) (cdr vals)))))\n    (if (eq? env the-empty-environment)\n        (error \"Unbound variable\" var)\n        (let ((frame (first-frame env)))\n          (scan (frame-variables frame)\n                (frame-values frame)))))\n  (env-loop env)) ","4.1.3#p9":"To set a variable\n      to a new value in a specified environment, we scan\n      for the variable, just as in \n      lookup-variable-value,\n      and change the corresponding value when we find it.\n      (define (set-variable-value! var val env)\n  (define (env-loop env)\n    (define (scan vars vals)\n      (cond ((null? vars)\n             (env-loop (enclosing-environment env)))\n            ((eq? var (car vars))\n             (set-car! vals val))\n            (else (scan (cdr vars) (cdr vals)))))\n    (if (eq? env the-empty-environment)\n        (error \"Unbound variable - - SET!\" var)\n        (let ((frame (first-frame env)))\n          (scan (frame-variables frame)\n                (frame-values frame)))))\n  (env-loop env)) ","4.1.3#p10":"\n        To define a variable,\n        we search the first frame for a binding for\n        the variable, and change the binding if it exists\n\t(just as in \n\tset-variable-value!.\n\tIf no such binding exists, we adjoin one to the first frame.\n\t(define (define-variable! var val env)\n  (let ((frame (first-frame env)))\n    (define (scan vars vals)\n      (cond ((null? vars)\n             (add-binding-to-frame! var val frame))\n            ((eq? var (car vars))\n             (set-car! vals val))\n            (else (scan (cdr vars) (cdr vals)))))\n    (scan (frame-variables frame)\n          (frame-values frame)))) ","4.1.3#p11":"\n    The method described here is only one of many plausible ways to represent\n    environments.  Since we used\n    \n    data abstraction to isolate the rest of the\n    evaluator from the detailed choice of representation, we could change the\n    environment representation if we wanted to.  (See\n    exercise 4.11.)  In a\n    production-quality\n    Lisp\n    system, the speed of the evaluator's environment\n    operations—especially that of\n    variable\n    lookup—has a major\n    impact on the performance of the system.  The representation described here,\n    although conceptually simple, is not efficient and would not ordinarily be\n    used in a production system.","4.1.3#footnote-link-2":"2","4.1.3#ex-4.11":"\n    Instead of representing a frame as a pair of lists, we can represent a frame\n    as a list of bindings, where each binding is a symbol-value pair. Rewrite the\n    environment operations to use this alternative representation.            \n  ","4.1.3#ex-4.12":"\n    The\n    proceduresset-variable-value!,\n\tdefine-variable!, and\n\tlookup-variable-value\n    can be expressed in terms of \n    more abstract procedures\n    for traversing the environment structure. \n    \n\tDefine abstractions that capture the common patterns and redefine the\n\tthree procedures in terms of these abstractions.\n      ","4.1.3#ex-4.13":"\n\tScheme allows us to create new bindings for variables by means of\n\tdefine, but provides no way to get rid of\n\tbindings.  Implement for the evaluator a special form\n\tmake-unbound! that removes the binding of a\n\tgiven symbol from the environment in which the\n\tmake-unbound! expression is evaluated. This\n\tproblem is not completely specified.  For example, should we remove only\n\tthe binding in the first frame of the environment?  Complete the\n\tspecification and justify any choices you make.\n\t","4.1.3#ex-4.14":"\n\t    JavaScript's specification requires an implementation to\n\t    signal a runtime error upon an attempt to access the\n\t    value of a name before its declaration is evaluated (see\n\t    the end of section 3.2.4).\n\t    To achieve this behavior in the evaluator,\n\t    \n\t    change lookup_symbol_value\n\t    to signal an error if the value it finds is\n\t    \"*unassigned*\".\n\t  \n\t    Similarly, we must not assign a new value to a variable if\n\t    we have not evaluated its let\n\t    declaration yet. Change the evaluation of assignment\n\t    such that assignment to a variable declared with\n\t    let signals an error\n\t    in this case.\n\t  ","4.1.3#footnote-1":"Frames are not really a data\n    \n\tabstraction in the following code:\n      set-variable-value!\n\tand \n\tdefine-variable!\n\tuse set-car!\n\tto directly modify the values in a frame.\n      \n    The purpose of the frame\n    procedures\n    is to make the environment-manipulation\n    procedures\n    easy to read.","4.1.3#footnote-2":"The drawback of this representation (as\n    well as the variant in\n    exercise 4.11) is that the\n    evaluator may have to search through many frames in order to find the binding\n    for a given variable.\n    (Such an approach is referred to as\n    deep binding.) One way to avoid\n    this inefficiency is to make use of a strategy called\n\tlexical addressing, which will be discussed in\n\tsection 5.5.6.\n  ","4.1.4":"4.1.4  \n    Running the Evaluator as a Program","4.1.4#p1":"\n    Given the evaluator, we have in our hands a description\n    (expressed in Lisp)\n    of the process by which\n    Lisp expressions\n    are evaluated.  One advantage of expressing the evaluator as a program is\n    that we can run the program.  This gives us, running within\n    Lisp,\n    a working model of how\n    Lisp\n    itself evaluates expressions.  This can serve as a framework for\n    experimenting with evaluation rules, as we shall do later in this chapter.\n  ","4.1.4#p2":"\n    Our evaluator program reduces expressions ultimately to the application of\n    \n    primitive\n    procedures.\n    Therefore, all that we need to run the evaluator is to create a mechanism\n    that calls on the underlying\n    Lisp\n    system to model the application of primitive\n    procedures.","4.1.4#p3":"\n    There must be a binding for each primitive\n    procedure\n    name and operator, so that when\n    eval\n    evaluates the\n    operator\n    of an application of a primitive, it will find an\n    object to pass to apply.  We thus set up a \n    \n    global environment that associates unique objects with the names of the\n    primitive\n    procedures\n    that can appear in the expressions we will be evaluating.  \n    \n\tThe global environment also includes bindings for the symbols \n\ttrue and false,\n      \n    so that they can be used as constants in expressions to be evaluated.\n    (define (setup-environment)\n  (let ((initial-env\n         (extend-environment (primitive-procedure-names)\n                             (primitive-procedure-objects)\n                             the-empty-environment)))\n    (define-variable! 'true true initial-env)\n    (define-variable! 'false false initial-env)\n    initial-env)) (define the-global-environment (setup-environment)) ","4.1.4#p4":"\n\tIt does not matter how we represent the primitive procedure objects, so\n\tlong as apply can identify and apply them\n\tby using the procedures primitive-procedure?\n\tand apply-primitive-procedure. We have\n\tchosen to represent a primitive procedure as a list beginning with the\n\tsymbol primitive and containing a procedure\n\tin the underlying Lisp that implements that primitive.\n\t(define (primitive-procedure? proc)\n  (tagged-list? proc 'primitive))\n\n(define (primitive-implementation proc) (cadr proc)) ","4.1.4#p5":"Setup-environment\n    will get the primitive names and implementation \n    procedures\n    from a list:\n(define primitive-procedures\n  (list (list 'car car)\n        (list 'cdr cdr)\n        (list 'cons cons)\n        (list 'null? null?)\n        (list 'display display)\n        (list 'read read)\n        (list '+ +)\n        (list '- -)\n        (list '* *)\n;;      more primitives\n        ))\n\n(define (primitive-procedure-names)\n  (map car\n       primitive-procedures))\n\n(define (primitive-procedure-objects)\n  (map (lambda (proc) (list 'primitive (cadr proc)))\n       primitive-procedures))\n      ","4.1.4#footnote-link-1":"1","4.1.4#p6":"\n    To apply a \n    primitive procedure,\n    we simply apply the implementation\n    procedure\n    to the arguments, using the underlying\n    Lisp\n    system:(define (apply-primitive-procedure proc args)\n  (apply-in-underlying-scheme\n   (primitive-implementation proc) args)) ","4.1.4#footnote-link-2":"2","4.1.4#p7":"\n\tFor convenience in running the metacircular evaluator, we provide a\n\tdriver loop that models the read-eval-print loop of the\n\tunderlying Lisp system.  It prints a \n\tprompt, reads an input expression, evaluates this expression in\n\tthe global environment, and prints the result.  We precede each printed\n\tresult by an output prompt so as to distinguish the value of\n\tthe expression from other output that may be printed.(define input-prompt \";;; M-Eval input:\\n\")\n(define output-prompt \";;; M-Eval value:\\n\")\n\n(define (driver-loop)\n  (prompt-for-input input-prompt)\n  (let ((input (read)))\n    (if (null? input)\n      'EVALUATOR-TERMINATED\n      (let ((output (eval input the-global-environment)))\n        (announce-output output-prompt)\n        (user-print output)\n        (driver-loop)))))\n\n(define (prompt-for-input string)\n  (newline) (display string))\n\n(define (announce-output string)\n  (newline) (display string)) \n    The function\n    prompt returns\n    null when the user cancels the\n    input. We use a special printing\n    procedure user-print,\n    to avoid printing the environment part of a compound\n    procedure,\n    which may be a very long list (or may even contain cycles).\n    (define (user-print object)\n  (if (compound-procedure? object)\n      (display (list 'compound-procedure\n                     (procedure-parameters object)\n                     (procedure-body object)\n                     '<-procedure-env->))\n      (display object))) ","4.1.4#footnote-link-3":"3","4.1.4#p8":"\n    Now all we need to do to run the evaluator is to initialize the global\n    environment and start the driver loop.  Here is a sample interaction:\n    (define the-global-environment (setup-environment))\n(driver-loop) (define (append x y)\n   (if (null? x)\n       y\n       (cons (car x)\n         (append (cdr x) y)))) (append '(a b c) '(d e f)) ","4.1.4#ex-4.15":"\n    Eva Lu Ator and Louis Reasoner are each experimenting with the\n    metacircular evaluator.  Eva types in the definition of\n    map, and runs some test programs that use it.\n    They work fine.  Louis, in contrast, has installed the system version of\n    map as a primitive for the metacircular\n    evaluator.  When he tries it, things go terribly wrong.  Explain why\n    Louis's map fails even though\n    Eva's works.\n    ","4.1.4#footnote-1":"Any\n    procedure\n    defined in the underlying\n    Lisp\n    can be used as a primitive for the metacircular evaluator.  The name of a\n    primitive installed in the evaluator need not be the same as the name of its\n    implementation in the underlying\n    Lisp;\n    the names are the same here because the metacircular evaluator implements\n    Scheme    \n    itself.\n    Thus, for example, we could put \n    (list 'first car) \n    or \n    (list 'square (lambda (x) (* x x)))\n    in the list of \n    primitive-procedures.","4.1.4#footnote-2":"Apply-in-underlying-scheme is the\n\tapply procedure we have used in earlier\n\tchapters.  The metacircular evaluator's\n\tapply procedure\n\t(section 4.1.1) models the\tworking\n\tof this primitive.  Having two different things called\n\tapply leads to a technical problem in\n\trunning the metacircular evaluator, because defining the metacircular\n\tevaluator's apply will mask the\n\tdefinition of the primitive.  One way around this is to\trename the\n\tmetacircular apply to avoid conflict with\n\tthe name of the primitive procedure. We have assumed instead that we\n\thave saved a reference to the underlying\n\tapply by doing\n\t(define apply-in-underlying-scheme apply) \n\tbefore defining the metacircular apply.\n\tThis allows us to access the original version of\n\tapply under a different name.\n      ","4.1.4#footnote-3":"The\n\tprimitive procedure\n\tread waits for input from the user, and\n\treturns the next complete expression that is typed. For example, if the\n\tuser types (+ 23 x),\n\tread returns a three-element list\n\tcontaining the symbol +, the number 23, and\n\tthe symbol x.\n\t\n\tIf the user types 'x,\n\tread returns a two-element list containing\n\tthe symbol quote and the symbol\n\tx.","4.1.5":"4.1.5  \n    Data as Programs","4.1.5#p1":"\n    In thinking about a\n    Lisp\n    program that evaluates\n    Lisp\n    expressions, an analogy might be helpful.  One operational view of the\n    meaning of a program is that a \n    \n    program is a description of an abstract (perhaps infinitely large) machine.\n    For example, consider the familiar program to compute factorials:\n    (define (factorial n)\n   (if (= n 1)\n       1\n       (* (factorial (- n 1)) n))) \n    We may regard this program as the description of a\n    \n    machine containing\n    parts that decrement, multiply, and test for equality, together with a\n    two-position switch and another factorial machine. (The factorial\n    machine is infinite because it contains another factorial machine\n    within it.)  Figure 4.3 is a flow\n    diagram for the factorial machine, showing how the parts are wired together.\n  ","4.1.5#fig-4.3":"","4.1.5#p2":"\n    In a similar way, we can regard the evaluator as a very special\n    \n    machine that takes as input a description of a machine.  Given this\n    input, the evaluator configures itself to emulate the machine\n    described.  For example, if we feed our evaluator the definition of\n    factorial, as shown in\n    figure ,\n    the evaluator will be able to compute factorials.\n    ","4.1.5#fig-":"","4.1.5#p3":"\n    From this perspective, our evaluator is seen to be a\n    universal machine.\n    It mimics other machines when these are described as \n    Lisp \n    programs.\n    This is striking. Try to imagine an analogous evaluator for electrical\n    circuits.  This would be a circuit that takes as input a signal encoding the\n    plans for some other circuit, such as a filter.  Given this input, the\n    circuit evaluator would then behave like a filter with the same description.\n    Such a universal electrical circuit is almost unimaginably complex.  It is\n    remarkable that the program evaluator is a rather simple\n    program.","4.1.5#footnote-link-1":"1","4.1.5#footnote-link-2":"2","4.1.5#p4":"\n    Another striking aspect of the evaluator is that it acts as a bridge between\n    the data objects that are manipulated by our programming language and the\n    programming language itself.  Imagine that the evaluator program\n    (implemented in Lisp)\n    is running, and that a user is typing\n    expressions\n    to the evaluator and\n    observing the results. From the perspective of the user, an input\n    expression\n    such as \n    (* x x)\n    is\n    an expression\n    in the programming language, which the evaluator should\n    execute.\n    \n\tFrom the perspective of the evaluator, however, the expression is simply\n\ta list (in this case, a list of three symbols:\n\t*, x, \n\tand x) that is to be manipulated according\n\tto a well-defined set of rules.\n      ","4.1.5#p5":"\n\tThat the user's programs are the evaluator's data need not\n\tbe a source of confusion.  In fact, it is sometimes convenient to ignore\n\tthis distinction, and to give the user the ability to explicitly\n\tevaluate a data object as a Lisp expression, by making\n\teval available for use in programs.  Many\n\tLisp dialects provide a \n\t\n\tprimitive eval procedure that takes as\n\targuments an expression and an environment and evaluates the expression\n\trelative to the environment.\n\tThus,\n\t(eval '(* 5 5) user-initial-environment) \n\tand\n\t(eval (cons '* (list 5 5)) user-initial-environment) \n\twill both return 25.","4.1.5#footnote-link-3":"3","4.1.5#footnote-link-4":"4","4.1.5#ex-4.16":"\n    Given a one-argument\n    procedurep\n    and an object a,\n    p\n    is said to \"halt\" on\n    a if evaluating the expression\n    (p a)\n    returns a value (as opposed to terminating with an error message or running\n    forever).\n    \n    Show that it is impossible to write a\n    procedurehalts?\n    that correctly determines whether\n    p\n    halts on\n    a for any\n    procedurep\n    and object a.\n    Use the following reasoning: If you had such a\n    procedurehalts?,\n    you could implement the following program:\n    (define (run-forever) (run-forever))\n\n(define (try p)\n   (if (halts? p p)\n      (run-forever)\n      'halted))\n    Now consider evaluating the expression\n    (try try)\n    and show that any possible outcome (either halting or running forever)\n    violates the intended behavior of\n    halts?.","4.1.5#footnote-link-5":"5","4.1.5#footnote-1":"The fact that the machines are described in \n    Lisp \n    is inessential.  If we give our evaluator a \n    Lisp \n    program that behaves as an evaluator for some other language, say C, the \n    Lisp \n    evaluator will emulate the C evaluator, which in turn can emulate any\n    machine described as a C program.  Similarly, writing a \n    Lisp \n    evaluator in C produces a C program that can execute any \n    Lisp \n    program.  The deep idea here is that any evaluator can emulate any other.\n    Thus, the notion of \"what can in principle be computed\"\n    (ignoring practicalities of time and memory required) is independent of the\n    language or the computer, and instead reflects an underlying notion of \n    computability.  This was first demonstrated in a clear way by \n    \n    Alan M. Turing (1912–1954), whose 1936 paper laid the foundations\n    for theoretical \n    \n    computer science.  In the paper, Turing presented a simple computational\n    model—now known as a \n    Turing machine—and argued that any \"effective\n    process\" can be formulated as a program for such a machine.  (This\n    argument is known as the \n    Church–Turing thesis.) Turing then implemented a universal machine,\n    i.e., a Turing machine that behaves as an evaluator for Turing-machine\n    programs.  He used this framework to demonstrate that there are well-posed\n    problems that cannot be computed by Turing machines (see\n    exercise 4.16), and so by implication\n    cannot be formulated as \"effective processes.\"  Turing went on\n    to make fundamental contributions to practical computer science as well.\n    For example, he invented the idea of \n    \n    structuring programs using general-purpose subroutines.  See \n    Hodges 1983 for a biography of Turing.","4.1.5#footnote-2":"Some people find it counterintuitive that an evaluator,\n    which is implemented by a relatively simple\n    procedure,\n    can emulate programs that are more complex than the evaluator itself.  The\n    existence of a universal evaluator machine is a deep and wonderful property\n    of computation.\n    Recursion theory, a branch of mathematical logic, is concerned with\n    logical limits of computation.  \n    \n    Douglas Hofstadter's beautiful book Gödel, Escher,\n    Bach (1979) explores some of these ideas.","4.1.5#footnote-3":"Warning:\n\t\n\tThis eval primitive is not\n\tidentical to the eval procedure we\n\timplemented in section 4.1.1,\n\tbecause it uses actual\tScheme environments rather than the\n\tsample environment structures we built in\n\tsection 4.1.3.  These actual\n\tenvironments cannot be manipulated by the user as ordinary lists; they\n\tmust be accessed via eval or other special\n\toperations.\n\tSimilarly, the\n\tapply primitive we saw\n\tearlier is not identical to the metacircular\n\tapply, because it uses actual Scheme\n\tprocedures rather than the procedure objects we constructed in\n\tsections 4.1.3\n\tand 4.1.4.","4.1.5#footnote-4":"The MIT\n\t\n\timplementation of Scheme includes eval, as\n\twell as a symbol user-initial-environment\n\tthat is bound to the initial environment in which the user's input\n\texpressions are evaluated.\n\t","4.1.5#footnote-5":"Although\n    we stipulated that\n    halts?\n    is given a\n    procedure\n    object, notice that this reasoning still applies even if\n    halts?\n    can gain access to the\n    procedure's\n    text and its environment.\n    \n    This is Turing's celebrated\n    Halting Theorem, which gave the\n    first clear example of a\n    noncomputable problem, i.e., a well-posed\n    task that cannot be carried out as a computational\n    procedure.","4.1.6":"4.1.6  \n    Internal","4.1.6#p1":"\n\tOur environment model of evaluation and our metacircular evaluator\n\texecute definitions in sequence, extending the environment frame one\n\tdefinition at a time. This is particularly convenient for interactive\n\tprogram development, in which the programmer needs to freely mix the\n\tapplication of procedures definition procedures. However, if we think\n\tcarefully about the internal definitions used to implement block\n\tstructure (introduced in section 1.1.8),\n\twe will find that name-by-name extension of the environment may not be\n\tthe best way to define local variables.\n      ","4.1.6#p2":"\n\tConsider a procedure with internal definitions,\tsuch as\n\t(define (f x)\n   (define (even? n)\n      (if (= n 0)\n          true\n          (odd? (- n 1))))\n   (define (odd? n)\n      (if (= n 0)\n          false\n          (even? (- n 1))))\n   $\\langle$ rest of body of$\\rangle$ f)","4.1.6#p3":"\n\tOur intention here is that the name odd?\n\tin the body of the procedure even?\tshould\n\trefer to the procedure odd? that is \n\tdefined after even?.\n\tThe scope of the name odd?\tis the entire\n\tbody of f, not just the portion of the body\n\tof f starting at the point where the\n\tdefine for odd?\n\toccurs. Indeed, when we consider that odd?\n\tis itself defined in terms of\n\teven?—so that\n\teven? and odd?\n\tare mutually recursive procedures—we see that the only\n\tsatisfactory interpretation of the two\n\tdefines is to regard them as if the names\n\teven? and odd?\n\twere being added to the environment simultaneously. More generally, in\n\tblock structure, the scope of a local name is the entire procedure\n\tbody in which the define is evaluated.\n      ","4.1.6#p4":"\n\tAs it happens, our interpreter will evaluate calls to\n\tf correctly, but for an\n\t\"accidental\" reason:    Since the definitions\n\tof the internal procedures come first, no calls to these procedures\n\twill be evaluated until all of them have been defined. Hence,\n\todd? will have been defined by the time\n\teven? is executed.  In fact, our sequential\n\tevaluation mechanism will give the same result as a mechanism that\n\tdirectly implements simultaneous definition for any\n\tprocedure in which the \n\t\n\tinternal definitions come first in a body and evaluation of the value\n\texpressions for the defined variables doesn't actually use any of\n\tthe defined variables. (For an example of a procedure that doesn't\n\tobey these restrictions, so that sequential definition isn't\n\tequivalent to simultaneous definition, see\n\texercise 4.20.)","4.1.6#footnote-link-1":"1","4.1.6#p5":"\n\tThere is, however, a simple way to treat definitions\n\tso that internally defined names have truly simultaneous\n\tscope—just create all local variables that will be in the\n\tcurrent environment before evaluating any of the value expressions.\n\tOne way to do this is by a syntax transformation on\n\tlambda expressions. Before evaluating the\n\tbody of a lambda expression, we \n\t\"scan out\" and eliminate all the internal definitions in\n\tthe body.  The internally defined variables will be created with a\n\tlet and then set to their values by\n\tassignment. For example, the procedure\n\t\n(lambda vars\n   (define u e1)\n   (define v e2)\n   e3)\n\t  \n\twould be transformed into\n\t\n(lambda vars\n   (let ((u '*unassigned*)\n         (v '*unassigned*))\n      (set! u e1)\n      (set! v e2)\n      e3))\n\t  \n\twhere *unassigned* is a special symbol that\n\tcauses looking up a variable to signal an error if an attempt is made to\n\tuse the\tvalue of the not-yet-assigned variable.\n      ","4.1.6#p6":"\n\tAn alternative strategy for scanning out internal definitions is shown\n\tin exercise 4.19. Unlike the\n\ttransformation shown above, this enforces the restriction that the\n\tdefined variables' values can be evaluated without using any of the\n\tvariables' values.","4.1.6#footnote-link-2":"2","4.1.6#ex-4.17":"\n\tIn this exercise we implement the method just described for interpreting\n\tinternal definitions.\n\tWe assume that the evaluator supports let\n\t(see exercise 4.6).\n\t\n\t    Change\n\t    lookup-variable-value\n\t    (section 4.1.3)\n\t    to signal an error if the value it finds is the symbol\n\t    *unassigned*.\n\t  \n\t    Write a procedure\n\t    scan-out-defines\n\t    that takes a procedure body and returns an equivalent one that has no\n\t    internal definitions, by making the transformation described above.\n\t  \n\t    Install scan-out-defines in the\n\t    interpreter, either in  make-procedure\n\t    or in procedure-body (see\n\t    section 4.1.3).\n\t    Which place is better? Why?\n\t  ","4.1.6#ex-4.18":"\n\tDraw diagrams of the environment in effect when evaluating the\n\texpression e3 in the procedure in the text,\n\tcomparing how this will be structured when definitions are interpreted\n\tsequentially with how it will be structured if definitions are scanned\n\tout as described. Why is there an extra frame in the transformed program?\n\tExplain why this difference in environment structure can never make a\n\tdifference in the behavior of a correct program.  Design a way to make\n\tthe interpreter implement the \"simultaneous\" scope rule for\n\tinternal definitions without constructing the extra frame.\n  ","4.1.6#ex-4.19":"\n\tConsider an alternative strategy for scanning out definitions that\n\ttranslates the example in the text to\n\t\n(lambda vars\n   (let ((u '*unassigned*)\n         (v '*unassigned*))\n      (let ((a e1)\n            (b e2))\n         (set! u a)\n   (set! v b))\n   e3))\n\t  \n\tHere a and b\n\tare meant to represent new variable names, created by the interpreter,\n\tthat do\tnot appear in the user's program. Consider the\n\tsolve procedure from\n\tsection 3.5.4:\n\t(define (solve f y0 dt)\n(define y (integral (delay dy) y0 dt))\n(define dy (stream-map f y))\n      y) \n\tWill this procedure work if internal definitions are scanned out as\n\tshown in this exercise? What if they are scanned out as shown in the\n\ttext?  Explain.\n      ","4.1.6#ex-4.20":"\n\tBen Bitdiddle, Alyssa P. Hacker, and Eva Lu Ator are arguing about\n\tthe desired result of evaluating the expression\n\t(let ((a 1))\n   (define (f x)\n      (define b (+ a x))\n      (define a 5)\n      (+ a b))\n   (f 10)) \n\tBen asserts that the result should be obtained using the sequential rule\n\tfor define: b\n\tis defined to be 11, then a is defined to\n\tbe 5, so the result is 16.  Alyssa objects that mutual recursion requires\n\tthe simultaneous scope rule for internal procedure definitions, and that\n\tit is unreasonable to treat procedure names differently from other names.\n\tThus, she argues for the mechanism implemented in\n\texercise 4.17. This would lead to\n\ta being unassigned at the time that the\n\tvalue for b is to be computed.  Hence, in\n\tAlyssa's view the procedure should produce an error.  Eva has a\n\tthird opinion.  She says that if the definitions of\n\ta and b are\n\ttruly meant to be simultaneous, then the value 5 for\n\ta should be used in evaluating\n\tb.  Hence, in Eva's view\n\ta should be 5,\n\tb should be 15, and the result should be 20.\n\tWhich (if any) of these\tviewpoints do you support?  Can you devise a way\n\tto implement internal definitions so that they behave as Eva\n\tprefers?","4.1.6#footnote-link-3":"3","4.1.6#ex-4.21":"  \n\tBecause internal definitions look sequential but are actually\n\tsimultaneous, some people prefer to avoid them entirely, and use the\n\tspecial form\n\tletrec instead.  \n\tLetrec looks like\n\tlet, so it is not surprising that the\n\tvariables it binds are bound simultaneously and have the same scope as\n\teach other.  The sample\tprocedure f above\n\tcan be written without internal definitions, but with exactly the same\n\tmeaning, as\n\t(define (f x)\n   (letrec ((even?\n               (lambda (n)\n                  (if (= n 0)\n                      true\n                      (odd? (- n 1)))))\n            (odd?\n               (lambda (n)\n                  (if (= n 0)\n                      false\n                      (even? (- n 1))))))\n      $\\langle$rest of body of$\\rangle$ f))Letrec expressions, which have the form\n\t(letrec ((var$_{1}$ exp$_{1}$) $\\ldots$ (var$_{n}$ exp$_{n}$))\n    body)\n\tare a variation on let in which the\n\texpressions exp$_{k}$ that provide the\n\tinitial values for the variables var$_{k}$\n\tare evaluated in an environment that includes all the\n\tletrec bindings.  This permits recursion in\n\tthe bindings, such as the mutual recursion of\n\teven? and odd?\n\tin the example above, or the evaluation of 10\n\t\n\tfactorial with\n\t(letrec ((fact\n            (lambda (n)\n               (if (= n 1)\n                   1\n                   (* n (fact (- n 1)))))))\n        (fact 10)) \n\t    Implement letrec as a derived\n\t    expression, by transforming a letrec\n\t    expression into a let expression as\n\t    shown in  the text above or in\n\t    exercise 4.19. That is,\n\t    the letrec variables should be created\n\t    with a let and then be assigned their\n\t    values with set!.\n\t  \n\t    Louis Reasoner is confused by all this fuss about internal\n\t    definitions.  The way he sees it, if you don't like to use\n\t    define inside a procedure, you can just\n\t    use let. Illustrate what is loose about\n\t    his reasoning by drawing an environment diagram that shows the\n\t    environment in which the\n\t    $\\langle$rest of body of$\\rangle$f\n\t    is evaluated during\n\t    evaluation of the expression (f 5), with\n\t    f defined as in this exercise. Draw an\n\t    environment diagram for the same evaluation, but with\n\t    let in place of\n\t    letrec in the definition of\n\t    f.\n\t  ","4.1.6#ex-4.22":"\n\tAmazingly, Louis's intuition in\n\texercise 4.21 is correct. It is indeed\n\t\n\tpossible to specify recursive procedures without using\n\tletrec (or even\n\tdefine), although the method for\n\taccomplishing this is much more subtle than Louis imagined.\n\tThe following expression computes 10\n\t\n\tfactorial by applying a recursive\n\tfactorial procedure:((lambda (n)\n    ((lambda (fact)\n        (fact fact n))\n     (lambda (ft k)\n        (if (= k 1)\n            1\n            (* k (ft ft (- k 1)))))))\n 10) \n\t    Check (by evaluating the expression) that this really does compute\n\t    factorials.  Devise an analogous expression for computing Fibonacci\n\t    numbers.\n\t  \n\t    Consider the following procedure, which includes mutually recursive\n\t    internal definitions:\n\t    (define (f x)\n   (define (even? n)\n      (if (= n 0)\n          true\n      (odd? (- n 1))))\n   (define (odd? n)\n      (if (= n 0)\n          false\n          (even? (- n 1))))\n   (even? x)) \n\t    Fill in the missing expressions to complete an alternative definition\n\t    of f, which uses neither internal\n\t    definitions nor letrec:\n\t    (define (f x)\n   ((lambda (even? odd?)\n  (even? even? odd? x))\n    (lambda (ev? od? n)\n  (if (= n 0) true (od? ?? ?? ??)))\n    (lambda (ev? od? n)\n  (if (= n 0) false (ev? ?? ?? ??)))))","4.1.6#footnote-link-4":"4","4.1.6#footnote-1":"Wanting\n\tprograms to not depend on this evaluation mechanism is the reason for the\n\t\"management is not responsible\" remark in\n\tfootnote 4 of chapter 1.\n\tBy insisting that internal definitions come first and do not use each\n\tother while the definitions are being evaluated, the IEEE standard\n\tfor Scheme leaves implementors some choice in the mechanism used to\n\tevaluate these definitions.  The choice of one evaluation rule rather\n\tthan another here may seem like a small issue, affecting only the\n\tinterpretation of \"badly formed\" programs. However, we will\n\tsee in section 5.5.6 that moving\n\tto a model of simultaneous scoping for internal definitions avoids some\n\tnasty difficulties that would otherwise arise in implementing a compiler.\n\t","4.1.6#footnote-2":"The IEEE standard for Scheme allows\n\tfor different implementation strategies by specifying that it is up to\n\tthe programmer to obey this restriction, not up to the implementation to\n\tenforce\tit.  Some Scheme implementations, including\n\t\n\tMIT Scheme, use the transformation shown above.  Thus, some programs that\n\tdon't obey this restriction will in fact run in such\n\timplementations.","4.1.6#footnote-3":"The MIT implementors of Scheme support Alyssa on the\n\tfollowing grounds: Eva is in principle correct—the definitions\n\tshould be regarded as simultaneous.  But it seems difficult to implement\n\ta general, efficient mechanism that does what Eva requires. In the\n\tabsence of such a mechanism, it is better to generate an error in the\n\tdifficult cases of simultaneous definitions (Alyssa's notion) than\n\tto produce an incorrect answer (as Ben would have it).","4.1.6#footnote-4":"This example illustrates a programming\n\ttrick for formulating recursive procedures without using\n\tdefine. The\n\tmost general trick of this sort is the\n\t$Y$operator, which can be used to give a \"pure\n\t$\\lambda$-calculus\" implementation of\n\trecursion.  (See\n\tStoy 1977 for details on the lambda\n\tcalculus, and Gabriel 1988 for an exposition of the\n\t$Y$ operator in Scheme.)","4.1.7":"4.1.7  \n    Separating Syntactic Analysis from Execution","4.1.7#p1":"\n    The evaluator implemented above is simple, but it is very\n    \n    inefficient, because the syntactic analysis of\n    \n\texpressions\n      \n    is interleaved\n    with their execution.  Thus if a program is executed many times, its\n    syntax is analyzed many times.  Consider, for example, evaluating\n    (factorial 4) using the following definition of\n    factorial:\n    (define (factorial n)\n  (if (= n 1)\n      1\n      (* (factorial (- n 1)) n))) ","4.1.7#p2":"\n    Each time factorial is called, the evaluator\n    must determine that the body is\n    an if\n    expression and extract the predicate. Only then can it evaluate the\n    predicate and dispatch on its value.  Each time it evaluates the expression\n    (* (factorial (- n 1)) n),\n      \n    or the subexpressions\n    (factorial (- n 1))\n    and \n    (- n 1),\n    the evaluator must perform the case analysis in\n    eval\n    to determine that the expression is an application, and must extract\n    its operator and operands.\n    This analysis is expensive.\n    Performing it repeatedly is wasteful.\n  ","4.1.7#p3":"\n    We can transform the evaluator to be significantly more efficient by\n    arranging things so that syntactic analysis is performed only\n    once. We split\n    eval,\n    which takes\n    an expression\n    and an environment, into two parts.  The\n    procedureanalyze takes only the\n    expression.\n    It performs the syntactic\n    analysis and returns a new\n    procedure, the \n    execution\n    procedure, that\n    encapsulates the work to be done in executing the analyzed\n    expression.\n    The execution\n    procedure\n    takes an environment as its\n    argument and completes the evaluation.  This saves work because\n    analyze will be called only once on\n    an expression,\n    while the execution\n    procedure\n    may be called many times.\n  ","4.1.7#footnote-link-1":"1","4.1.7#p4":"\n    With the separation into analysis and execution,\n    eval\n    now becomes\n    (define (eval exp env)\n  ((analyze exp) env)) ","4.1.7#p5":"\n    The result of calling analyze is the execution\n    procedure\n    to be applied to the environment.  The analyzeprocedure\n    is the same case analysis as performed by the original\n    eval\n    of section 4.1.1, except that the\n    procedures\n    to which we dispatch perform only analysis, not full evaluation:\n    (define (analyze exp)\n      (cond ((self-evaluating? exp) \n            (analyze-self-evaluating exp))\n            ((quoted? exp) (analyze-quoted exp))\n            ((variable? exp) (analyze-variable exp))\n            ((assignment? exp) (analyze-assignment exp))\n            ((definition? exp) (analyze-definition exp))\n            ((if? exp) (analyze-if exp))\n            ((lambda? exp) (analyze-lambda exp))\n            ((begin? exp) (analyze-sequence (begin-actions exp)))\n            ((cond? exp) (analyze (cond->if exp)))\n            ((application? exp) (analyze-application exp))\n            (else\n            (error \"Unknown expression type - - ANALYZE\" exp)))) ","4.1.7#p6":"\n    Here is the simplest syntactic analysis\n    \n\tprocedure,\n\twhich handles self-evaluating expressions.\n      \n    It returns an execution\n    procedure\n    that ignores its environment argument and just returns the\n    expression:(define (analyze-self-evaluating exp)\n  (lambda (env) exp)) ","4.1.7#p7":"\n\tFor a quoted expression, we can gain a little efficiency by\n\textracting the text of the quotation only once, in the analysis phase,\n\trather than in the execution phase.\n\t(define (analyze-quoted exp)\n  (let ((qval (text-of-quotation exp)))\n    (lambda (env) qval))) ","4.1.7#p8":"\n    Looking up\n    \n\ta variable value\t    \n      \n    must still be done in the execution phase, since this depends upon knowing\n    the environment.(define (analyze-variable exp)\n  (lambda (env) (lookup-variable-value exp env))) ","4.1.7#footnote-link-2":"2","4.1.7#p9":"Analyze-assignment\n    must defer actually setting the variable until the execution, when the\n    environment has been supplied.  However, the fact that the\n    assignment-value expression \n      \n    can be analyzed (recursively) during analysis is a major gain in\n    efficiency, because the\n    assignment-value expression\n      \n    will now be analyzed only once.  The same holds true for\n    definitions.(define (analyze-assignment exp)\n  (let ((var (assignment-variable exp))\n        (vproc (analyze (assignment-value exp))))\n    (lambda (env)\n      (set-variable-value! var (vproc env) env)\n      'ok)))\n\n(define (analyze-definition exp)\n  (let ((var (definition-variable exp))\n        (vproc (analyze (definition-value exp))))\n    (lambda (env)\n      (define-variable! var (vproc env) env)\n      'ok))) ","4.1.7#p10":"\n\tFor if expressions,\n\twe extract and analyze the predicate, consequent, and alternative at\n\tanalysis time.\n\t(define (analyze-if exp)\n  (let ((pproc (analyze (if-predicate exp)))\n        (cproc (analyze (if-consequent exp)))\n        (aproc (analyze (if-alternative exp))))\n    (lambda (env)\n      (if (true? (pproc env))\n          (cproc env)\n          (aproc env)))))","4.1.7#p11":"\n\tAnalyzing a\n\tlambda\n\texpression also achieves a major gain in efficiency: We analyze the\n\tlambda\n\tbody only once, even though procedures\n\tresulting from evaluation of the\n\tlambda\n\tmay be applied many times.\n\t(define (analyze-lambda exp)\n  (let ((vars (lambda-parameters exp))\n        (bproc (analyze-sequence (lambda-body exp))))\n  (lambda (env) (make-procedure vars bproc env))))","4.1.7#p12":"\n\tAnalysis of a sequence of\n\texpressions (as in a\n\tbegin or the body\n\tof a lambda expression)\n\tis more \n\tinvolved. Each\n\t    expression\n\tin the sequence is analyzed, yielding an execution\n\tprocedure. These execution\n\tprocedures are combined to produce an execution\n\tprocedure\n\tthat takes an environment as argument and sequentially\n\tcalls each individual execution\n\tprocedure with the environment as argument.\n\t(define (analyze-sequence exps)\n  (define (sequentially proc1 proc2)\n    (lambda (env) (proc1 env) (proc2 env)))\n  (define (loop first-proc rest-procs)\n    (if (null? rest-procs)\n        first-proc\n        (loop (sequentially first-proc (car rest-procs))\n              (cdr rest-procs))))\n  (let ((procs (map analyze exps)))\n    (if (null? procs)\n        (error \"Empty sequence - - ANALYZE\"))\n        (loop (car procs) (cdr procs))))","4.1.7#footnote-link-3":"3","4.1.7#p13":"\n\tTo analyze an application, we analyze the\n\toperator and operands\n\tand construct an execution\n\tprocedure\n\tthat calls the\n\toperator execution function\n\t(to obtain the actual\n\tprocedure\n\tto be applied) and the\n\toperand\n\texecution\n\tprocedures\n\t(to obtain the actual arguments).  We then pass these to\n\texecute-application,\n\twhich is the analog of apply in\n\tsection 4.1.1.\n\tExecute-application\n\tdiffers from apply in that the\n\tprocedure\n\tbody for a compound\n\tprocedure\n\thas already been analyzed, so there is no need to do further analysis.\n\tInstead, we just call the execution\n\tprocedure\n\tfor the body on the extended environment.\n      (define (analyze-application exp)\n  (let ((fproc (analyze (operator exp)))\n        (aprocs (map analyze (operands exp))))\n    (lambda (env)\n      (execute-application (fproc env)\n                           (map (lambda (aproc) (aproc env))\n                                aprocs)))))\n\n(define (execute-application proc args)\n  (cond ((primitive-procedure? proc)\n         (apply-primitive-procedure proc args))\n        ((compound-procedure? proc)\n         ((procedure-body proc)\n          (extend-environment (procedure-parameters proc)\n                              args\n                              (procedure-environment proc))))\n        (else\n         (error\n          \"Unknown procedure type - - EXECUTE-APPLICATION\"\n          proc))))","4.1.7#p14":"\n    Our new evaluator uses the same data structures, syntax\n    procedures,\n    and\n    run-time support procedures\n    as in sections 4.1.2,\n     4.1.3,\n    and 4.1.4.\n  ","4.1.7#ex-4.23":"\n\tExtend the evaluator in this section to support the special form\n\tlet. (See\n\texercise 4.6.)\n\t","4.1.7#ex-4.24":"\n    Alyssa P. Hacker doesn't understand why\n    analyze-sequence\n    needs to be so complicated.  All the other analysis\n    procedures\n    are straightforward transformations of the corresponding evaluation\n    procedures\n    (or\n    eval\n    clauses) in\n    section 4.1.1.\n    She expected\n    analyze-sequence\n    to look like this:\n    (define (analyze-sequence exps)\n  (define (execute-sequence procs env)\n      (cond ((null? (cdr procs)) ((car procs) env))\n            (else ((car procs) env)\n                  (execute-sequence (cdr procs) env))))\n  (let ((procs (map analyze exps)))\n    (if (null? procs)\n        (error \"Empty sequence - - ANALYZE\"))\n        (lambda (env) (execute-sequence procs env)))) \n    Eva Lu Ator explains to Alyssa that the version in the text does more of the\n    work of evaluating a sequence at analysis time. Alyssa's \n    sequence-execution procedure,\n      \n    rather than having the calls to the individual execution\n    procedures\n    built in, loops through the\n    procedures\n    in order to call them: In effect, although the individual\n    expressions\n    in the sequence have been analyzed, the sequence itself has not been.\n    \n    Compare the two versions of\n    analyze-sequence.\n      \n    For example, consider the common case (typical of\n    procedure\n    bodies) where the sequence has just one\n    expression.\n    What work will the\n    execution\n    procedure\n    produced by Alyssa's program do?  What about the execution\n    procedure\n    produced by the program in the text above?  How do the two versions compare\n    for a sequence with two expressions?\n    ","4.1.7#ex-4.25":"\n    Design and carry out some experiments to compare the speed of the original\n    metacircular evaluator with the version in this section.  Use your results\n    to estimate the fraction of time that is spent in analysis versus execution\n    for various\n    procedures.","4.1.7#footnote-1":"This technique is an integral part of the compilation\n    process, which we shall discuss in chapter 5.  Jonathan Rees wrote\n    a Scheme\n    interpreter like this in about 1982 for the T project \n    \n    (Rees and Adams 1982).  \n    \n    Marc Feeley 1986\n    (see also\n    Feeley and Lapalme 1987)\n    independently invented this technique\n    in his master's thesis.","4.1.7#footnote-2":"There is, however, an important part of the\n    \n\tvariable search    \n      \n    that can be done as part of the syntactic analysis.\n       As we will show in section 5.5.6,\n\tone can determine the position in the environment structure where the\n\tvalue of the variable will be found, thus obviating the need to scan the\n\tenvironment for the entry that matches the variable.\n","4.1.7#footnote-3":"See exercise 4.24 for\n\tsome insight into the processing of sequences.","4.2":"4.2  \n    \n      \n    \n    Lazy Evaluation","4.2#p1":"\n    Now that we have an evaluator expressed as a\n    Lisp\n    program, we can experiment with alternative choices in\n    \n    language design\n    simply by modifying the evaluator.  Indeed, new languages are often\n    invented by first writing an evaluator that embeds the new language\n    within an existing high-level language.  For example, if we wish to\n    discuss some aspect of a proposed modification to\n    Lisp\n    with another member of the\n    Lisp\n    community, we can supply an evaluator that embodies\n    the change.  The recipient can then experiment with the new\n    evaluator and send back comments as further modifications.  Not only\n    does the high-level implementation base make it easier to test and\n    debug the evaluator; in addition, the embedding enables the designer\n    to snarf features\n    from the underlying language, just as our embedded\n    Lisp\n    evaluator uses primitives and control structure from the underlying\n    Lisp.\n    Only later (if ever) need the designer go to the trouble of building a\n    complete implementation in a low-level language or in hardware.  In\n    this section and the next we explore some variations on\n    Scheme\n    that provide significant additional expressive power.\n  ","4.2#footnote-link-1":"1","4.2#footnote-1":"Snarf: \"To grab, especially a large document or\n    file for the purpose of using it either with or without the owner's\n    permission.\"  Snarf down: \"To snarf, sometimes with the\n    connotation of absorbing, processing, or understanding.\"\n    (These definitions were\n    \n    snarfed from\n    Steele et al. 1983.  \n    See also\n    Raymond 1996.)","4.2.1":"4.2.1  \n    Normal Order and Applicative Order","4.2.1#p1":"\n    In section 1.1, where we began\n    our discussion of models of evaluation, we noted that\n    Scheme\n    is an applicative-order language, namely, that all the arguments to\n    Scheme procedures\n    are evaluated when the\n    procedure\n    is applied. In contrast, normal-order languages delay evaluation of\n    procedure\n    arguments until the actual argument values are needed. Delaying evaluation of\n    procedure\n    arguments until the last possible moment (e.g., until they are required by a\n    primitive operation) is called \n    lazy evaluation. Consider the\n    procedure(define (try a b)\n  (if (= a 0) 1 b)) \n    Evaluating\n    (try 0 (/ 1 0))\n\tgenerates\n      \n    an error in\n    Scheme.\n    With lazy evaluation, there would be no error.  Evaluating the\n    expression\n    would return 1, because the argument\n    (/ 1 0)\n    would never be evaluated.\n  ","4.2.1#footnote-link-1":"1","4.2.1#p2":"\n    An example that exploits lazy evaluation is the\n    definition\n    of a\n    procedureunless(define (unless condition usual-value exceptional-value)\n  (if condition exceptional-value usual-value)) \n    that can be used in\n    expressions\n    such as\n    (unless (= b 0)\n        (/ a b)\n        (begin (display \"exception: returning 0\")\n               0)) \n    This won't work in an applicative-order language because both the\n    usual value and the exceptional value will be evaluated before\n    unless is called (compare\n    exercise 1.6). An advantage of lazy evaluation is\n    that some\n    procedures,\n    such as unless, can do useful computation\n    even if evaluation of some of their arguments would produce errors or\n    would not terminate.\n  ","4.2.1#p3":"\n    If the body of a\n    procedure\n    is entered before an argument has been evaluated we say that the\n    procedure\n    is \n    non-strict in that argument.  If the argument is evaluated before\n    the body of the\n    procedure\n    is entered we say that the\n    procedure\n    is \n    strict in that\n    argument.\n    In a purely applicative-order language, all\n    procedures\n    are strict in each argument.  In a purely normal-order language, all compound\n    procedures\n    are non-strict in each argument, and primitive\n    procedures\n    may be either strict or non-strict.  There are also languages (see\n    exercise 4.32) that give\n    programmers detailed control over the strictness of the\n    procedures\n    they define.\n  ","4.2.1#footnote-link-2":"2","4.2.1#p4":"\n    A striking example of a\n    procedure\n    that can usefully be made non-strict is\n    cons\n    (or, in general, almost any constructor for data structures).\n    One can do useful computation, combining elements to form\n    data structures and operating on the resulting data structures,\n    even if the values of the elements are not known.  It makes perfect\n    sense, for instance, to compute the length of a list without knowing\n    the values of the individual elements in the list.  We will exploit\n    this idea in section 4.2.3 to implement the\n    streams of chapter 3 as lists formed of non-strict\n    cons pairs.","4.2.1#ex-4.26":"\n\tSuppose that (in ordinary applicative-order Scheme) we define\n      unless as shown above and then define\n    factorial in terms\n    of unless as\n\n    (define (factorial n)\n  (unless (= n 1)\n          (* n (factorial (- n 1)))\n          1)) \n    What happens if we attempt to evaluate\n    (factorial 5)?\n    Will our\n    definitions\n    work in a normal-order language?\n    ","4.2.1#ex-4.27":"\n    Ben Bitdiddle and Alyssa P. Hacker\n    \n    disagree over the importance of lazy\n    evaluation for implementing things such as\n    unless. Ben points out that it's possible\n    to implement unless in applicative order as a\n    special\n    form. Alyssa counters that, if one did that,\n    unless would be merely syntax, not a\n    procedure\n    that could be used in conjunction with higher-order\n    procedures.\n    Fill in the details on both sides of the argument.\n    \n\tShow how to implement unless as a derived\n\texpression (like cond or\n\tlet), and give an example of a situation\n\twhere it might be useful to have unless\n\tavailable as a procedure, rather than as a special form.\n      ","4.2.1#footnote-1":"The difference between the\n    \"lazy\" terminology and the \"normal-order\"\n    terminology is somewhat fuzzy.  Generally, \"lazy\" refers to the\n    mechanisms of particular evaluators, while \"normal-order\"\n    refers to the semantics of languages, independent of any particular\n    evaluation strategy.  But this is not a hard-and-fast distinction, and the\n    two terminologies are often used interchangeably.","4.2.1#footnote-2":"The \"strict\" versus \"non-strict\"\n    terminology means essentially the same as\n    \"applicative-order\" versus \"normal-order,\" except\n    that it refers to individual\n    procedures\n    and arguments rather than to the language as a whole.  At a conference on\n    programming languages you might hear someone say, \"The normal-order\n    language \n    \n    Hassle has certain strict primitives.  Other\n    procedures\n    take their arguments by lazy evaluation.\"","4.2.2":"4.2.2  \n    An Interpreter with Lazy Evaluation","4.2.2#p1":"\n    In this section we will implement a normal-order language that is\n    the same as\n    Scheme\n    except that compound\n    procedures\n    are non-strict in each argument.  Primitive\n    procedures\n    will still be strict. It is not difficult to modify the evaluator of\n    section 4.1.1 so that the language it\n    interprets behaves this way.  Almost all the required changes center around\n    procedure\n    application.\n  ","4.2.2#p2":"\n    The basic idea is that, when applying a\n    procedure,\n    the interpreter must determine which arguments are to be evaluated and which\n    are to be delayed.  The delayed arguments are not evaluated; instead, they\n    are transformed into objects called \n    thunks.\n    The thunk must contain the information required to produce the value\n    of the argument when it is needed, as if it had been evaluated at\n    the time of the application.  Thus, the thunk must contain the\n    argument expression and the environment in\n    which the\n    procedure\n    application is being evaluated.\n  ","4.2.2#footnote-link-1":"1","4.2.2#p3":"\n    The process of evaluating the expression in a thunk is called \n    forcing.\n    In general, a thunk will be forced only when its value is needed:\n    when it is passed to a primitive\n    procedure\n    that will use the value of the thunk; when it is the value of a predicate of\n    a conditional; and when it is the value of\n    an operator\n    that is about to be\n    applied as a\n    procedure.\n    One design choice we have available is whether or not to \n    memoize thunks, similar to the optimization for streams in\n    section 3.5.1.  With memoization, the first\n    time a thunk is forced, it stores the value that is computed.  Subsequent\n    forcings simply return the stored value without repeating the computation.\n    We'll make our interpreter memoize, because this is more efficient for\n    many applications.  There are tricky considerations here,\n    however.","4.2.2#footnote-link-2":"2","4.2.2#footnote-link-3":"3","4.2.2#h1":"Modifying the evaluator","4.2.2#p4":"\n    The main difference between the lazy evaluator and the one in\n    section 4.1 is in the handling of\n    procedure\n    applications in\n    eval\n    and\n    apply.\n  ","4.2.2#p5":"\n\tThe application? clause of\n\teval becomes\n      ((application? exp)\n(apply (actual-value (operator exp) env)\n       (operands exp)\n       env)) \n    This is almost the same as the\n    application?\n    clause of\n    eval\n    in section 4.1.1.  For lazy evaluation,\n    however, we call apply with the\n    operand\n    expressions, rather than the arguments produced by evaluating them. Since\n    we will need the environment to construct thunks if the arguments are to be\n    delayed, we must pass this as well.  We still evaluate the\n    operator,\n    because\n    apply needs the actual\n    procedure\n    to be applied in order to dispatch on its type (primitive versus compound)\n    and apply it.\n  ","4.2.2#p6":"\n    Whenever we need the actual value of an expression, we use\n    (define (actual-value exp env)\n  (force-it (eval exp env))) \n    instead of just\n    eval,\n      \n    so that if the expression's value is a thunk, it will be forced.\n  ","4.2.2#p7":"\n    Our new version of apply is also almost the\n    same as the version in section 4.1.1.\n    The difference is that\n    eval\n    has passed in unevaluated\n    \n\toperand\n      \n    expressions: For primitive\n    procedures\n    (which are strict), we evaluate all the arguments before applying the\n    primitive; for compound\n    procedures\n    (which are non-strict) we delay all the\n    arguments before applying the\n    procedure.(define (apply procedure arguments env)\n  (cond ((primitive-procedure? procedure)\n         (apply-primitive-procedure\n          procedure\n          (list-of-arg-values arguments env)))  \n        ((compound-procedure? procedure)\n         (eval-sequence\n          (procedure-body procedure)\n          (extend-environment\n           (procedure-parameters procedure)\n           (list-of-delayed-args arguments env) \n           (procedure-environment procedure))))\n        (else\n         (error\n          \"Unknown procedure type - - APPLY\" procedure)))) \n    The\n    procedures\n    that process the arguments are just like\n    list-of-values\n    from section 4.1.1,\n    except that\n    list-of-delayed-args\n    delays the arguments instead of evaluating them, and\n    list-of-arg-values\n    uses\n    actual-value\n    instead of\n    eval:\n      (define (list-of-arg-values exps env)\n  (if (no-operands? exps)\n      '()\n      (cons (actual-value (first-operand exps) env)\n            (list-of-arg-values (rest-operands exps)\n                                env))))\n\n(define (list-of-delayed-args exps env)\n  (if (no-operands? exps)\n      '()\n      (cons (delay-it (first-operand exps) env)\n            (list-of-delayed-args (rest-operands exps)\n                                  env))))","4.2.2#p8":"\n    The other place we must change the evaluator is in the handling of\n    if,\n      \n    where we must use\n    actual-value\n    instead of\n    eval\n    to get the value of the predicate\n    expression before testing whether it is true or false:\n    (define (eval-if exp env)\n  (if (true? (actual-value (if-predicate exp) env))\n      (eval (if-consequent exp) env)\n      (eval (if-alternative exp) env))) ","4.2.2#p9":"\n    Finally, we must change the\n    driver-loopprocedure\n    (from section 4.1.4) to use\n    actual-value\n    instead of\n    eval,\n      \n    so that if a delayed value is propagated back to the\n    \n\tread-eval-print loop,\n      \n    it will be forced before being printed.\n    We also change the prompts to indicate that\n    this is the lazy evaluator:\n    (define input-prompt \";;; L-Eval input:\")\n(define output-prompt \";;; L-Eval value:\")\n\n(define (driver-loop)\n  (prompt-for-input input-prompt)\n  (let ((input (read)))\n    (let ((output\n           (actual-value input the-global-environment)))\n      (announce-output output-prompt)\n      (user-print output)))\n  (driver-loop)) ","4.2.2#p10":"\n    With these changes made, we can start the evaluator and test it.  The\n    successful evaluation of the\n    try\n    expression\n    discussed in section 4.2.1 indicates\n    that the interpreter is performing lazy evaluation:\n    (define the-global-environment (setup-environment))\n\n(driver-loop) ","4.2.2#h2":"Representing thunks","4.2.2#p11":"\n    Our evaluator must arrange to create thunks when\n    procedures\n    are applied to arguments and to force these thunks later.  A thunk must\n    package an expression together with the environment, so that the argument\n    can be produced later. To force the thunk, we simply extract the expression\n    and environment from the thunk and evaluate the expression in the\n    environment. We use\n    actual-value\n    rather than\n    eval\n    so that in case the value of the expression is itself a thunk, we will force\n    that, and so on, until we reach something that is not a thunk:\n    (define (force-it obj)\n  (if (thunk? obj)\n      (actual-value (thunk-exp obj) (thunk-env obj))\n      obj))","4.2.2#p12":"\n    One easy way to package an expression with an environment is to make a list\n    containing the expression and the environment. Thus, we create a thunk as\n    follows:\n    (define (delay-it exp env)\n  (list 'thunk exp env))\n\n(define (thunk? obj)\n  (tagged-list? obj 'thunk))\n\n(define (thunk-exp thunk) (cadr thunk))\n\n(define (thunk-env thunk) (caddr thunk)) ","4.2.2#p13":"\n    Actually, what we want for our interpreter is not quite this, but\n    rather thunks that have been memoized.\n    \n    When a thunk is forced, we will turn it into an evaluated thunk by replacing\n    the stored expression with its value and changing the\n    thunk tag so that it can be recognized as\n    already evaluated.(define (evaluated-thunk? obj)\n  (tagged-list? obj 'evaluated-thunk))\n\n(define (thunk-value evaluated-thunk) (cadr evaluated-thunk))\n\n(define (force-it obj)\n  (cond ((thunk? obj)\n         (let ((result (actual-value\n                        (thunk-exp obj)\n                        (thunk-env obj))))\n           (set-car! obj 'evaluated-thunk)\n           (set-car! (cdr obj) result)  \n           (set-cdr! (cdr obj) '())     \n           result))\n        ((evaluated-thunk? obj)\n         (thunk-value obj))\n        (else obj))) \n    Notice that the same\n    delay-itprocedure\n    works both with and\n    without memoization.","4.2.2#footnote-link-4":"4","4.2.2#ex-4.28":"\n  Suppose we type in the following\n  \n      definitions\n    \n  to the lazy evaluator:\n  (define count 0)\n\n(define (id x)\n  (set! count (+ count 1))\n  x) \n  Give the missing values in the following sequence of interactions, and explain\n  your answers.(define w (id (id 10))) \n    \n    \n    \n    \n  \n;;; L-Eval input:\ncount\n;;; L-Eval value:\nresponse\n    \n    \n    \n    \n    \n  \n;;; L-Eval input:\nw\n;;; L-Eval value:\nresponse\n    \n    \n    \n    \n    \n  \n;;; L-Eval input:\ncount\n;;; L-Eval value:\nresponse\n    ","4.2.2#footnote-link-5":"5","4.2.2#ex-4.29":"Eval\n    uses\n    actual-value\n    rather than\n    eval\n    to evaluate the\n    operator\n    before passing it to\n    apply, in order to force the value of the\n    operator.\n    Give an example that demonstrates the need for this forcing.\n    ","4.2.2#ex-4.30":"\n    Exhibit a program that you would expect to run much more slowly without\n    memoization than with memoization.  Also, consider the following\n    interaction, where the idprocedure\n    is defined as in exercise 4.28 and\n    count starts at 0:\n    (define (square x)\n  (* x x)) \n      \n      \n      \n    \n      \n      \n      \n    \n    Give the responses both when the evaluator memoizes and when it does not.\n    ","4.2.2#ex-4.31":"\n\tCy D. Fect, a reformed C programmer, is worried that some side effects\n\tmay never take place, because the lazy evaluator doesn't force the\n\texpressions in a sequence.\n\tSince the value of an expression in a sequence\n\tother than the last one is not used (the expression is there only for\n\tits effect, such as assigning to a variable or printing), there can be\n\tno subsequent use of this value (e.g., as an argument to a primitive\n\tprocedure) that will cause it to be forced.  Cy thus thinks that when\n\tevaluating sequences, we must force all expressions in the sequence\n\texcept the final one. He proposes to modify\n\teval-sequence\n\tfrom section 4.1.1 to use\n\tactual-value\n\trather than eval:\n      (define (eval-sequence exps env)\n  (cond ((last-exp? exps) (actual-value (first-exp exps) env))\n        (else (actual-value (first-exp exps) env)\n              (eval-sequence (rest-exps exps) env))))\n\tBen Bitdiddle thinks Cy is wrong. He shows Cy the\n\tfor-eachprocedure\n\tdescribed in exercise 2.23,\twhich gives an\n\timportant example of a sequence with side effects:\n\t(define (for-each proc items)\n  (if (null? items)\n      'done\n      (begin (proc (car items))\n             (for-each proc (cdr items))))) \n\tHe claims that the evaluator in the text (with the original\n\teval-sequence)\n\t  \n\thandles this correctly:\n        (for-each (lambda (x) (newline) (display x))\n          (list 57 321 88))\n\tExplain why Ben is right about the behavior of\n\tfor-each.\n\t  \n\tCy agrees that Ben is right about the\n\tfor-each\n\texample, but says that that's not the kind of program he\n\twas thinking about when he proposed his change to\n\teval-sequence.\n\t  \n\tHe\n\tdefines\n\tthe following two\n\tprocedures\n\tin the lazy evaluator:\n\t(define (p1 x)\n  (set! x (cons x '(2)))\n  x)\n\n(define (p2 x)\n  (define (p e)\n    e\n    x)\n  (p (set! x (cons x '(2))))) \n\tWhat are the values of\n\t(p1 1)\n\tand\n\t(p2 1)\n\twith the original\n\teval-sequence?\n\t  \n\tWhat would the values be with Cy's proposed change to\n\teval-sequence?\n\t  \n\tCy also points out that changing\n\teval-sequence\n\tas he proposes does not affect the behavior of the example in part a.\n\tExplain why this is true.\n      \n\tHow do you think sequences\n\tought to be treated in the lazy evaluator?\n\tDo you like Cy's approach, the approach in the text, or some other\n\tapproach?\n      ","4.2.2#ex-4.32":"\n    The approach taken in this section is somewhat unpleasant, because it\n    makes an incompatible change to\n    Scheme.\n    It might be nicer to implement lazy evaluation as an\n    upward-compatible extension, that is, so that ordinary\n    Scheme\n    programs will work as before.  We can do this by \n    extending the syntax of procedure\n    declarations to let the user control whether or not arguments are to be\n    delayed.  While we're at it, we may as well also give the user the\n    choice between delaying with and without memoization.  For example, the\n    \n\tdefinition\n      \n(define (f a (b lazy) c (d lazy-memo))\n  $\\ldots$)\n      \n    would define f to be a\n    procedure\n    of four arguments, where the first and third arguments are evaluated when the\n    procedure\n    is called, the second argument is delayed, and the fourth argument is both\n    delayed and memoized.\n    \n\tThus, ordinary procedure definitions will produce the same behavior as\n\tordinary Scheme,\n\twhile adding the\n\tlazy-memo\n\tdeclaration to each parameter of every compound procedure\n\twill produce the behavior of the lazy evaluator defined in this section.\n\tDesign and implement the changes required to produce such an extension to\n\tScheme. You will have to implement new syntax procedures\n\tto handle the new syntax for define.\n      \n    You must also arrange for\n    eval\n    or apply to determine when arguments are to be\n    delayed, and to force or delay arguments accordingly, and you must arrange\n    for forcing to memoize or not, as appropriate.\n    ","4.2.2#footnote-1":"The word thunk was invented by an informal\n      \n      working group that was discussing the implementation of call-by-name\n      \n      in Algol 60.  They observed that most of the analysis of (\"thinking\n      about\") the expression could be done at compile time; thus, at run\n      time, the expression would already have been \"thunk\" about \n      \n      (Ingerman et al. 1960).","4.2.2#footnote-2":"This is analogous to the\n    \n\tuse of\n\tforce\n\ton \n      \n    the delayed objects that were introduced in chapter 3 to\n    represent streams.  The critical difference between what we are\n    doing here and what we did in chapter 3 is that we are building\n    delaying and forcing into the evaluator, and thus making this uniform\n    and automatic throughout the language.","4.2.2#footnote-3":"Lazy evaluation combined with memoization is sometimes\n    referred to as\n    call-by-need argument passing, in contrast to\n    call-by-name argument passing.  \n    \n    (Call-by-name, introduced in\n    \n    Algol 60, is similar to non-memoized lazy\n    evaluation.) As language designers, we can build our evaluator to memoize,\n    not to memoize, or leave this an option for programmers\n    (exercise 4.32).  As you might\n    expect from chapter 3, these choices raise issues that become both\n    subtle and confusing in the presence of assignments.  (See\n    exercises 4.28\n    and 4.30.)\n    An excellent article by\n    \n    Clinger (1982) attempts to clarify the\n    multiple dimensions of confusion that arise here.","4.2.2#footnote-4":"Notice that we also erase the\n    env from the thunk once the expression's\n    value has been computed.  This makes no difference in the values returned by\n    the interpreter.  It does help save space, however, because removing the\n    reference from the thunk to the env once it is\n    no longer needed allows this structure to be\n    garbage-collected and its space \n    recycled, as we will discuss in\n    section 5.3.\n    ","4.2.2#p14":"\n      Similarly, we could have allowed unneeded environments in the memoized\n      delayed objects of section 3.5.1\n      to be garbage-collected, by having\n      memo-proc\n      do something like\n      (set! proc '())\n      to discard the\n      procedure\n\tproc\n      (which includes the environment in which the\n      delay\n      was evaluated) after storing its\n      value.\n    ","4.2.2#footnote-5":"This exercise demonstrates that the interaction between\n  lazy evaluation and side effects can be very confusing.  This is just what you\n  might expect from the discussion in chapter 3.","4.2.3":"4.2.3  \n    Streams as Lazy Lists","4.2.3#p1":"\n    In section 3.5.1, we showed how to\n    implement streams as delayed lists.\n    \n\tWe introduced special forms delay and\n\tcons-stream, which allowed us\n      \n    to construct a\n    \"promise\" to compute the\n    cdr\n    of a stream, without actually fulfilling that promise until later.\n    \n\tWe could use this general technique of introducing special forms\n\twhenever we need more control over the evaluation process, but this is\n\tawkward.  For one thing, a special form is not a first-class object\n\tlike a procedure, so we cannot use it together with higher-order\n\tprocedures.\n\tAdditionally, we were forced to create streams as a new kind of data\n\tobject similar but not identical to lists, and this required us to\n    reimplement many ordinary list operations (map,\n    append, and so on) for use with streams.\n      ","4.2.3#footnote-link-1":"1","4.2.3#p2":"\n    With lazy evaluation, streams and lists can be identical, so there is\n    no need for special forms or for\n    separate list and stream operations. All we need to do is to arrange matters\n    so that\n    cons\n    is non-strict.  One way to accomplish this is to extend the lazy evaluator\n    to allow for non-strict primitives, and to implement\n    cons\n    as one of these.  An easier way is to recall\n    (section 2.1.3) that there is no fundamental need\n    to implement\n    cons\n    as a primitive at all.  Instead, we can represent\n    \n    pairs as\n    procedures:(define (cons x y)\n  (lambda (m) (m x y)))\n\n(define (car z)\n  (z (lambda (p q) p)))\n\n(define (cdr z)\n  (z (lambda (p q) q)))","4.2.3#footnote-link-2":"2","4.2.3#p3":"\n    In terms of these basic operations, the standard definitions of the list\n    operations will work with infinite lists (streams) as well as finite ones,\n    and the stream operations can be implemented as list operations. Here are\n    some examples:\n    (define (list-ref items n)\n  (if (= n 0)\n      (car items)\n      (list-ref (cdr items) (- n 1))))\n\n(define (map proc items)\n  (if (null? items)\n      '()\n      (cons (proc (car items))\n            (map proc (cdr items)))))\n\n(define (scale-list items factor)\n  (map (lambda (x) (* x factor))\n       items))\n\n(define (add-lists list1 list2)\n  (cond ((null? list1) list2)\n         ((null? list2) list1)\n         (else (cons (+ (car list1) (car list2))\n                     (add-lists (cdr list1) (cdr list2))))))\n\n(define ones (cons 1 ones))\n\n(define integers (cons 1 (add-lists ones integers)))(list-ref integers 17)","4.2.3#p4":"\n    Note that these lazy lists are even lazier than the streams of\n    chapter 3:  The\n    car\n    of the list, as well as the\n    cdr,\n    is delayed.\n    In fact, even accessing the\n    car\n    or\n    cdr\n    of a lazy pair need not force the value of a list element.  The value will be\n    forced only when it is really needed—e.g., for use as the argument\n    of a primitive, or to be printed as an answer.\n  ","4.2.3#footnote-link-3":"3","4.2.3#p5":"\n    Lazy pairs also help with the problem that arose with streams in\n    section 3.5.4, where we\n    found that formulating stream models of systems with loops may require us to\n    sprinkle our programs with\n    \n\texplicit\n\tdelay\n\toperations, beyond the ones supplied by\n\tcons-stream.\n      \n    With lazy evaluation, all arguments to\n    procedures\n    are delayed uniformly.  For instance, we can implement\n    procedures\n    to integrate lists and solve differential equations as we originally\n    intended in section 3.5.4:\n    (define (integral integrand initial-value dt)\n  (define int\n          (cons initial-value\n                (add-lists (scale-list integrand dt)\n                           int)))\n  int)\n\n      (define (solve f y0 dt)\n      (define y (integral dy y0 dt))\n      (define dy (map f y))\n      y)","4.2.3#ex-4.33":"\n    Give some examples that illustrate the difference between the streams\n    of chapter 3 and the \"lazier\" lazy lists described in\n    this section. How can you take advantage of this extra laziness?\n  ","4.2.3#ex-4.34":"\n    Ben Bitdiddle tests the lazy list implementation given above by\n    evaluating the expression\n    (car '(a b c))\n    To his surprise, this produces an error.  After some thought, he realizes\n    that the \"lists\" obtained\n    \n\tby reading in quoted expressions\n      \n    are different from the lists manipulated by the new definitions of\n    cons,car,\n    and\n    cdr.\n    Modify \n    \n\tthe evaluator's treatment of\t\n\tquoted expressions so that quoted lists\n      \n    typed at the driver loop will produce true lazy lists.\n    ","4.2.3#ex-4.35":"\n    Modify the driver loop for the evaluator so that lazy pairs and lists will\n    print in some reasonable way.  (What are you going to do about infinite\n    lists?)  You may also need to modify the representation of lazy pairs so\n    that the evaluator can identify them in order to print them.\n    ","4.2.3#footnote-1":"This is precisely the issue with the\n\tunless procedure,\n\tas in\n\texercise 4.27.","4.2.3#footnote-2":"This\n    is the \n    procedural\n    representation described in exercise 2.4.\n    Essentially any \n    procedural\n    representation (e.g., a message-passing implementation) would do as well.\n    Notice that we can install these definitions in the lazy evaluator simply by\n    typing them at the driver loop.  If we had originally included\n    cons,car,\n    and\n    cdr\n    as primitives in the global environment, they will be redefined.  (Also see\n    exercises 4.34\n    and 4.35.)","4.2.3#footnote-3":"This permits us to create delayed versions of more\n    general kinds of\n    list structures, not just sequences.\n    Hughes 1990\n    discusses some\n    applications of\n    \"lazy trees.\"","4.3":"4.3  \n    \n      \n    \n    Nondeterministic Computing","4.3#p1":"\n    In this section, we extend the\n    Scheme\n    evaluator to support a\n    programming paradigm called nondeterministic computing by\n    building into the evaluator a facility to support\n    \n    automatic search.\n    This is a much more profound change to the language than the\n    introduction of lazy evaluation in\n    section 4.2.\n  ","4.3#p2":"\n    Nondeterministic computing, like stream processing, is useful for\n    \"generate and test\" applications.  Consider the task of\n    starting with two lists of positive integers and finding a pair of\n    integers—one from the first list and one from the second\n    list—whose sum is prime. We saw how to handle this with finite\n    sequence operations in section 2.2.3 and\n    with infinite streams in section 3.5.3.\n    Our approach was to generate the sequence of all possible pairs and filter\n    these to select the pairs whose sum is prime.  Whether we actually generate\n    the entire sequence of pairs first as in chapter 2, or interleave the\n    generating and filtering as in chapter 3, is immaterial to the\n    essential image of how the computation is organized.\n  ","4.3#p3":"\n    The nondeterministic approach evokes a different image.  Imagine simply\n    that we choose (in some way) a number from the first list and a number\n    from the second list and require (using some mechanism) that their\n    \n    sum be prime.  This is expressed by the following\n    procedure:(define (prime-sum-pair list1 list2)\n  (let ((a (an-element-of list1))\n        (b (an-element-of list2)))\n    (require (prime? (+ a b)))\n    (list a b))) \n    It might seem as if this\n    procedure\n    merely restates the problem,\n    rather than specifying a way to solve it.  Nevertheless, this is a\n    legitimate nondeterministic program.","4.3#footnote-link-1":"1","4.3#p4":"\n    The key idea here is that\n    expressions\n    in a nondeterministic language\n    can have more than one possible value.  For instance,\n    an-element-of\n    might return any element of the given list.  Our nondeterministic program\n    evaluator will work by automatically choosing a possible value and keeping\n    track of the choice.  If a subsequent requirement is not met, the evaluator\n    will try a different choice, and it will keep trying new choices until the\n    evaluation succeeds, or until we run out of choices.  Just as the lazy\n    evaluator freed the programmer from the details of how values are delayed\n    and forced, the nondeterministic program evaluator will free the programmer\n    from the details of how choices are made.\n  ","4.3#p5":"\n    It is instructive to contrast the different images of\n    \n    time evoked by\n    nondeterministic evaluation and stream processing.  Stream processing\n    uses lazy evaluation to decouple the time when the stream of possible\n    answers is assembled from the time when the actual stream elements are\n    produced.  The evaluator supports the illusion that all the possible\n    answers are laid out before us in a timeless sequence.  With\n    nondeterministic evaluation,\n    an expression\n    represents the exploration\n    of a set of possible worlds, each determined by a set of choices.\n    Some of the possible worlds lead to dead ends, while others have\n    useful values.  The nondeterministic program evaluator supports the\n    illusion that time branches, and that our programs have different\n    possible execution histories.  When we reach a dead end, we can\n    revisit a previous choice point and proceed along a different branch.\n  ","4.3#p6":"\n    The nondeterministic program evaluator implemented below is called the\n    amb evaluator because it is based on \n    a new special form\n    called amb.  We can type the above\n    definition\n    of\n    prime-sum-pair\n    at the amb evaluator driver loop (along with\n    definitions\n    of\n    prime?,an-element-of,\n    and require) and run the\n    procedure\n    as follows:\n    (prime-sum-pair '(1 3 5 8) '(20 35 110)) \n    The value returned was obtained after the evaluator repeatedly chose\n    elements from each of the lists, until a successful choice was made.\n  ","4.3#p7":"\n    Section 4.3.1 introduces\n    amb and explains how it supports nondeterminism\n    through the evaluator's automatic search mechanism.\n    Section 4.3.2 presents examples of\n    nondeterministic programs, and\n    section 4.3.3 gives the details of how\n    to implement the amb evaluator by modifying the\n    ordinary\n    Scheme\n    evaluator.\n  ","4.3#footnote-1":"We assume that we have\n    previously defined a\n    procedureprime?\n    that tests whether numbers are prime.  Even with\n    prime?\n    defined, the \n    prime-sum-pairprocedure\n    may look suspiciously like the unhelpful\n    \"pseudo-Lisp\"\n    attempt to define the square-root function, which we described at the\n    beginning of section 1.1.7.  In fact, a square-root\n    procedure\n    along those lines can actually be formulated as a nondeterministic program.\n    By incorporating a search mechanism into the evaluator, we are eroding the \n    \n    distinction between purely declarative descriptions and imperative\n    specifications of how to compute answers.  We'll go even farther in\n    this direction in\n    section 4.4.","4.3.1":"4.3.1","4.3.1#p1":"\n    To extend\n    Scheme\n    to support nondeterminism, we introduce a new\n    special form \n    called amb.\n    The expression\n    (amb $e_1\\ e_2\\ldots e_n$)\n    returns the value of one of the $n$ expressions\n    $e_i$\"ambiguously.\" For example,\n    the expression\n    (list (amb 1 2 3) (amb 'a 'b)) \n    can have six possible values:\n    (1 a) (1 b) (2 a) (2 b) (3 a) (3 b)Amb\n    with a single choice produces an ordinary (single) value.\n  ","4.3.1#footnote-link-1":"1","4.3.1#p2":"Amb\n    with no choices—the expression\n    (amb)—is\n    an expression with no acceptable values.  Operationally, we can think of\n    (amb)\n    as an expression that when evaluated causes the computation to\n    \"fail\": The computation aborts and no value is produced.\n    Using this idea, we can express the requirement that a particular predicate\n    expression p must be true as follows:\n    (define (require p)\n  (if (not p) (amb))) ","4.3.1#p3":"\n    With amb and\n    require, we can implement the\n    an-element-of procedure\n    used above:\n    (define (an-element-of items)\n  (require (not (null? items)))\n  (amb (car items) (an-element-of (cdr items)))) An-element-of\n    fails if the list is empty.  Otherwise it ambiguously returns either the\n    first element of the list or an element chosen from the rest of the list.\n  ","4.3.1#p4":"\n    We can also express infinite ranges of choices.  The following\n    procedure\n    potentially returns any integer greater than or equal to some\n    given $n$:\n    (define (an-integer-starting-from n)\n  (amb n (an-integer-starting-from (+ n 1)))) \n    This is like the stream\n    procedure\n      integers-starting-from  \n    described in section 3.5.2, but with an\n    important difference: The stream\n    procedure\n    returns an object that represents the sequence of all integers beginning\n    with $n$, whereas the\n    ambprocedure\n    returns a single integer.","4.3.1#footnote-link-2":"2","4.3.1#p5":"\n    Abstractly, we can imagine that evaluating an\n    amb expression causes\n    \n    time to split into\n    branches, where the computation continues on each branch with one of the\n    possible values of the expression.  We say that\n    amb represents a \n    nondeterministic choice point. If we had a machine with a\n    sufficient number of processors that could be dynamically allocated, we\n    could implement the search in a straightforward way.  Execution would\n    proceed as in a sequential machine, until an amb\n    expression is encountered.  At this point, more processors would be allocated\n    and initialized to continue all of the parallel executions implied by the\n    choice.  Each processor would proceed sequentially as if it were the only\n    choice, until it either terminates by encountering a failure, or it further\n    subdivides, or it finishes.","4.3.1#footnote-link-3":"3","4.3.1#p6":"\n    On the other hand, if we have a machine that can execute only one process\n    (or a few concurrent processes), we must consider the alternatives\n    \n    sequentially. One could imagine modifying an evaluator to pick at random a\n    branch to follow whenever it encounters a choice point.  Random choice,\n    however, can easily lead to failing values. We might try running the\n    evaluator over and over, making random choices and hoping to find a\n    non-failing value, but it is better to \n    systematically search all possible execution paths. The\n    amb evaluator that we will develop and work\n    with in this section implements a systematic search as follows: When the\n    evaluator encounters an application of amb, it\n    initially selects the first alternative.  This selection may itself lead to\n    a further choice.  The evaluator will always initially choose the first\n    alternative at each choice point.  If a choice results in a failure, then\n    the evaluator\n    \n    automagicallybacktracks to the most recent choice point and tries the next\n    alternative.  If it runs out of alternatives at any choice point, the\n    evaluator will back up to the previous choice point and resume from there.\n    This process leads to a search strategy known as \n    depth-first search or\n    chronological\n    backtracking.","4.3.1#footnote-link-4":"4","4.3.1#footnote-link-5":"5","4.3.1#h1":"Driver loop","4.3.1#p7":"\n    The\n    \n    driver loop for the amb evaluator has some\n    unusual properties.  It reads\n    an expression    \n    and prints the value of the\n    first non-failing execution, as in the\n    prime-sum-pair\n    example shown above.  If we want to see the value of the next successful\n    execution, we can ask the interpreter to backtrack and attempt to generate a\n    second non-failing execution.\n    \n\tThis is signaled by typing the symbol\n\ttry-again. If any expression except\n\ttry-again\n\tis given, the interpreter will start a new problem, discarding the\n\tunexplored alternatives in the previous problem.\n      \n    Here is a sample interaction:\n    (prime-sum-pair '(1 3 5 8) '(20 35 110)) try-again try-again try-again (prime-sum-pair '(19 27 30) '(11 36 58)) ","4.3.1#ex-4.36":"\n    Write a\n    procedure\n      an-integer-between\n    that returns an integer between two given bounds.  This can be used to\n    implement a\n    procedure\n    that finds\n    \n    Pythagorean triples, i.e., triples of integers\n    $(i,j,k)$ between the given bounds such\n    that $i \\leq j$ and\n    $i^2 + j^2 =k^2$, as follows:\n    (define (a-pythagorean-triple-between low high)\n  (let ((i (an-integer-between low high)))\n    (let ((j (an-integer-between i high)))\n      (let ((k (an-integer-between j high)))\n        (require (= (+ (* i i) (* j j)) (* k k)))\n        (list i j k))))) ","4.3.1#ex-4.37":"\n    Exercise 3.69 discussed how to\n    generate the stream of all\n    Pythagorean triples, with no upper bound\n    on the size of the integers to be searched.  Explain why simply replacing\n    an-integer-between\n    by\n    an-integer-starting-from\n    in the\n    procedure\n    in\n    exercise 4.36 is not an adequate way to\n    generate arbitrary Pythagorean triples.  Write a\n    procedure\n    that actually will accomplish this.  (That is, write a\n    procedure\n    for which repeatedly typing\n    try-again\n    would in principle eventually generate all Pythagorean triples.)\n  ","4.3.1#ex-4.38":"\n    Ben Bitdiddle claims that the following method for generating\n    \n    Pythagorean\n    triples is more efficient than the one in\n    exercise 4.36.  Is he correct?\n    (Hint: Consider the number of possibilities that must be explored.)\n    (define (a-pythagorean-triple-between low high)\n  (let ((i (an-integer-between low high))\n        (hsq (* high high)))\n    (let ((j (an-integer-between i high)))\n      (let ((ksq (+ (* i i) (* j j))))\n        (require (>= hsq ksq))\n        (let ((k (sqrt ksq)))\n          (require (integer? k))\n          (list i j k)))))) ","4.3.1#footnote-1":"The idea of\n    amb for nondeterministic programming was\n    first described in 1961 by\n    \n    John McCarthy (see\n    McCarthy 1967).","4.3.1#footnote-2":"In actuality, the distinction between\n    nondeterministically returning a single choice and returning all choices\n    depends somewhat on our point of view.  From the perspective of the code\n    that uses the value, the nondeterministic choice returns a single value.\n    From the perspective of the programmer designing the code, the\n    nondeterministic choice potentially returns all possible values, and the\n    computation branches so that each value is investigated\n    separately.","4.3.1#footnote-3":"One might object that this is a\n    hopelessly inefficient mechanism.  It might require millions of processors\n    to solve some easily stated problem this way, and most of the time most\n    of those processors would be idle.  This objection should be taken in\n    the context of history.  Memory used to be considered just such an\n    expensive commodity.  \n    \n    In 1965 a megabyte of RAM cost about $400,000. Now every personal\n    computer has many gigabytes of RAM, and most of the time most of that RAM is\n    unused.  It is hard to underestimate the cost of mass-produced\n    electronics.","4.3.1#footnote-4":"Automagically: \"Automatically, but in a way\n    which, for some reason (typically because it is too complicated, or too ugly,\n    or perhaps even too trivial), the speaker doesn't feel like\n    explaining.\"\n    (Steele 1983,\n    Raymond 1996)","4.3.1#footnote-5":"The integration of\n    \n    automatic search strategies\n    into programming languages has had a long and checkered history.  The first\n    suggestions that nondeterministic algorithms might be elegantly encoded in a\n    programming language with search and automatic backtracking came from\n    \n    Robert Floyd (1967).  \n    \n    Carl Hewitt (1969) invented a programming language called \n    \n    Planner that explicitly supported automatic chronological backtracking,\n    providing for a built-in depth-first search strategy.  \n    \n    Sussman, Winograd, and Charniak (1971) implemented a subset of this language,\n    called \n    \n    MicroPlanner, which was used to support work in problem solving and robot\n    planning. Similar ideas, arising from logic and theorem proving, led to the\n    genesis in Edinburgh and Marseille of the elegant language \n    \n    Prolog (which we will discuss in\n    section 4.4).  After sufficient\n    frustration with automatic search, \n    \n    McDermott and Sussman (1972) developed a language called \n    \n    Conniver, which included mechanisms for placing the search strategy under\n    programmer control.  This proved unwieldy, however, and \n    \n    Sussman and Stallman (1975) found a more tractable approach while\n    investigating methods of symbolic analysis for electrical circuits.  They\n    developed a nonchronological backtracking scheme that was based on tracing\n    out the logical dependencies connecting facts, a technique that has come to\n    be known as \n    dependency-directed backtracking.  Although their method was\n    complex, it produced reasonably efficient programs because it did little\n    redundant search.  \n    Doyle (1979) and\n    McAllester (1978, 1980)\n    generalized and clarified the methods of Stallman and Sussman, developing a\n    new paradigm for formulating search that is now called\n    truth maintenance.\n    \n\tModern problem-solving systems all\n      \n    use some form of truth-maintenance system as a substrate.  See \n    Forbus and de Kleer 1993 for a discussion of elegant\n    ways to build truth-maintenance systems and applications using truth\n    maintenance.\n    Zabih, McAllester, and Chapman 1987 describes a\n    \n    nondeterministic extension to Scheme that is based on\n    amb; it is similar to the interpreter described\n    in this section, but more sophisticated, because it uses dependency-directed\n    backtracking rather than chronological\n    backtracking.\n    Winston 1992 gives an introduction to\n    both kinds of backtracking.","4.3.2":"4.3.2  \n    Examples of Nondeterministic Programs","4.3.2#p1":"\n    Section 4.3.3 describes the\n    implementation of the amb evaluator.  First,\n    however, we give some examples of how it can be used.  The advantage of\n    nondeterministic programming is that we can suppress the details of how\n    search is carried out, thereby\n    expressing our programs at a higher level of\n    \n    abstraction.\n  ","4.3.2#h1":"Logic Puzzles","4.3.2#p2":"\n    The following puzzle (adapted from\n    Dinesman 1968) \n    is typical of a large class of simple logic puzzles:\n    \n      The software company\n      \n      Gargle is expanding, and Alyssa, Ben, Cy,  Lem, and Louis\n      are moving into a row of five private offices in a\n      new building. Alyssa does not move into the last office. Ben does not\n      move into the first office. Cy takes neither the first nor the last office.\n      Lem moves into an office after Ben's. Louis's office is not next to\n      Cy's. Cy's office is not next to Ben's. Who moves into which office?\n      ","4.3.2#p3":"\n    We can determine who moves into which office in a straightforward way by\n    enumerating all the possibilities and imposing the given\n    restrictions:(define (multiple-dwelling)\n  (let ((baker (amb 1 2 3 4 5))\n        (cooper (amb 1 2 3 4 5))\n        (fletcher (amb 1 2 3 4 5))\n        (miller (amb 1 2 3 4 5))\n        (smith (amb 1 2 3 4 5)))\n    (require\n     (distinct? (list baker cooper fletcher miller smith)))\n    (require (not (= baker 5)))\n    (require (not (= cooper 1)))\n    (require (not (= fletcher 5)))\n    (require (not (= fletcher 1)))\n    (require (> miller cooper))\n    (require (not (= (abs (- smith fletcher)) 1)))\n    (require (not (= (abs (- fletcher cooper)) 1)))\n    (list (list 'baker baker)\n          (list 'cooper cooper)\n          (list 'fletcher fletcher)\n          (list 'miller miller)\n          (list 'smith smith)))) ","4.3.2#footnote-link-1":"1","4.3.2#p4":"\n    Evaluating the expression\n    (multiple-dwelling)\n    produces the result\n    ((baker 3) (cooper 2) (fletcher 4) (miller 5) (smith 1))\n    Although this simple\n    procedure\n    works, it is very slow.\n    Exercises 4.40\n    and 4.41 discuss some possible\n    improvements.\n  ","4.3.2#ex-4.39":"\n    Modify the office-move\n    procedure\n    to omit the requirement that Louis's office is not next to Cy's.\n    How many solutions are there to this modified puzzle?\n  ","4.3.2#ex-4.40":"\n    Does the order of the restrictions in the office-move\n    procedure\n    affect the answer? Does it affect the time to find an answer?  If you\n    think it matters, demonstrate a faster program obtained from the given\n    one by reordering the restrictions.  If you think it does not matter,\n    argue your case.\n  ","4.3.2#ex-4.41":"\n    In the office move problem, how many sets of assignments are\n    there of people to offices, both before and after the requirement that\n    office assignments be distinct?  It is very inefficient to generate all\n    possible assignments of people to offices and then leave it to\n    backtracking to eliminate them.  For example, most of the restrictions\n    depend on only one or two of the person-office\n    variables,\n    and can thus be imposed before offices have been selected for all the people.\n    Write and demonstrate a much more efficient nondeterministic\n    procedure\n    that solves this problem based upon generating only those possibilities that\n    are not already ruled out by previous restrictions.\n    \n\t(Hint: This will require a nest of let\n\texpressions.)\n      ","4.3.2#ex-4.42":"\n    Write an ordinary\n    Scheme\n    program to solve the office move puzzle.\n  ","4.3.2#ex-4.43":"\n    Solve the following \"Liars\" puzzle (adapted from\n    Phillips 1934):\n    \n      Alyssa, Cy, Eva, Lem, and Louis meet for a business lunch at SoSoService.\n      Their meals arrive one after the other, a considerable time after they\n      placed their orders. To entertain Ben, who expects them back at the office\n      for a meeting, they decide to each make one true statement and one false\n      statement about their orders:\n      \n          Alyssa:\n\t  \"Lem's meal arrived second. Mine arrived third.\"\n\t  Cy:\n\t  \"Mine arrived first. Eva's arrived second.\"\n\t  Eva:\n\t  \"Mine arrived third, and poor Cy's arrived last.\"\n\t  Lem:\n\t  \"Mine arrived second. Louis's arrived fourth.\"\n\t  Louis:\n\t  \"Mine arrived fourth. Alyssa's meal arrived first.\"\n      What was the real order in which the five diners received their meals?\n      ","4.3.2#ex-4.44":"\n    Use the amb evaluator to solve the following\n    puzzle (adapted from\n    Phillips 1961):\n      \n\tAlyssa, Ben, Cy, Eva, and Louis each pick a different chapter of SICP JS\n\tand solve all the exercises in that chapter.\n\tLouis solves the exercises in the \"Functions\" chapter,\n\tAlyssa the ones in the \"Data\" chapter, and\n\tCy the ones in the \"State\" chapter.\n\tThey decide to check each other's work, and\n\tAlyssa volunteers to check the exercises in the \"Meta\" chapter.\n\tThe exercises in the \"Register Machines\" chapter are solved by Ben\n\tand checked by Louis.\n\tThe person who checks the exercises in the \"Functions\" chapter\n\tsolves the exercises that are checked by Eva.\n\tWho checks the exercises in the\t\"Data\" chapter?\n      \n\t   Try to write the program so that it runs efficiently (see\n\t   exercise 4.41).  Also determine\n\t   how many solutions there are if we are not told that Alyssa checks the\n\t   exercises in the \"Meta\" chapter.\n\t   ","4.3.2#ex-4.45":"\n    Exercise 2.42 described the\n    \"eight-queens puzzle\" of placing queens on a chessboard so that\n    no two attack each other. Write a nondeterministic program to solve this\n    puzzle.\n  ","4.3.2#h2":"Parsing natural language","4.3.2#p5":"\n    Programs designed to accept natural language as input usually start by\n    attempting to parse the input, that is, to match the input\n    against some grammatical structure.  For example, we might try to\n    recognize simple sentences consisting of an article followed by a noun\n    followed by a verb, such as \"The cat eats.\"  To accomplish\n    such an analysis, we must be able to identify the parts of speech of\n    individual words.  We could start with some lists that classify various\n    words:(define nouns '(noun student professor cat class))\n\n(define verbs '(verb studies lectures eats sleeps))\n\n(define articles '(article the a)) \n    We also need a\n    grammar, that is, a set of rules describing how\n    grammatical elements are composed from simpler elements.  A very\n    simple grammar might stipulate that a sentence always consists of two\n    pieces—a noun phrase followed by a verb—and that a noun\n    phrase consists of an article followed by a noun.  With this grammar, the\n    sentence \"The cat eats\" is parsed as follows:\n    (sentence (noun-phrase (article the) (noun cat))\n          (verb eats))","4.3.2#footnote-link-2":"2","4.3.2#p6":"\n    We can generate such a parse with a simple program that has separate\n    procedures\n    for each of the grammatical rules.  To parse a sentence, we identify its\n    two constituent pieces and return a list of these two elements, tagged with\n    the symbol sentence:\n    (define (parse-sentence)\n  (list 'sentence\n        (parse-noun-phrase)\n        (parse-word verbs))) \n    A noun phrase, similarly, is parsed by finding an article followed by a\n    noun:\n    (define (parse-noun-phrase)\n  (list 'noun-phrase\n        (parse-word articles)\n        (parse-word nouns))) ","4.3.2#p7":"\n    At the lowest level, parsing boils down to repeatedly checking that\n    the next\n    unparsed\n    word is a member of the list of words for the\n    required part of speech.  To implement this, we maintain a global\n    variable\n    *unparsed*,\n      \n    which is the input that has not yet been parsed.  Each time we check a word,\n    we require that\n    *unparsed*\n    must be nonempty and that it should begin with a word from the designated\n    list.  If so, we remove that word from\n    *unparsed*\n    and return the word together with its part of speech (which is found at\n    the head of the list):(define (parse-word word-list)\n  (require (not (null? *unparsed*)))\n  (require (memq (car *unparsed*) (cdr word-list)))\n  (let ((found-word (car *unparsed*)))\n    (set! *unparsed* (cdr *unparsed*))\n    (list (car word-list) found-word))) ","4.3.2#footnote-link-3":"3","4.3.2#p8":"\n    To start the parsing, all we need to do is set\n    *unparsed*\n    to be\n    the entire input, try to parse a sentence, and check that nothing is\n    left over:\n    (define *unparsed* '()) (define (parse input)\n  (set! *unparsed* input)\n  (let ((sent (parse-sentence)))\n    (require (null? *unparsed*))\n    sent)) ","4.3.2#p9":"\n    We can now try the parser and verify that it works for our simple test\n    sentence:\n    (parse '(the cat eats)) ","4.3.2#p10":"\n    The amb evaluator is useful here because it is\n    convenient to express the parsing constraints with the aid of\n    require. Automatic search and backtracking\n    really pay off, however, when we consider more complex grammars where there\n    are choices for how the units can be decomposed.\n  ","4.3.2#p11":"\n    Let's add to our grammar a list of prepositions:\n    (define prepositions '(prep for to in by with)) \n    and define a prepositional phrase (e.g., \"for the cat\") to be\n    a preposition followed by a noun phrase:\n    (define (parse-prepositional-phrase)\n  (list 'prep-phrase\n        (parse-word prepositions)\n        (parse-noun-phrase))) \n    Now we can define a sentence to be a noun phrase followed by a verb\n    phrase, where a verb phrase can be either a verb or a verb phrase\n    extended by a prepositional phrase:(define (parse-sentence)\n  (list 'sentence\n        (parse-noun-phrase)\n        (parse-verb-phrase)))\n\n(define (parse-verb-phrase)\n  (define (maybe-extend verb-phrase)\n    (amb verb-phrase\n         (maybe-extend (list 'verb-phrase\n                             verb-phrase\n                             (parse-prepositional-phrase)))))\n  (maybe-extend (parse-word verbs))) ","4.3.2#footnote-link-4":"4","4.3.2#p12":"\n    While we're at it, we can also elaborate the definition of noun\n    phrases to permit such things as \"a cat in the class.\"  What\n    we used to call a noun phrase, we'll now call a simple noun phrase,\n    and a noun phrase will now be either a simple noun phrase or a noun phrase\n    extended by a prepositional phrase:\n    (define (parse-simple-noun-phrase)\n  (list 'simple-noun-phrase\n        (parse-word articles)\n        (parse-word nouns)))\n\n(define (parse-noun-phrase)\n  (define (maybe-extend noun-phrase)\n    (amb noun-phrase\n         (maybe-extend (list 'noun-phrase\n                             noun-phrase\n                             (parse-prepositional-phrase)))))\n  (maybe-extend (parse-simple-noun-phrase))) ","4.3.2#p13":"\n    Our new grammar lets us parse more complex sentences.  For example\n    (parse '(the student with the cat sleeps in the class)) \n  produces\n  (sentence\n (noun-phrase\n  (simple-noun-phrase (article the) (noun student))\n  (prep-phrase (prep with)\n   (simple-noun-phrase\n    (article the) (noun cat))))\n (verb-phrase\n  (verb sleeps)\n  (prep-phrase (prep in)\n   (simple-noun-phrase\n    (article the) (noun class)))))","4.3.2#p14":"\n    Observe that a given input may have more than one legal parse.  In the\n    sentence \"The professor lectures to the student with the cat,\"\n    it may be that the professor is lecturing with the cat, or that the student\n    has the cat.  Our nondeterministic program finds both possibilities:\n    (parse '(the professor lectures to the student with the cat)) \n    produces\n    (sentence\n (simple-noun-phrase (article the) (noun professor))\n (verb-phrase\n  (verb-phrase\n   (verb lectures)\n   (prep-phrase (prep to)\n    (simple-noun-phrase\n     (article the) (noun student))))\n  (prep-phrase (prep with)\n   (simple-noun-phrase\n    (article the) (noun cat)))))\n    Asking the evaluator to retry yields\n    (sentence\n (simple-noun-phrase (article the) (noun professor))\n (verb-phrase\n  (verb lectures)\n  (prep-phrase (prep to)\n   (noun-phrase\n    (simple-noun-phrase\n     (article the) (noun student))\n    (prep-phrase (prep with)\n     (simple-noun-phrase\n      (article the) (noun cat)))))))","4.3.2#ex-4.46":"\n    With the grammar given above, the following sentence can be parsed in five\n    different ways: \"The professor lectures to the student in the class\n    with the cat.\" Give the five parses and explain the differences in\n    shades of meaning among them.\n  ","4.3.2#ex-4.47":"\n    The\n    \n    evaluators in sections 4.1 and\n    4.2 do not determine what order\n    operands\n    are\n    evaluated in. We will see that the amb evaluator\n    evaluates them from left to right. Explain why our parsing program\n    wouldn't work if the\n    operands\n    were evaluated in some other order.\n  ","4.3.2#ex-4.48":"\n    Louis Reasoner suggests that, since a verb phrase is either a verb or\n    a verb phrase followed by a prepositional phrase, it would be much more\n    straightforward to\n    define\n    the\n    procedureparse-verb-phrase\n    as follows (and similarly for noun phrases):\n    (define (parse-verb-phrase)\n  (amb (parse-word verbs)\n       (list 'verb-phrase\n             (parse-verb-phrase)\n             (parse-prepositional-phrase))))\n    Does this work?  Does the program's behavior change if we interchange\n    the order of expressions in the amb?\n  ","4.3.2#ex-4.49":"\n    Extend the grammar given above to handle more complex sentences.  For\n    example, you could extend noun phrases and verb phrases to include adjectives\n    and adverbs, or you could handle compound sentences.","4.3.2#footnote-link-5":"5","4.3.2#ex-4.50":"\n    Alyssa P. Hacker is more interested in\n    \n    generating interesting sentences\n    than in parsing them.  She reasons that by simply changing the\n    procedure\n      parse-word\n    so that it ignores the \"input sentence\" and instead always\n    succeeds and generates an appropriate word, we can use the programs we had\n    built for parsing to do generation instead.  Implement Alyssa's idea,\n    and show the first half-dozen or so sentences generated.","4.3.2#footnote-link-6":"6","4.3.2#footnote-1":"Our program uses the following\n    procedure\n    to determine if the elements of a list are distinct:\n    (define (distinct? items)\n  (cond ((null? items) true)\n        ((null? (cdr items)) true)\n        ((member (car items) (cdr items)) false)\n        (else (distinct? (cdr items))))) Member \n\tis like\n\tmemq\n\texcept that it uses equal? instead\n\tof eq? to test for equality.\n      ","4.3.2#footnote-2":"Here we use the convention that the first element of each\n    list designates the part of speech for the rest of the words in the\n    list.","4.3.2#footnote-3":"Notice that\n    parse-word\n    uses\n    set!\n    to modify the\n    unparsed\n    input list.  For this to work, our\n    amb evaluator must undo the effects of\n    set! operations\n    when it backtracks.","4.3.2#footnote-4":"Observe that this\n    definition is recursive—a verb may be followed by any number\n    of prepositional phrases.","4.3.2#footnote-5":"This kind of\n    grammar can become arbitrarily complex, but it\n    is only a\n    \n    toy as far as real language understanding is concerned.\n    Real natural-language understanding by computer requires an elaborate\n    mixture of syntactic analysis and interpretation of meaning.  On the\n    other hand, even toy parsers can be useful in supporting flexible\n    command languages for programs such as information-retrieval systems.\n    Winston 1992 discusses computational approaches to\n    real language understanding and also the applications of simple grammars\n    to command languages.","4.3.2#footnote-6":"Although\n    Alyssa's idea works just fine (and is surprisingly simple), the\n    sentences that it generates are a bit boring—they don't\n    sample the possible sentences of this language in a very interesting way.\n    In fact, the grammar is highly recursive in many places, and\n    Alyssa's technique \"falls into\" one of these recursions\n    and gets stuck.  See exercise 4.51 for a way to deal\n    with this.","4.3.3":"4.3.3  \n    Implementing the\n    \n      \n      \n    \n    Evaluator","4.3.3#p1":"\n    The evaluation of an ordinary\n    Scheme expression\n    may return a value, may never terminate, or may signal an error.\n    In nondeterministic\n    Scheme\n    the evaluation of\n    an expression\n    may in addition result in the discovery of\n    a dead end, in which case evaluation must backtrack to a previous choice\n    point.  The interpretation of nondeterministic\n    Scheme\n    is complicated by this extra case.\n  ","4.3.3#p2":"\n    We will construct the amb evaluator for\n    nondeterministic\n    Scheme\n    by modifying the\n    \n    analyzing evaluator of\n    section 4.1.7. As in the analyzing evaluator, evaluation of\n    an expression\n    is accomplished by calling an \n    \n    execution\n    procedure\n    produced by analysis of that\n    expression.\n    The difference between the interpretation of ordinary\n    Scheme\n    and the interpretation of nondeterministic\n    Scheme\n    will be entirely\n    in the execution\n    procedures.","4.3.3#footnote-link-1":"1","4.3.3#h1":"\n      Execution\n      procedures\n      and continuations\n    ","4.3.3#p3":"\n    Recall that the\n    \n    execution\n    procedures\n    for the ordinary evaluator take one argument: the environment of execution.\n    In contrast, the execution\n    procedures\n    in the amb evaluator take three arguments:\n    the environment, and two\n    procedures\n    called\n    continuation procedures.\n    The evaluation of\n    an expression\n    will finish by calling one of these two\n    continuations: If the evaluation results in a value, the \n    success continuation is called with that value; if the evaluation\n    results in the discovery of a dead end, the \n    failure continuation is called.  Constructing and calling\n    appropriate continuations is the mechanism by which the nondeterministic\n    evaluator implements backtracking.\n  ","4.3.3#p4":"\n    It is the job of the success continuation to receive a value and proceed\n    with the computation.  Along with that value, the success continuation is\n    passed another failure continuation, which is to be called subsequently if\n    the use of that value leads to a dead end.\n  ","4.3.3#p5":"\n    It is the job of the failure continuation to try another branch of the\n    nondeterministic process.  The essence of the nondeterministic\n    language is in the fact that\n    expressions\n    may represent choices among\n    alternatives.  The evaluation of such\n    an expression\n    must proceed with\n    one of the indicated alternative choices, even though it is not known\n    in advance which choices will lead to acceptable results.  To deal\n    with this, the evaluator picks one of the alternatives and passes this\n    value to the success continuation.  Together with this value, the\n    evaluator constructs and passes along a failure continuation that can\n    be called later to choose a different alternative.\n  ","4.3.3#p6":"\n    A failure is triggered during evaluation (that is, a failure\n    continuation is called) when a user program explicitly rejects the\n    current line of attack (for example, a call to\n    require may result in execution of\n    (amb),\n    an expression that always\n    fails—see section 4.3.1).  The failure\n    continuation in hand at that point will cause the most recent choice point\n    to choose another alternative.  If there are no more alternatives to be\n    considered at that choice point, a failure at an earlier choice point\n    is triggered, and so on.  Failure continuations are also invoked by\n    the driver loop in response to a\n    try-again\n    request, to find another value of the\n    expression.","4.3.3#p7":"\n    In addition, if a side-effect operation (such as assignment to a\n    variable) occurs on a branch of the process resulting from a choice,\n    it may be necessary, when the process finds a dead end, to undo the\n    side effect before making a new choice.  This is accomplished by\n    having the side-effect operation produce a failure continuation that\n    undoes the side effect and propagates the failure.\n  ","4.3.3#p8":"\n    In summary, failure continuations are constructed by\n    amb expressions—to provide a\n\tmechanism to make alternative choices if the current choice made by the\n\tamb expression leads to a dead end;\n      \n\tthe top-level driver—to provide a mechanism to report failure\n\twhen the choices are exhausted;\n      \n\tassignments—to intercept failures and undo assignments\n\tduring backtracking.\n      ","4.3.3#p9":"\n    Failures are initiated only when a dead end is encountered. This occurs\n    \n\tif the user program executes\n\t(amb);\n\tif the user types\n\ttry-again\n\tat the top-level driver.\n      ","4.3.3#p10":"\n    Failure continuations are also called during processing of a failure:\n    When the failure continuation created by an assignment finishes\n      undoing a side effect, it calls the failure continuation it intercepted,\n      in order to propagate the failure back to the choice point that\n      led to this assignment or to the top level.\n\n      When the failure continuation for an amb\n      runs out of choices, it calls the failure continuation that was originally\n      given to the amb, in order to propagate the\n      failure back to the previous choice point or to the top level.\n      ","4.3.3#h2":"Structure of the evaluator","4.3.3#p11":"\n    The syntax- and data-representation\n    procedures\n    for the amb evaluator, and also the basic\n    analyzeprocedure,\n    are identical to those in the evaluator of\n    section 4.1.7, except for the fact\n    that we need additional syntax\n    procedures\n    to recognize\n    \n\tthe amb special\n\tform:(define (amb? exp) (tagged-list? exp 'amb))\n\n(define (amb-choices exp) (cdr exp)) ","4.3.3#footnote-link-2":"2","4.3.3#p12":"\n    We must also add to the dispatch in analyze a\n    clause that will recognize\n    this special form and generate an appropriate execution procedure:\n      \n((amb? exp) (analyze-amb exp))\n      ","4.3.3#p13":"\n    The top-level\n    procedureambeval (similar to the version of\n    eval\n    given in section 4.1.7) analyzes the\n    given\n    expression\n    and applies the resulting execution\n    procedure\n    to the given environment, together with two given continuations:\n    (define (ambeval exp env succeed fail)\n  ((analyze exp) env succeed fail)) ","4.3.3#p14":"\n    A success\n    \n    continuation is a\n    procedure\n    of two arguments: the value just obtained and another failure continuation to\n    be used if that value leads to a subsequent failure. A\n    \n    failure continuation\n    is a\n    procedure\n    of no arguments.  So \n    the general form of an\n    \n    execution\n    procedure\n    is\n    \n(lambda (env succeed fail)\n  ;; succeed is (lambda (value fail) $\\ldots$)\n  ;; fail is (lambda () $\\ldots$)\n  $\\ldots$)\n      ","4.3.3#p15":"\n    For example, executing\n    (ambeval exp\n         the-global-environment\n         (lambda (value fail) value)\n         (lambda () 'failed))\n    will attempt to evaluate the given\n    expression\n    and will return either the\n    expression's\n    value (if the evaluation succeeds) or the\n    \n\tsymbol failed\n    (if the evaluation fails).\n    The call to ambeval in the driver loop shown\n    below uses much more complicated continuation\n    procedures,\n    which continue the loop and support the\n    try-again\n    request.\n  ","4.3.3#p16":"\n    Most of the complexity of the amb evaluator\n    results from the mechanics of passing the continuations around as the\n    execution\n    procedures\n    call each other.  In going through the following code, you should compare\n    each of the execution\n    procedures\n    with the corresponding\n    procedure\n    for the ordinary evaluator given in\n    section 4.1.7.\n  ","4.3.3#h3":"Simple expressions","4.3.3#p17":"\n    The execution\n    procedures\n    for the simplest kinds of expressions are\n    essentially the same as those for the ordinary evaluator, except for the\n    need to manage the continuations.  The execution\n    procedures\n    simply succeed with the value of the expression, passing along the failure\n    continuation that was passed to them.\n    (define (analyze-self-evaluating exp)\n  (lambda (env succeed fail)\n    (succeed exp fail))) (define (analyze-variable exp)\n  (lambda (env succeed fail)\n    (succeed (lookup-variable-value exp env)\n             fail))) (define (analyze-lambda exp)\n  (let ((vars (lambda-parameters exp))\n        (bproc (analyze-sequence (lambda-body exp))))\n    (lambda (env succeed fail)\n      (succeed (make-procedure vars bproc env)\n               fail)))) ","4.3.3#p18":"\n    Notice that looking up a\n    variable\n    always \"succeeds.\"\n    If\n    lookup-variable-value\n    fails to find the\n    variable,\n    it signals an\n    error, as usual.  Such a \"failure\" indicates a program\n    bug—a reference to an unbound\n    variable;\n    it is not an indication\n    that we should try another nondeterministic choice instead of the one that\n    is currently being tried.\n  ","4.3.3#h4":"Conditionals and sequences","4.3.3#p19":"\n    Conditionals are also handled in a similar way as in the ordinary\n    evaluator.  The execution\n    procedure\n    generated by\n    analyze-if\n    invokes the predicate execution\n    procedure\n      pproc\n    with a success continuation that checks whether the predicate value is true\n    and goes on to execute either the consequent or the alternative.  If the\n    execution of\n    pproc\n    fails, the original failure continuation for\n    the\n    if\n    expression is called.\n    \n(define (analyze-if exp)\n  (let ((pproc (analyze (if-predicate exp)))\n        (cproc (analyze (if-consequent exp)))\n        (aproc (analyze (if-alternative exp))))\n   (lambda (env succeed fail)\n           (pproc env\n                  \n                  \n                  (lambda (pred-value fail2)\n                    (if (true? pred-value)\n                        (cproc env succeed fail2)\n                        (aproc env succeed fail2)))\n                  \n                  fail))))\n      ","4.3.3#p20":"\n\tSequences are also handled in the same way as in the previous\n\tevaluator, except for the machinations in the\n      subprocedure\n\tsequentially that are required for passing the\n\tcontinuations. Namely, to sequentially execute a\n\tand then b, we call\n\ta with a success continuation that calls\n\tb.\n\t(define (analyze-sequence exps)\n\t    (define (sequentially a b)\n\t    (lambda (env succeed fail)\n\t    (a env\n            ;; success continuation for calling a\n            (lambda (a-value fail2)\n            (b env succeed fail2))\n            ;; failure continuation for calling a\n            fail)))\n\t    (define (loop first-proc rest-procs)\n\t    (if (null? rest-procs)\n            first-proc\n            (loop (sequentially first-proc (car rest-procs))\n            (cdr rest-procs))))\n\t    (let ((procs (map analyze exps)))\n\t    (if (null? procs)\n            (error \"Empty sequence - - ANALYZE\"))\n\t    (loop (car procs) (cdr procs)))) ","4.3.3#h5":"Definitions\n      and assignments\n    ","4.3.3#p21":"Definitions\n    are another case where we must go to some trouble to\n    manage the continuations, because it is necessary to evaluate the\n    \n\tdefinition-value expression before actually defining the new variable.\n      \n    To accomplish this, the\n    definition-value\n    execution\n    procedure\n      vproc\n    is called with the environment, a success continuation, and the\n    failure continuation.  If the execution of\n    vproc\n    succeeds, obtaining a value val for the\n    \n\tdefined variable, the variable is defined and the success is propagated:\n      (define (analyze-definition exp)\n  (let ((var (definition-variable exp))\n        (vproc (analyze (definition-value exp))))\n    (lambda (env succeed fail)\n      (vproc env                        \n             (lambda (val fail2)\n               (define-variable! var val env)\n               (succeed 'ok fail2))\n             fail)))) ","4.3.3#p22":"\n    Assignments\n    \n    are more interesting.  This is the first place where we\n    really use the continuations, rather than just passing them around.\n    The execution\n    procedure\n    for assignments starts out like the one for\n    definitions.\n    It first attempts\n    to obtain the new value to be assigned to the\n    variable.\n    If this evaluation of\n    vproc  \n    fails, the assignment fails.\n  ","4.3.3#p23":"\n    If\n    vproc  \n    succeeds, however, and we go on to make the assignment, we must consider the\n    possibility that this branch of the computation might later fail, which will\n    require us to backtrack out of the assignment.  Thus, we must arrange to\n    undo the assignment as part of the backtracking  process.","4.3.3#footnote-link-3":"3","4.3.3#p24":"\n    This is accomplished by giving\n    vproc  \n    a success continuation (marked with the comment \"*1*\" below)\n    that saves the old value of the variable before assigning the new value to\n    the variable and proceeding from the assignment.  The failure continuation\n    that is passed along with the value of the assignment (marked with the\n    comment \"*2*\" below) restores the old value of the variable\n    before continuing the failure. That is, a successful assignment provides a\n    failure continuation that will intercept a subsequent failure; whatever\n    failure would otherwise have called fail2 calls\n    this\n    procedure\n    instead, to undo the assignment before actually calling\n    fail2.\n    (define (analyze-assignment exp)\n  (let ((var (assignment-variable exp))\n        (vproc (analyze (assignment-value exp))))\n    (lambda (env succeed fail)\n      (vproc env\n             (lambda (val fail2)        \n               (let ((old-value\n                     (lookup-variable-value var env))) \n                 (set-variable-value! var val env)\n                 (succeed 'ok\n                          (lambda ()    \n                            (set-variable-value! var\n                                                 old-value\n                                                 env)\n                            (fail2)))))\n             fail)))) ","4.3.3#h6":"Procedure\n      applications\n    ","4.3.3#p25":"\n    The execution\n    procedure\n    for applications contains no new ideas except for the technical complexity\n    of managing the continuations.  This complexity arises in\n    analyze-application,  \n    due to the need to keep track of the success and failure continuations as\n    we evaluate the\n    operands.\n    We use a\n    procedure get-args  \n    to evaluate the list of\n    operands,\n    rather than a simple\n    map as in the ordinary evaluator.\n    (define (analyze-application exp)\n  (let ((fproc (analyze (operator exp)))\n        (aprocs (map analyze (operands exp))))\n    (lambda (env succeed fail)\n      (fproc env\n             (lambda (proc fail2)\n               (get-args aprocs\n                         env\n                         (lambda (args fail3)\n                           (execute-application\n                            proc args succeed fail3))\n                         fail2))\n             fail)))) ","4.3.3#p26":"\n\tIn  get-args, notice how\n\tcdring down the list of\n\taproc\n\texecution procedures and\n\tconsing\n\tup the resulting list of\n\targs is accomplished by calling each\n\taproc\n\tin the list with a success continuation that recursively calls\n\tget-args.\n      \n    Each of these recursive calls to\n    get-args  \n    has a success continuation whose value is the\n    cons\n\tof the newly obtained argument onto the list of accumulated arguments:\n      \n(define (get-args aprocs env succeed fail)\n  (if (null? aprocs)\n      (succeed '() fail)\n      ((car aprocs) env\n                    \n                    (lambda (arg fail2)\n                      (get-args (cdr aprocs)\n                                env\n                                \n                                \n                                (lambda (args fail3)\n                                  (succeed (cons arg args)\n                                           fail3))\n                                fail2))\n                    fail)))\n      ","4.3.3#p27":"\n    The actual\n    procedure\n    application, which is performed by\n    execute-application,  \n    is accomplished in the same way as for the ordinary evaluator, except for\n    the need to manage the continuations.\n    (define (execute-application proc args succeed fail)\n  (cond ((primitive-procedure? proc)\n         (succeed (apply-primitive-procedure proc args)\n                  fail))\n        ((compound-procedure? proc)\n         ((procedure-body proc)\n          (extend-environment (procedure-parameters proc)\n                              args\n                              (procedure-environment proc))\n          succeed\n          fail))\n        (else\n         (error\n          \"Unknown procedure type - - EXECUTE-APPLICATION\"\n          proc)))) ","4.3.3#h7":"Evaluating amb expressions","4.3.3#p28":"\n    The ambspecial\n    form is the key element in the nondeterministic language.  Here we see the\n    essence of the interpretation process and the reason for keeping track of\n    the continuations.  The execution\n    procedure\n    for amb defines a loop\n    try-next\n    that cycles through the execution\n    procedures\n    for all the possible values of the amb\n    expression.  Each execution\n    procedure\n    is called with a\n    \n    failure continuation that will try the next one.  When\n    there are no more alternatives to try, the entire\n    amb expression fails.\n    (define (analyze-amb exp)\n  (let ((cprocs (map analyze (amb-choices exp))))\n    (lambda (env succeed fail)\n      (define (try-next choices)\n        (if (null? choices)\n            (fail)\n            ((car choices) env\n             succeed\n             (lambda ()\n               (try-next (cdr choices))))))\n      (try-next cprocs)))) ","4.3.3#h8":"Driver loop","4.3.3#p29":"\n    The driver loop for the amb evaluator is\n    complex, due to the mechanism that permits the user to retry in evaluating\n    an expression.\n    The driver uses a\n    procedure\n    called\n    internal-loop,\n    which takes as argument a\n    proceduretry-again.\n      \n    The intent is that calling\n    try-again\n    should go on to the next untried alternative in the nondeterministic\n    evaluation.\n    Internal-loop\n    either calls\n    try-again\n    in response to the user typing\n    try-again\n    at the driver loop, or else starts a new evaluation by calling\n    ambeval.  \n  ","4.3.3#p30":"\n    The failure continuation for this call to\n    ambeval\n    informs the user that there are no more values and reinvokes the driver\n    loop.\n  ","4.3.3#p31":"\n    The success continuation for the call to ambeval\n    is more subtle.  We print the obtained value and then\n    \n\tinvoke the internal loop again\n      \n    with a\n    try-againprocedure\n    that will be able to try the next alternative.  This\n    next-alternativeprocedure\n    is the second argument that was passed to the success continuation.\n    Ordinarily, we think of this second argument as a failure continuation to\n    be used if the current evaluation branch later fails.  In this case,\n    however, we have completed a successful evaluation, so we can invoke the\n    \"failure\" alternative branch in order to search for additional\n    successful evaluations.\n    (define input-prompt \";;; Amb-Eval input:\")\n(define output-prompt \";;; Amb-Eval value:\")\n\n(define (driver-loop)\n  (define (internal-loop try-again)\n    (prompt-for-input input-prompt)\n    (let ((input (read)))\n      (if (eq? input 'try-again)\n          (try-again)\n          (begin\n            (newline)\n            (display \";;; Starting a new problem \")\n            (ambeval input\n                     the-global-environment\n                     \n                     (lambda (val next-alternative)\n                       (announce-output output-prompt)\n                       (user-print val)\n                       (internal-loop next-alternative))\n                    \n                    (lambda ()\n                      (announce-output\n                       \";;; There are no more values of\")\n                      (user-print input)\n                      (driver-loop)))))))\n  (internal-loop\n   (lambda ()\n     (newline)\n     (display \";;; There is no current problem\")\n     (driver-loop)))) \n    The initial call to\n    internal-loop\n    uses a\n    try-againprocedure\n    that complains that there is no current problem and restarts the driver loop.\n    This is the behavior that will happen if the user types\n    try-again\n    when there is no evaluation in progress.\n  ","4.3.3#ex-4.51":"\n    Implement a new\n    special\n    form ramb that is like\n    amb except that it searches alternatives in a\n    random order, rather than from left to right.  Show how this can help with\n    Alyssa's problem in exercise 4.50.\n    ","4.3.3#ex-4.52":"\n\tImplement a new kind of assignment called\n\tpermanent-set! that\n      \n    is not undone upon failure. For example, we can choose two distinct\n    elements from a list and count the number of trials required to make a\n    successful choice as follows:\n    (define count 0)\n\n(let ((x (an-element-of '(a b c)))\n      (y (an-element-of '(a b c))))\n  (permanent-set! count (+ count 1))\n  (require (not (eq? x y)))\n  (list x y count))try-again\n\tWhat values would have been displayed if we had used\n\tset! here rather than\n\tpermanent-set!?\n      ","4.3.3#ex-4.53":"\n\tImplement a new construct called if-fail\n\tthat permits the user to catch the failure of an expression.\n\tIf-fail takes two expressions. It evaluates\n\tthe first expression as usual and returns as usual if the evaluation\n\tsucceeds. If the evaluation fails, however, the value of the second\n\texpression is returned, as in the following example:\n\t(if-fail (let ((x (an-element-of '(1 3 5))))\n           (require (even? x))\n           x)\n         'all-odd)(if-fail (let ((x (an-element-of '(1 3 5 8))))\n           (require (even? x))\n           x)\n         'all-odd)","4.3.3#ex-4.54":"\n    With\n    permanent-set!\n    as described in exercise 4.52 and\n    if-fail\n    as in exercise , what will be the result of\n    evaluating\n    (let ((pairs '()))\n  (if-fail (let ((p (prime-sum-pair '(1 3 5 8) '(20 35 110))))\n             (permanent-set! pairs (cons p pairs))\n             (amb))\n           pairs))","4.3.3#ex-4.55":"\n    If we had not realized that\n    require could be\n    implemented as an ordinary\n    procedure\n    that uses amb, to be defined by the user as\n    part of a nondeterministic program, we would have had to implement it\n    as a \n    special\n    form. This would require syntax\n    procedures(define (require? exp) (tagged-list? exp 'require))\n\n(define (require-predicate exp) (cadr exp))\n    and a new clause in the dispatch in analyze((require? exp) (analyze-require exp))\n    as well the\n    procedureanalyze-require\n    that handles require\n    expressions.  Complete the following definition of\n    analyze-require.\n      \n(define (analyze-require exp)\n  (let ((pproc (analyze (require-predicate exp))))\n    (lambda (env succeed fail)\n      (pproc env\n             (lambda (pred-value fail2)\n               (if ??\n                   ??\n                   (succeed 'ok fail2)))\n             fail))))\n      ","4.3.3#footnote-1":"We chose to\n    implement the lazy evaluator in\n    section 4.2 as a modification of the\n    ordinary metacircular evaluator of\n    section 4.1.1.  In contrast, we will\n    base the amb evaluator on the analyzing\n    evaluator of section 4.1.7, because\n    the execution\n    procedures\n    in that evaluator provide a convenient framework for implementing\n    backtracking.","4.3.3#footnote-2":"We\n\tassume that the evaluator supports let\n\t(see exercise 4.23),\n\twhich we have used in our nondeterministic programs.","4.3.3#footnote-3":"We\n    \n\tdidn't worry about undoing definitions,\n\tsince we can\n\tassume that\n\t\n\tinternal definitions\n\tare scanned out\n\t(section 4.1.6).\n      ","4.4":"4.4  Logic Programming","4.4#p1":"\n    In chapter 1 we stressed that computer science deals with\n    \n    imperative\n    (how to) knowledge, whereas mathematics deals with declarative (what\n    is) knowledge.  Indeed, programming languages require that the\n    programmer express knowledge in a form that indicates the step-by-step\n    methods for solving particular problems.  On the other hand,\n    high-level languages provide, as part of the language implementation,\n    a substantial amount of methodological knowledge that frees\n    the user from concern with numerous details of how a specified\n    computation will progress.\n  ","4.4#p2":"\n    Most programming languages, including\n    Lisp,\n    are organized around\n    computing the values of mathematical functions.  Expression-oriented\n    languages\n    \n\t(such as Lisp, Fortran, Algol and JavaScript)\n      \n    capitalize on the\n    \"pun\" that an expression that describes the value of a\n    function may also be interpreted as a means of computing that value.\n    Because of this, most programming languages are strongly biased toward\n    unidirectional computations (computations with well-defined inputs and\n    outputs). There are, however, radically different programming languages\n    that relax this bias.  We saw one such example in\n    section 3.3.5, where the objects of\n    computation were arithmetic constraints.  In a constraint system the\n    direction and the order of computation are not so well specified; in\n    carrying out a computation the system must therefore provide more detailed\n    \"how to\" knowledge than would be the case with an ordinary\n    arithmetic computation.  This does not mean, however, that the user is\n    released altogether from the responsibility of providing imperative\n    knowledge. There are many constraint networks that implement the same set\n    of constraints, and the user must choose from the set of mathematically\n    equivalent networks a suitable network to specify a particular computation.\n  ","4.4#p3":"\n    The nondeterministic program evaluator of\n    section 4.3 also moves\n    away from the view that programming is about constructing algorithms for\n    computing unidirectional functions.  In a nondeterministic language,\n    expressions can have more than one value, and, as a result, the\n    computation is\n    dealing with\n    \n    relations rather than with single-valued functions.  Logic\n    programming extends this idea by combining a relational vision of programming\n    with a powerful kind of symbolic pattern matching called \n    unification.","4.4#footnote-link-1":"1","4.4#p4":"\n    This approach, when it works, can be a very\n    \n    powerful way to write programs.\n    Part of the power comes from the fact that a single \"what is\"\n    fact can be used to solve a number of different problems that would have\n    different \"how to\" components.  As an example, consider the \n    append operation, which takes two lists as\n    arguments and combines their elements to form a single list.  In a procedural\n    language such as\n    Lisp,\n    we could define append in terms of the\n    basic list constructor\n    cons,\n    as we did in section 2.2.1:\n    (define (append x y)\n  (if (null? x)\n      y\n      (cons (car x) (append (cdr x) y)))) This\n    procedure\n    can be regarded as a translation into\n    Lisp\n    of the following two rules, the first of which covers the case where the\n    first list is empty and the second of which handles the case of a nonempty\n    list, which is a\n    cons\n    of two parts:\n    \n\tFor any list y, the empty list and\n\tyappend to\n\tform y.\n      \n\tFor any u, v,\n\ty, and z,\n\t(cons u v)\n\tand yappend\n\tto form\n\t(cons u z)\n\tif v and yappend to form\n\tz.\n    Using the appendprocedure,\n    we can answer questions such as\n    \n      Find the append of\n      (a b)\n      and\n      (c d).\n    But the same two rules are also sufficient for answering the following\n    sorts of questions, which the\n    procedure\n    can't answer:\n    \n      Find a list y\n      that\n      appends with\n      (a b)\n      to produce\n      (a b c d).\n      Find all x and y\n      that append to form\n      (a b c d).\n    In a\n    \n    logic programming language, the programmer writes an\n    append\"procedure\"\n    by stating the two rules about append given\n    above.\n    \"How to\" knowledge is provided automatically by the\n    interpreter to allow this single pair of rules to be used to answer all\n    three types of questions about\n    append.","4.4#footnote-link-2":"2","4.4#footnote-link-3":"3","4.4#p5":"\n    Contemporary logic programming languages (including the one we\n    implement here) have substantial deficiencies, in that their general\n    \"how to\" methods can lead them into spurious infinite loops or\n    other undesirable behavior. Logic programming is an active field of research\n    in computer science.","4.4#footnote-link-4":"4","4.4#p6":"\n    Earlier in this chapter we explored the technology of implementing\n    interpreters and described the elements that are essential to an\n    interpreter for a\n    Lisp-like\n    language (indeed, to an interpreter for any conventional language).  Now we\n    will apply these ideas to discuss an interpreter for a logic programming\n    language.  We call this\n    language the\n    query language, because it is very useful for\n    retrieving information from data bases by formulating \n    queries, or questions, expressed in the language.  Even though the\n    query language is very different from\n    Lisp,\n    we will find it convenient to describe the language in terms of the same\n    general framework we have been using all along: as a collection of primitive\n    elements, together with means of combination that enable us to combine\n    simple elements to create more complex elements and means of abstraction\n    that enable us to regard complex elements as single conceptual units.  An\n    interpreter for a logic programming language is considerably more complex\n    than an interpreter for a language like\n    Lisp.\n    Nevertheless, we will see\n    that our\n    \n    query-language interpreter contains many of the same elements\n    found in the interpreter of section 4.1.  In\n    particular, there will be an \"evaluate\" part that classifies\n    expressions according to type and an \"apply\" part that\n    implements the language's abstraction mechanism\n    (procedures\n    in the case of\n    Lisp,\n    and rules in the case of logic programming).  Also, a central role\n    is played in the implementation by a frame data structure, which determines\n    the correspondence between symbols and their associated values.  One\n    additional interesting aspect of our query-language implementation is\n    that we make substantial use of streams, which were introduced in\n    chapter 3.\n  ","4.4#footnote-1":"Logic programming has grown out of a long\n    \n    history of research in\n    \n    automatic theorem proving.  Early theorem-proving\n    programs could accomplish very little, because they exhaustively searched\n    the space of possible proofs.  The major breakthrough that made such a\n    search plausible was the discovery in the early 1960s of the \n    unification algorithm and the \n    resolution principle (Robinson 1965).  \n    Resolution was used, for example, by \n    \n    Green and Raphael (1968) (see also Green 1969) as the\n    basis for a deductive question-answering system.  During most of this period,\n    researchers concentrated on algorithms that are guaranteed to find a proof if\n    one exists.  Such algorithms were difficult to control and to direct toward\n    a proof.  \n    \n    Hewitt (1969) recognized the possibility of merging the control structure of\n    a programming language with the operations of a logic-manipulation system,\n    leading to the work in automatic search mentioned in\n    section 4.3.1\n    (footnote 5).  At the same time that this\n    was being done,\n    \n    Colmerauer, in Marseille, was developing rule-based systems for manipulating\n    natural language (see Colmerauer et al. 1973).\n    He invented a programming language called \n    \n    Prolog for representing those rules.  \n    Kowalski (1973; 1979)\n    in Edinburgh, recognized that execution of a Prolog program could be\n    interpreted as proving theorems (using a proof technique called linear \n    \n    Horn-clause resolution).  The merging of the last two strands led to the\n    logic-programming movement.  Thus, in assigning credit for the development\n    of logic programming, the French can point to Prolog's genesis at the \n    \n    University of Marseille, while the British can highlight the work at the \n    \n    University of Edinburgh. According to people at \n    \n    MIT, logic programming was developed by these groups in an attempt to figure\n    out what Hewitt was talking about in his brilliant but impenetrable Ph.D.\n    thesis.  For a history of logic\n    programming, see\n    Robinson 1983.","4.4#footnote-2":"To see the correspondence\n\tbetween the rules and the\n\tprocedure,\n\tlet x in the\n\tprocedure\n\t(where x is nonempty) correspond to\n\t(cons u v)\n\tin the rule.  Then z in the rule corresponds\n\tto the append of\n\t(cdr x)\n\tand y.","4.4#footnote-3":"This certainly does not\n    relieve the user of the entire problem of how to compute the answer. There\n    are many different mathematically equivalent sets of rules for formulating\n    the append relation, only some of which can be\n    turned into effective devices for computing in any direction.  In addition,\n    sometimes \"what is\" information gives no clue\n    \"how to\" compute an answer.  For example, consider the problem\n    of computing the $y$ such that\n    $y^2 = x$.","4.4#footnote-4":"Interest in logic programming peaked\n    \n    during the early 1980s when the Japanese government began an ambitious\n    project aimed at building superfast computers optimized to run logic\n    programming languages.  The speed of such computers was to be measured\n    in LIPS (Logical Inferences Per Second) rather than the usual FLOPS\n    (FLoating-point Operations Per Second).  Although the project\n    succeeded in developing hardware and software as originally planned,\n    the international computer industry moved in a different direction.\n    See \n    Feigenbaum and Shrobe 1993 for an overview evaluation\n    of the Japanese project.  The logic programming community has also moved on\n    to consider relational programming based on techniques other than\n    simple pattern matching, such as the ability to deal with numerical\n    constraints such as the ones illustrated in the constraint-propagation\n    system of section 3.3.5.","4.4.1":"4.4.1  \n    Deductive Information Retrieval","4.4.1#p1":"\n    Logic programming excels in providing interfaces to\n    \n    data bases for\n    information retrieval.  The query language we shall implement in this\n    chapter is designed to be used in this way.\n  ","4.4.1#p2":"\n    In order to illustrate what the query system does, we will show how it\n    can be used to manage the data base of personnel records for\n    \n    Gargle, a thriving high-technology company in the\n    Boston area.  The language provides pattern-directed access to\n    personnel information and can also take advantage of general rules in\n    order to make logical deductions.\n  ","4.4.1#h1":"A sample data base","4.4.1#p3":"\n    The personnel data base for Gargle\n    contains\n    assertions about company personnel.  Here is the\n    information about Ben Bitdiddle, the resident computer wizard:\n    (address (Bitdiddle Ben) (Slumerville (Ridge Road) 10))\n(job (Bitdiddle Ben) (computer wizard))\n(salary (Bitdiddle Ben) 60000) \n\tEach assertion is a list (in this case a triple) whose elements can\n\tthemselves be lists.\n      ","4.4.1#p4":"\n    As resident wizard, Ben is in charge of the company's computer\n    division, and he supervises two programmers and one technician.  Here\n    is the information about them:\n    (address (Hacker Alyssa P) (Cambridge (Mass Ave) 78))\n(job (Hacker Alyssa P) (computer programmer))\n(salary (Hacker Alyssa P) 40000)\n(supervisor (Hacker Alyssa P) (Bitdiddle Ben))\n\n(address (Fect Cy D) (Cambridge (Ames Street) 3))\n(job (Fect Cy D) (computer programmer))\n(salary (Fect Cy D) 35000)\n(supervisor (Fect Cy D) (Bitdiddle Ben))\n\n(address (Tweakit Lem E) (Boston (Bay State Road) 22))\n(job (Tweakit Lem E) (computer technician))\n(salary (Tweakit Lem E) 25000)\n(supervisor (Tweakit Lem E) (Bitdiddle Ben)) There is also a programmer trainee, who is supervised by Alyssa:\n    (address (Reasoner Louis) (Slumerville (Pine Tree Road) 80))\n(job (Reasoner Louis) (computer programmer trainee))\n(salary (Reasoner Louis) 30000)\n(supervisor (Reasoner Louis) (Hacker Alyssa P)) \n    All these people are in the computer division, as indicated by the\n    word\n    computer\n    as the first item in their job\n    descriptions.\n  ","4.4.1#p5":"\n    Ben is a high-level employee.  His supervisor is the company's big\n    wheel himself:\n    (supervisor (Bitdiddle Ben) (Warbucks Oliver))\n\n(address (Warbucks Oliver) (Swellesley (Top Heap Road)))\n(job (Warbucks Oliver) (administration big wheel))\n(salary (Warbucks Oliver) 150000) ","4.4.1#p6":"\n    Besides the computer division supervised by Ben, the company has an\n    accounting division, consisting of a chief accountant and his\n    assistant:\n    (address (Scrooge Eben) (Weston (Shady Lane) 10))\n(job (Scrooge Eben) (accounting chief accountant))\n(salary (Scrooge Eben) 75000)\n(supervisor (Scrooge Eben) (Warbucks Oliver))\n\n(address (Cratchet Robert) (Allston (N Harvard Street) 16))\n(job (Cratchet Robert) (accounting scrivener))\n(salary (Cratchet Robert) 18000)\n(supervisor (Cratchet Robert) (Scrooge Eben)) \n    There is also\n    \n\ta secretary\n      \n    for the big wheel:\n    (address (Aull DeWitt) (Slumerville (Onion Square) 5))\n(job (Aull DeWitt) (administration secretary))\n(salary (Aull DeWitt) 25000)\n(supervisor (Aull DeWitt) (Warbucks Oliver)) ","4.4.1#p7":"\n    The data base also contains assertions about which kinds of jobs can\n    be done by people holding other kinds of jobs.  For instance, a\n    computer wizard can do the jobs of both a computer programmer and a\n    computer technician:\n    (can-do-job (computer wizard) (computer programmer))\n(can-do-job (computer wizard) (computer technician)) \n    A computer programmer could fill in for a trainee:\n    (can-do-job (computer programmer)\n            (computer programmer trainee)) \n    Also, as is well known,\n    (can-do-job (administration secretary)\n            (administration big wheel)) ","4.4.1#h2":"Simple queries","4.4.1#p8":"\n    The query language allows users to retrieve information from the data\n    base by posing queries in response to the system's prompt.\n    For example, to find all computer programmers one can say\n    (job ?x (computer programmer)) \n    The system will respond with the following items:\n    ","4.4.1#p9":"\n    The input query specifies that we are looking for entries in the data\n    base that match a certain\n    pattern.\n    \n\tIn this example, the pattern\n\tspecifies entries consisting of three items, of which the first is the\n\tliteral symbol job, the second can be\n\tanything, and the third is the literal list\n\t(computer programmer).\n\tThe \"anything\" that can be the second item in the matching\n\tlist is specified by a \n\tpattern variable,\n\t    ?x.\n\tThe general form of a pattern variable is a symbol, taken to be the name\n\tof the variable, preceded by a question mark. We will see below why it\n\tis useful to specify names for pattern variables rather than just putting\n\t?\n\tinto patterns to represent \"anything.\"\n    The system responds to a simple query by showing all entries in the data\n    base that match the specified pattern.\n  ","4.4.1#p10":"\n    A pattern can have more than one variable.  For example, the query\n    (address ?x ?y) \n    will list all the employees' addresses.\n  ","4.4.1#p11":"\n    A pattern can have no variables, in which case the query simply\n    determines whether that pattern is an entry in the data base.  If so,\n    there will be one match; if not, there will be no matches.\n  ","4.4.1#p12":"\n    The same pattern variable can appear more than once in a query,\n    specifying that the same \"anything\" must appear in each\n    position. This is why variables have names.  For example,\n    (supervisor ?x ?x) \n    finds all people who supervise themselves (though there are no\n    such assertions in our sample data base).\n  ","4.4.1#p13":"\n    The query\n    (job ?x (computer ?type)) \n    matches all job entries whose\n    \n\tthird\n      \n    item is a two-element list whose\n    first item is\n    computer:\n      (job (Bitdiddle Ben) (computer wizard))\n(job (Hacker Alyssa P) (computer programmer))\n(job (Fect Cy D) (computer programmer))\n(job (Tweakit Lem E) (computer technician))\n\tThis same pattern does not match\n\t(job (Reasoner Louis) (computer programmer trainee))\n\tbecause the third item in the entry is a list of three elements, and\n\tthe pattern's third item specifies that there should be two\n\telements. If we wanted to change the pattern so that the third item\n\tcould be any\n\t\n\tlist beginning with\n\tcomputer,\n\twe could\n\tspecify(job ?x (computer . ?type)) \n    For example,\n    (computer . ?type)\n    matches the data\n    (computer programmer trainee)\n    with\n    ?type\n    as \n    the list (programmer trainee).\n    It also matches the data\n    (computer programmer)\n    with\n    ?type\n    as \n    the list (programmer),\n    and matches the data\n    (computer)\n    with\n    ?type\n    as the empty \n    list ().","4.4.1#footnote-link-1":"1","4.4.1#p14":"\n    We can describe the query language's processing of simple queries as\n    follows:\n    \n\tThe system finds all assignments to variables in the query\n\tpattern that\n\tsatisfy the pattern—that is, all sets\n\tof values for the variables such that if the pattern variables are \n\tinstantiated with (replaced by) the values, the result is\n\tin the data base.\n      \n\tThe system responds to the query by listing all instantiations of the\n\tquery pattern with the variable assignments that satisfy it.\n      \n    Note that if the pattern has no variables, the query reduces to a\n    determination of whether that pattern is in the data base.  If so, the\n    empty assignment, which assigns no values to variables, satisfies that\n    pattern for that data base.\n  ","4.4.1#ex-4.56":"\n    Give simple queries that retrieve the following information from the\n    data base:\n    \n\tall people supervised by Ben Bitdiddle;\n      \n\tthe names and jobs of all people in the accounting division;\n      \n\tthe names and addresses of all people who live \n\tin Slumerville.\n      ","4.4.1#h3":"Compound queries","4.4.1#p15":"\n    Simple queries form the primitive operations of the query language.\n    In order to form compound operations, the query language provides\n    means of combination.  One thing that makes the query language a logic\n    programming language is that the means of combination mirror the means\n    of combination used in forming logical expressions:\n    and, or, and\n    not.\n    \n\t(Here and,\tor,\n\tand not are not the Lisp primitives, but\n\trather operations built into the query language.)\n      ","4.4.1#p16":"\n    We can use\n    and as follows to find the addresses\n    of all the computer programmers:\n    (and (job ?person (computer programmer))\n     (address ?person ?where)) \n    The resulting output is\n    (and (job (Hacker Alyssa P) (computer programmer))\n     (address (Hacker Alyssa P) (Cambridge (Mass Ave) 78)))\n\n(and (job (Fect Cy D) (computer programmer))\n     (address (Fect Cy D) (Cambridge (Ames Street) 3)))\n\tIn general,\n\t\n(and $\\langle \\textit{query}_{1}\\rangle$ $\\langle \\textit{query}_{2} \\rangle$ $\\ldots$ $\\langle \\textit{query}_{n} \\rangle$)\n\t  \n\tis\n\t\n\tsatisfied by all sets of values for the pattern variables that\n\tsimultaneously satisfy\n\t$\\langle\\textit{query}_{1}\\rangle\\ldots \\langle\\textit{query}_{n}\\rangle$.\n      ","4.4.1#p17":"\n    As for simple queries, the system processes a compound query by\n    finding all assignments to the pattern variables that satisfy the\n    query, then displaying instantiations of the query with those values.\n  ","4.4.1#p18":"\n    Another means of constructing compound queries is through\n    or. For example,\n    (or (supervisor ?x (Bitdiddle Ben))\n    (supervisor ?x (Hacker Alyssa P))) \n    will find all employees supervised by Ben Bitdiddle or Alyssa P.\n    Hacker:\n    (or (supervisor (Hacker Alyssa P) (Bitdiddle Ben))\n    (supervisor (Hacker Alyssa P) (Hacker Alyssa P)))\n\n(or (supervisor (Fect Cy D) (Bitdiddle Ben))\n    (supervisor (Fect Cy D) (Hacker Alyssa P)))\n\n(or (supervisor (Tweakit Lem E) (Bitdiddle Ben))\n    (supervisor (Tweakit Lem E) (Hacker Alyssa P)))\n\n(or (supervisor (Reasoner Louis) (Bitdiddle Ben))\n    (supervisor (Reasoner Louis) (Hacker Alyssa P)))\n\tIn general,\n\t\n(or $\\langle \\textit{query}_{1}\\rangle$ $\\langle \\textit{query}_{2}\\rangle$ $\\ldots$ $\\langle \\textit{query}_{n}\\rangle$)\n\t  \n\tis satisfied by all sets of values for the pattern variables that\n\tsatisfy at least one of\n\t$\\langle \\textit{query}_{1}\\rangle \\ldots \\langle \\textit{query}_{n}\\rangle$.\n      ","4.4.1#p19":"\n    Compound queries can also be formed with\n    not.\n    For example,\n    (and (supervisor ?x (Bitdiddle Ben))\n     (not (job ?x (computer programmer)))) \n    finds all people supervised by Ben Bitdiddle who are not computer\n    programmers.  In general,\n    \n(not $\\langle \\textit{query}_{1}\\rangle$)\n\t  \n    is satisfied by all assignments to the pattern variables that do not\n    satisfy\n    $\\langle \\textit{query}_{1} \\rangle$.","4.4.1#footnote-link-2":"2","4.4.1#p20":"\n\tThe final combining form is called\n\tlisp-value. When\n\tlisp-value is the first element of a\n\tpattern, it specifies that the next element is a Lisp predicate to be\n\tapplied to the rest of the (instantiated) elements as arguments.\n\tIn general,\n\t\n(lisp-value $\\langle \\textit{predicate}\\rangle$ $\\langle \\textit{arg}_{1}\\rangle$ $\\ldots$ $\\langle \\textit{arg}_{n} \\rangle$)\n\t  \n\twill be satisfied by assignments to the pattern variables for which the\n\t$\\langle \\textit{predicate} \\rangle$ applied\n\tto the instantiated\n\t$\\langle \\textit{arg}_{1} \\rangle, \\ldots, \\langle \\textit{arg}_{n}\\rangle$\n\tis true.  For example, to find all people whose salary is greater than\n\t$30,000 we could write(and (salary ?person ?amount)\n     (lisp-value > ?amount 30000))","4.4.1#footnote-link-3":"3","4.4.1#ex-4.57":"\n    Formulate compound queries that retrieve the following information:\n    \n\tthe names of all people who are supervised by Ben Bitdiddle, together\n\twith their addresses;\n      \n\tall people whose salary is less than Ben Bitdiddle's, together\n\twith their salary and Ben Bitdiddle's salary;\n      \n\tall people who are supervised by someone who is not in the computer\n\tdivision, together with the supervisor's name and job.\n      ","4.4.1#h4":"Rules","4.4.1#p21":"\n    In addition to primitive queries and compound queries, the query\n    language provides means for\n    \n    abstracting queries.  These are given by\n    rules.  The rule\n    (rule (lives-near ?person-1 ?person-2)\n      (and (address ?person-1 (?town . ?rest-1))\n           (address ?person-2 (?town . ?rest-2))\n           (not (same ?person-1 ?person-2)))) \n    specifies that two people live near each other if they live in the\n    same town.  The final not clause prevents the\n    rule from saying that all people live near themselves.  The\n    same relation is defined by a very simple\n    rule:(rule (same ?x ?x)) ","4.4.1#footnote-link-4":"4","4.4.1#p22":"\n    The following rule declares that a person is a \"wheel\" in an\n    organization if he supervises someone who is in turn a supervisor:\n    (rule (wheel ?person)\n      (and (supervisor ?middle-manager ?person)\n           (supervisor ?x ?middle-manager))) ","4.4.1#p23":"\n\tThe general form of a rule is\n\t\n(rule $\\langle \\textit{conclusion} \\rangle$ $\\langle \\textit{body} \\rangle$)\n\t  \n\twhere $\\langle \\textit{conclusion}\\rangle$ is\n\ta pattern and $\\langle \\textit{body} \\rangle$\n\tis any query.\n    We can think of a rule as representing a large (even\n    infinite) set of assertions, namely all instantiations of the rule conclusion\n    with variable assignments that satisfy the rule body.  When we described\n    simple queries (patterns), we said that an assignment to variables satisfies\n    a pattern if the instantiated pattern is in the data base.  But the pattern\n    needn't be explicitly in the data base as an assertion.  It\n    can be an\n    \n    implicit assertion implied by a rule.  For example, the\n    query(lives-near ?x (Bitdiddle Ben)) \n    results in\n    (lives-near (Reasoner Louis) (Bitdiddle Ben))\n(lives-near (Aull DeWitt) (Bitdiddle Ben))\n    To find all computer programmers who live near Ben Bitdiddle, we can\n    ask\n    (and (job ?x (computer programmer))\n     (lives-near ?x (Bitdiddle Ben))) ","4.4.1#footnote-link-5":"5","4.4.1#p24":"\n    As in the case of compound\n    procedures,\n    rules can be used as parts of other rules (as we saw with the\n    lives-near\n    rule above) or even be defined\n    \n    recursively.  For instance, the rule\n    (rule (outranked-by ?staff-person ?boss)\n      (or (supervisor ?staff-person ?boss)\n          (and (supervisor ?staff-person ?middle-manager)\n               (outranked-by ?middle-manager ?boss)))) \n    says that a staff person is outranked by a boss in the organization if\n    the boss is the person's supervisor or (recursively) if the\n    \n    person's supervisor is outranked by the boss.\n  ","4.4.1#ex-4.58":"\n    Define a rule that says that person 1 can replace person 2 if either\n    person 1 does the same job as person 2 or someone who does person 1's\n    job can also do person 2's job, and if person 1 and person 2\n    are not the same person. Using your rule, give queries that find the\n    following:\n    \n\tall people who can replace Cy D. Fect;\n      \n\tall people who can replace someone who is being paid more than they\n\tare, together with the two salaries.\n      ","4.4.1#ex-4.59":"\n    Define a rule that says that a person is a \"big shot\" in a\n    division if the person works in the division but does not have a supervisor\n    who works in the division.\n    ","4.4.1#ex-4.60":"\n    Ben Bitdiddle has missed one meeting too many. Fearing that his habit of\n    forgetting meetings could cost him his job, Ben decides to do something about\n    it.  He adds all the weekly meetings of the firm to the Gargle data base\n    by asserting the following:\n    (meeting accounting (Monday 9am))\n(meeting administration (Monday 10am))\n(meeting computer (Wednesday 3pm))\n(meeting administration (Friday 1pm))Each of the above assertions is for a meeting of an entire division.\n    Ben also adds an entry for the company-wide meeting that spans all the\n    divisions.  All of the company's employees attend this meeting.\n    (meeting whole-company (Wednesday 4pm))\n\tOn Friday morning, Ben wants to query the data base for all the meetings\n\tthat occur that day.  What query should he use?\n      \n\tAlyssa P. Hacker is unimpressed.  She thinks it would be much more\n\tuseful to be able to ask for her meetings by specifying her name.  So\n\tshe designs a rule that says that a person's meetings include all\n\twhole-company\n\tmeetings plus all meetings of that person's division.\n\tFill in the body of Alyssa's rule.\n\t\n(rule (meeting-time ?person ?day-and-time)\n      rule-body)\n              \n\tAlyssa arrives at work on Wednesday morning and wonders what meetings she\n\thas to attend that day.  Having defined the above rule, what query should\n\tshe make to find this out?\n      ","4.4.1#ex-4.61":"\n    By giving the query\n    (lives-near ?person (Hacker Alyssa P))\n    Alyssa P. Hacker is able to find people who live near her, with whom\n    she can ride to work.  On the other hand, when she tries to find all\n    pairs of people who live near each other by querying\n    (lives-near ?person-1 ?person-2)\n    she notices that each pair of people who live near each other is\n    listed twice; for example,\n    (lives-near (Hacker Alyssa P) (Fect Cy D))\n(lives-near (Fect Cy D) (Hacker Alyssa P))\n    Why does this happen?\n    Is there a way to find a list of people who live near each other, in\n    which each pair appears only once?  Explain.\n    ","4.4.1#h5":"Logic as programs","4.4.1#p25":"\n    We can regard a rule as a kind of logical implication: If an\n    assignment of values to pattern variables satisfies the body, \n    then it satisfies the conclusion.  Consequently, we can regard the\n    query language as having the ability to perform logical\n    deductions based upon the rules.  As an example, consider the\n    append operation described at the beginning of\n    section 4.4.  As we said,\n    append can be characterized by the following\n    two rules:\n    \n\tFor any list y, the empty list and\n\tyappend to\n\tform y.\n      \n\tFor any u, v,\n\ty, and z,\n        (cons u v)\n\tand yappend\n\tto form\n\t(cons u z)\n\tif v and yappend to form\n\tz.\n      ","4.4.1#p26":"\n    To express this in our query language, we define two rules for a relation\n    (append-to-form x y z)\n    which we can interpret to mean \"x and\n    yappend to\n    form z\":\n    (rule (append-to-form () ?y ?y))\n\n(rule (append-to-form (?u . ?v) ?y (?u . ?z))\n      (append-to-form ?v ?y ?z)) \n    The first rule has\n    \n    no body, which means that the conclusion holds for\n    any value of\n    ?y.\n    Note how the second rule makes use of\n    \n\tdotted-tail notation to name the\n      car\n    and\n    cdr\n    of a list.\n  ","4.4.1#p27":"\n    Given these two rules, we can formulate queries that compute the\n    append of two lists:\n    (append-to-form (a b) (c d) ?z) \n    What is more striking, we can use the same rules to ask the question\n    \"\n\t  Which list, when appended to\n\t  (a b), yields\n\t  (a b c d)?\"\n    This is done as follows:\n    (append-to-form (a b) ?y (a b c d)) \n    We can\n    \n\talso\n      \n    ask for all pairs of lists that\n    append to form\n    (a b c d):(append-to-form ?x ?y (a b c d)) ","4.4.1#p28":"\n    The query system may seem to exhibit quite a bit of intelligence in\n    using the rules to deduce the answers to the queries above.  Actually,\n    as we will see in the next section, the system is following a\n    well-determined algorithm in unraveling the rules.  Unfortunately,\n    although the system works impressively in the\n    append case, the general methods may break down\n    in more complex cases, as we will see\n    in section 4.4.3.\n  ","4.4.1#ex-4.62":"\n    The following rules implement a\n    next-to\n    relation that finds adjacent elements of a list:\n    (rule (?x next-to ?y in (?x ?y . ?u)))\n\n(rule (?x next-to ?y in (?v . ?z))\n      (?x next-to ?y in ?z))What will the response be to the following queries?\n    (?x next-to ?y in (1 (2 3) 4))\n\n(?x next-to 1 in (2 1 3 1))","4.4.1#ex-4.63":"\n    Define rules to implement the\n    last-pair\n    operation of exercise 2.17,\n    which returns a list\n    containing the last element of a nonempty list.  Check your rules on\n    \n\tqueries such as\n\t(last-pair (3) ?x),\n\t(last-pair (1 2 3) ?x),\n\tand\n\t(last-pair (2 ?x) (3)).\n      \n    Do your rules work correctly on queries such as\n    (last-pair ?x (3))?","4.4.1#ex-4.64":"\n    The following data base (see Genesis 4) traces the genealogy of the\n    descendants of\n    \n    Ada back to Adam, by way of Cain:\n\n    (son Adam Cain)\n(son Cain Enoch)\n(son Enoch Irad)\n(son Irad Mehujael)\n(son Mehujael Methushael)\n(son Methushael Lamech)\n(wife Lamech Ada)\n(son Ada Jabal)\n(son Ada Jubal)\n    Formulate rules such as \"If S is the son of F, and\n    F is the son of G, then S is the grandson of\n    G\" and \"If W is the wife of M, and\n    S is the son of W, then S is the son of\n    M\" (which was supposedly more true in biblical times than\n    today) that will enable the query system to find the grandson of Cain; the\n    sons of Lamech; the grandsons of Methushael.\n    (See\n    \n\texercise 4.70\n    for some rules to deduce more complicated relationships.)\n    ","4.4.1#footnote-1":"This uses the dotted-tail\n\tnotation introduced in\n\texercise 2.20.","4.4.1#footnote-2":"Actually, this description of\n    not is valid only for simple cases. The real\n    behavior of not is more complex.  We will\n    examine not's  peculiarities\n    in sections 4.4.2\n    and 4.4.3.","4.4.1#footnote-3":"Lisp-value\n\tshould be used only to perform an operation not\n\tprovided in the query language.  In particular, it should not\n\tbe used to\n\t\n\ttest equality (since that is what the matching in the\n\tquery language is designed to do) or inequality (since that can\n\tbe done with the same rule shown\n\tbelow).","4.4.1#footnote-4":"Notice that we do not need same\n    in order to make two things be the same: We just use the same pattern\n    variable for each—in effect, we have one thing instead of two things\n    in the first place.  For example, see\n    ?town\n    in the\n    lives-near\n    rule and\n    ?middle-manager\n    in the\n    wheel\n    rule below.\n    Same\n    is useful when we want to force two things to be\n    different, such as\n    ?person-1\n    and\n    ?person-2\n    in the\n    lives-near\n    rule.  Although using the same pattern variable in two\n    parts of a query forces the same value to appear in both places, using\n    different pattern variables does not force different values to appear.\n    (The values assigned to different pattern variables may be the same or\n    different.)","4.4.1#footnote-5":"We will also allow\n\t\n\trules without bodies, as in\n\tsame, and we will interpret such a rule to\n\tmean that the rule conclusion is satisfied by any values of the\n\tvariables.","4.4.2":"4.4.2  \n    How the Query System Works","4.4.2#p1":"\n    In section 4.4.4 we will\n    present an implementation of the query interpreter as a collection of\n    procedures.\n    In this section we give an overview that explains the general\n    structure of the system independent of low-level implementation\n    details.  After describing the implementation of the interpreter, we\n    will be in a position to understand some of its limitations and some\n    of the subtle ways in which the query language's logical operations\n    differ from the operations of mathematical logic.\n  ","4.4.2#p2":"\n    It should be apparent that the query evaluator must perform some kind\n    of search in order to match queries against facts and rules in the\n    data base.  One way to do this would be to implement the query system\n    as a nondeterministic program, using the amb\n    evaluator of section 4.3\n    (see exercise 4.79).  Another possibility\n    is to manage the search with the aid of streams.  Our implementation follows\n    this second approach.\n  ","4.4.2#p3":"\n    The query system is organized around two central operations, called\n    pattern matching and unification.  We first describe\n    pattern matching and explain how this operation, together with the\n    organization of information in terms of streams of frames, enables us\n    to implement both simple and compound queries.  We next discuss\n    unification, a generalization of pattern matching needed to implement\n    rules.  Finally, we show how the entire query interpreter fits\n    together through a\n    procedure\n    that classifies\n    expressions\n    in a manner analogous to the way\n    eval\n    classifies expressions for the interpreter described in\n    section 4.1.\n  ","4.4.2#h1":"Pattern matching","4.4.2#p4":"\n    A pattern matcher is a program that tests whether some datum\n    fits a specified pattern.  For example, the \n    \n\tdata list\n\t((a b) c (a b))\n    matches the pattern\n    (?x c ?x)\n    with the pattern variable\n    ?x\n    bound to\n    (a b).\n    The same\n    \n\tdata list\n      \n    matches the pattern\n    (?x ?y ?z)\n    with\n    ?x\n    and\n    ?z\n    both bound to\n    (a b)\n    and\n    ?y\n    bound to\n    c.\n    It also matches the pattern\n    ((?x ?y) c (?x ?y))\n    with\n    ?x\n    bound to\n    a\n    and\n    ?y\n    bound to\n    b.\n    However, it does not match the pattern\n    (?x a ?y),\n    since that pattern specifies a list whose second element is the\n    symbol a.","4.4.2#p5":"\n    The pattern matcher used by the query system takes as inputs a\n    pattern, a datum, and a\n    frame that specifies bindings for\n    various pattern variables.  It checks whether the datum matches the\n    pattern in a way that is consistent with the bindings already in the\n    frame.  If so, it returns the given frame augmented by any bindings\n    that may have been determined by the match.  Otherwise, it indicates\n    that the match has failed.\n  ","4.4.2#p6":"\n\tFor example, using the pattern\n      (?x ?y ?x)\n    to match\n    (a b a)\n\tgiven an empty frame\n      \n    will return a frame specifying that\n    ?x\n    is bound to\n    a\n    and\n    ?y\n    is bound to\n    b.\n    Trying the match with the same pattern, the same datum, and a frame\n    specifying that\n    ?y\n    is bound to\n    a\n    will fail.  Trying the match with the same pattern, the same datum, and a\n    frame in which\n    ?y\n    is bound to\n    b\n    and\n    ?x\n    is unbound will return the given frame augmented by a binding of\n    ?x\n    to a.","4.4.2#p7":"\n    The pattern matcher is all the mechanism that is needed to process\n    \n    simple\n    queries that don't involve rules.  For instance, to process the query\n    (job ?x (computer programmer))\n    we scan through all assertions in the data base and select those that\n    match the pattern with respect to an initially empty frame.  For each\n    match we find, we use the frame returned by the match to instantiate\n    the pattern with a value for\n    ?x.","4.4.2#h2":"Streams of frames","4.4.2#p8":"\n    The testing of patterns against frames is organized through the use of\n    \n    streams.  Given a single frame, the matching process runs through the\n    data-base entries one by one.  For each data-base entry, the matcher\n    generates either a special symbol indicating that the match has failed\n    or an extension to the frame.  The results for all the data-base\n    entries are collected into a stream, which is passed through a filter\n    to weed out the failures.  The result is a stream of all the frames\n    that extend the given frame via a match to some assertion in the data\n    base.","4.4.2#footnote-link-1":"1","4.4.2#p9":"\n    In our system, a query takes an input stream of frames and performs\n    the above matching operation for every frame in the stream, as\n    indicated in\n    \n\tfigure .\n      \n    That is, for\n    each frame in the input stream, the query generates a new stream consisting\n    of all extensions to that frame by matches to assertions in the data base.\n    All these streams are then combined to form one huge stream, which contains\n    all possible extensions of every frame in the input stream. This stream is\n    the output of the query.\n    ","4.4.2#fig-":"","4.4.2#p10":"\n    To answer a\n    \n    simple query, we use the query with an input stream\n    consisting of a single empty frame.  The resulting output stream\n    contains all extensions to the empty frame (that is, all answers to\n    our query).  This stream of frames is then used to generate a stream\n    of copies of the original query pattern with the variables\n    instantiated by the values in each frame, and this is the stream that\n    is finally printed.\n  ","4.4.2#h3":"Compound queries","4.4.2#p11":"\n    The real elegance of the stream-of-frames implementation is evident\n    when we deal with compound queries.  The processing of compound\n    queries makes use of the ability of our matcher to demand that a match\n    be consistent with a specified frame.  For example, to handle the\n    and of two queries, such as\n    (and (can-do-job ?x (computer programmer trainee))\n     (job ?person ?x))\n    (informally, \"Find all people who can do the job of a computer\n    programmer trainee\"), we first find all entries that match the\n    pattern\n    (can-do-job ?x (computer programmer trainee))\n    This produces a stream of frames, each of which contains a binding for\n    ?x.\n    Then for each frame in the stream we find all entries that\n    match\n    (job ?person ?x)\n    in a way that is consistent with the given binding for\n    ?x.\n    Each such match will produce a frame containing bindings for\n    ?x\n    and\n    ?person.\n    The and of two queries can be viewed as a series\n    combination of the two component queries, as shown in\n    \n\tfigure .\n      \n    The frames that pass through the\n    first query filter are filtered and further extended by the second query.\n    ","4.4.2#p12":"\n    Figure \n    shows the analogous method for\n    computing the\n    or of two queries as a parallel\n    combination of the two component queries.  The input stream of frames is\n    extended separately by each query.  The two resulting streams are then\n    merged to produce the final output stream.\n\n    ","4.4.2#p13":"\n    Even from this high-level description, it is apparent that the\n    processing of compound queries can be slow.\n    \n    For example, since a query may produce more than one output frame for each\n    input frame, and each query in an and gets its\n    input frames from the previous query, an and\n    query could, in the worst case, have to perform a number of matches that is\n    exponential in the number of queries (see \n    exercise 4.77).\n    Though systems for handling only simple queries are quite practical, dealing\n    with complex queries is extremely difficult.","4.4.2#footnote-link-2":"2","4.4.2#footnote-link-3":"3","4.4.2#p14":"\n    From the stream-of-frames viewpoint, the\n    not of\n    some query acts as a filter that removes all frames for which the query can\n    be satisfied.  For instance, given the pattern\n    (not (job ?x (computer programmer)))\n    we attempt, for each frame in the input stream, to produce extension\n    frames that satisfy\n    (job ?x (computer programmer)).\n      \n    We remove from the input stream all frames for which such extensions exist.\n    The result is a stream consisting of only those frames in which the binding\n    for\n    ?x\n    does not satisfy\n    (job ?x (computer programmer)).\n      \n    For example, in processing the query\n    (and (supervisor ?x ?y)\n     (not (job ?x (computer programmer))))\n    the first clause will generate frames with bindings for\n    ?x\n    and\n    ?y.\n    The not clause will then filter these by\n    removing all frames in which the binding for\n    ?x\n    satisfies the restriction that\n    ?x\n    is a computer programmer.","4.4.2#footnote-link-4":"4","4.4.2#p15":"\n    The\n    lisp-value\n\tspecial\tform\n      \n    is implemented as a similar filter on frame streams.  We use each frame in\n    the stream to instantiate any variables in the pattern, then apply the\n    Lisp\n    predicate.  We remove from the input stream all frames for which the\n    predicate fails.\n  ","4.4.2#h4":"Unification","4.4.2#p16":"\n    In order to handle rules in the query language, we must be able to\n    find the rules whose conclusions match a given query pattern.  Rule\n    conclusions are like assertions except that they can contain\n    variables, so we will need a generalization of pattern\n    matching—called unification—in which both the\n    \"pattern\" and the \"datum\" may contain variables.\n  ","4.4.2#p17":"\n    A unifier takes two patterns, each containing constants and variables,\n    and determines whether it is possible to assign values to the\n    variables that will make the two patterns equal.  If so, it returns a\n    frame containing these bindings.  For example, unifying\n    (?x a ?y)\n    and\n    (?y ?z a)\n    will specify a frame in which\n    ?x,?y,\n    and\n    ?z\n    must all be bound to\n    a.\n    On the other hand, unifying\n    (?x ?y a)\n    and\n    (?x b ?y)\n    will fail, because there is no value for\n    ?y\n    that can make the two patterns equal. (For the second elements of the\n    patterns to be equal,\n    ?y\n    would have to be\n    b;\n    however, for the third elements to be equal,\n    ?y\n    would have to be\n    a.)\n    The unifier used in the query system, like the pattern matcher, takes a\n    frame as input and performs unifications that are consistent with this frame.\n  ","4.4.2#p18":"\n    The unification algorithm is the most technically difficult part of\n    the query system.  With complex patterns, performing unification may\n    seem to require deduction.\n    To unify\n    (?x ?x)\n    and\n    ((a ?y c) (a b ?z)),\n    for example,\n    the algorithm must infer that\n    ?x\n    should be\n    (a b c),?y\n    should be\n    b,\n    and\n    ?z\n    should be\n    c.\n    We may think of this process as solving a set of equations among the pattern\n    components.  In general, these are simultaneous equations, which may require\n    substantial manipulation to solve.  For example,\n    unifying\n    (?x ?x)\n    and\n    ((a ?y c) (a b ?z))\n    may be thought of as specifying the simultaneous equations\n    \n\t  \\[\\begin{array}{lll}\n\t  \\texttt{?x} & = & \\texttt{(a ?y c)} \\\\\n\t  \\texttt{?x} & = & \\texttt{(a b ?z)}\n\t  \\end{array}\\]\n\t\n    These equations imply that\n    \n\t  \\[ (a ?y c) =  (a b ?z) \\]\n\t\n    which in turn implies that\n    \n\t  \\[ \\texttt{a} =  \\texttt{a},\\ \n             \\texttt{?y} = \\texttt{b},\\ \n\t     \\texttt{c} = \\texttt{?z} \\]\n\t\n    and hence that\n    \n\t  \\[\\begin{array}{lll}\n\t  \\texttt{?x} & = & \\texttt{(a b c)}\n\t  \\end{array}\\]\n\t","4.4.2#footnote-link-5":"5","4.4.2#p19":"\n    In a successful pattern match, all pattern variables become bound, and\n    the values to which they are bound contain only constants.  This is\n    also true of all the examples of unification we have seen so far.\n    \n    In general, however, a successful unification may not completely\n    determine the variable values; some variables may remain unbound and\n    others may be bound to values that contain variables.\n  ","4.4.2#p20":"\n    Consider the unification of\n    (?x a)\n    and\n    ((b ?y) ?z).\n    We can deduce that\n    ?x$=$(b ?y)\n    and\n    a$=$?z,\n    but we cannot further solve for\n    ?x\n    or\n    ?y.\n    The unification doesn't fail, since it is certainly possible to make\n    the two patterns equal by assigning values to\n    ?x\n    and\n    ?y.\n    Since this match in no way restricts the values\n    ?y\n    can take on, no binding for\n    ?y\n    is put into the result frame. The match does, however, restrict the value of\n    ?x.\n    Whatever value\n    ?y\n    has,\n    ?x\n    must be\n    (b ?y).\n    A binding of\n    ?x\n    to the pattern\n    (b ?y)\n    is thus put into the frame.  If a value for\n    ?y\n    is later determined and added to the frame (by a pattern match or\n    unification that is required to be consistent with this frame), the\n    previously bound\n    ?x\n    will refer to this value.","4.4.2#footnote-link-6":"6","4.4.2#h5":"Applying rules","4.4.2#p21":"\n    Unification is the key to the component of the query system that makes\n    inferences from rules. To see how this is accomplished, consider\n    processing a query that involves applying a rule, such as\n    (lives-near ?x (Hacker Alyssa P))\n    To process this query, we first use the ordinary pattern-match\n    procedure\n    described above to see if there are any assertions in the data base that\n    match this pattern.  (There will not be any in this case, since our data\n    base includes no direct assertions about who lives near whom.)  The next\n    step is to attempt to unify the query pattern with the conclusion of each\n    rule.  We find that the pattern unifies with the conclusion of the rule\n    (rule (lives-near ?person-1 ?person-2)\n      (and (address ?person-1 (?town . ?rest-1))\n           (address ?person-2 (?town . ?rest-2))\n           (not (same ?person-1 ?person-2))))\n    resulting in a frame specifying that\n    ?person-2\n\tis bound to\n\t(Hacker Alyssa P)\n\tand that\n\t?x\n\tshould be bound to (have the same value as)\n\t?person-1.\n      \n    Now, relative to this frame, we evaluate the compound query given by the body\n    of the rule.  Successful matches will extend this frame by providing a\n    binding for\n    ?person-1,\n    and consequently a value for\n    ?x,\n    which we can use to instantiate the original query pattern.\n  ","4.4.2#p22":"\n    In general, the query evaluator uses the following method to apply a\n    rule when trying to establish a query pattern in a frame that\n    specifies bindings for some of the pattern variables:\n\n    \n\tUnify the query with the conclusion of the rule to form, if\n\tsuccessful, an extension of the original frame.\n      \n\tRelative to the extended frame, evaluate the query formed by\n\tthe body of the rule.\n      ","4.4.2#p23":"\n    Notice how similar this is to the method for applying a\n    procedure\n    in the\n    eval/apply\n    evaluator for\n    Lisp:\n\tBind the\n\tprocedure's\n\tparameters to its arguments to form a frame that extends the original\n\tprocedure\n\tenvironment.\n      \n\tRelative to the extended environment, evaluate the expression\n\tformed by the body of the\n\tprocedure.\n    The similarity between the two evaluators should come as no surprise.\n    Just as\n    procedure\n    definitions are the means of abstraction in\n    Lisp,\n    rule definitions are the means of abstraction in the query language.\n    In each case, we unwind the abstraction by creating appropriate\n    bindings and evaluating the rule or\n    procedure\n    body relative to these.\n  ","4.4.2#h6":"Simple queries","4.4.2#p24":"\n    We saw earlier in this section how to evaluate simple queries in the\n    absence of rules.  Now that we have seen how to apply rules, we can\n    describe how to evaluate simple queries by using both rules and\n    assertions.\n  ","4.4.2#p25":"\n    Given the query pattern and a stream of frames, we produce, for each\n    frame in the input stream, two streams:\n    a stream of extended frames obtained by matching the pattern\n      against all assertions in the data base (using the pattern matcher),\n      and\n\n      a stream of extended frames obtained by applying all\n      possible rules (using the unifier).\n    Appending these two streams produces a stream that consists of all the\n    ways that the given pattern can be satisfied consistent with the\n    original frame.  These streams (one for each frame in the input\n    stream) are now all combined to form one large stream, which therefore\n    consists of all the ways that any of the frames in the original input\n    stream can be extended to produce a match with the given pattern.\n  ","4.4.2#footnote-link-7":"7","4.4.2#h7":"The query evaluator and the driver loop","4.4.2#p26":"\n    Despite the complexity of the underlying matching operations, the\n    system is organized much like an\n    \n    evaluator for any language.  The\n    procedure\n    that coordinates the matching operations is called \n    qeval,\n    and it plays a role analogous to that of the\n    evalprocedure\n    for\n    Lisp.Qeval\n    takes as inputs a query and a stream of frames.  Its output is a stream of\n    frames, corresponding to successful matches to the query pattern, that\n    extend some frame in the input stream, as indicated in\n    \n\tfigure .\n      \n    Like\n    eval,qeval\n    classifies the different types of expressions (queries) and dispatches to an\n    appropriate\n    procedure\n    for each.  There is a\n    procedure\n    for each\n    special\n    form\n    (and, or,\n    not, and\n    lisp-value)\n      \n    and one for simple queries.\n    ","4.4.2#p27":"\n    The\n    \n    driver loop, which is analogous to the\n    driver-loopprocedure\n    for the other evaluators in this chapter, reads queries\n    from the terminal.\n    For each query, it calls\n    qeval\n    with the query and a stream that consists of a single empty frame.  This\n    will produce the stream of all possible matches (all possible extensions to\n    the empty frame).  For each frame in the resulting stream, it instantiates\n    the original query using the values of the variables found in the frame.\n    This stream of instantiated queries is then printed.","4.4.2#footnote-link-8":"8","4.4.2#p28":"\n    The driver also checks for the special command\n    assert!,\n    which signals that the input is not a query but rather an assertion or rule\n    to be added to the data base.  For instance,\n    (assert! (job (Bitdiddle Ben) (computer wizard)))\n\n(assert! (rule (wheel ?person)\n         (and (supervisor ?middle-manager ?person)\n              (supervisor ?x ?middle-manager))))","4.4.2#footnote-1":"Because matching is generally very\n    \n    expensive, we would\n    like to avoid applying the full matcher to every element of the data\n    base.  This is usually arranged by breaking up the process into a\n    fast, coarse match and the final match.  The coarse match filters the\n    data base to produce a small set of candidates for the final match.\n    With care, we can arrange our data base so that some of the work of\n    coarse matching can be done when the data base is constructed rather\n    then when we want to select the candidates.  This is called\n    indexing the data base.  There is a vast technology built around\n    data-base-indexing schemes.  Our implementation, described in\n    section 4.4.4, contains a\n    simpleminded form of such an optimization.","4.4.2#footnote-2":"But this kind\n    of exponential explosion is not common in and\n    queries because the added conditions tend to reduce rather than expand\n    the number of frames produced.","4.4.2#footnote-3":"There is a large\n    literature on data-base-management systems that is concerned with how to\n    handle complex queries efficiently.","4.4.2#footnote-4":"There is a subtle difference between this\n    filter implementation of not and the usual\n    meaning of not in mathematical logic.  See\n    section 4.4.3.","4.4.2#footnote-5":"In one-sided pattern matching,\n    all the equations that contain pattern variables are explicit and already\n    solved for the unknown (the pattern variable).","4.4.2#footnote-6":"Another way to think of unification is\n    that it generates the most general pattern that is a specialization of the\n    two input patterns.\n    \n\tThat is, the unification of\n\t(?x a)\n    and\n    ((b ?y) ?z)\n    is\n    ((b ?y) a),\n    and\n    \n    the unification of\n    (?x a ?y)\n    and\n    (?y ?z a),\n    discussed above, is\n    (a a a).\n    For our implementation, it is more convenient to think of the result\n    of unification as a frame rather than a pattern.","4.4.2#footnote-7":"Since unification is a\n      \n      generalization of matching, we could simplify the system by using the\n      unifier to produce both streams.  Treating the easy case with the\n      simple matcher, however, illustrates how matching (as opposed to\n      full-blown unification) can be useful in its own right.","4.4.2#footnote-8":"The reason we\n    use\n    \n    streams (rather than lists) of frames is that the\n    recursive application of rules can generate infinite numbers of values that\n    satisfy a query.  The delayed evaluation embodied in streams is crucial\n    here: The system will print responses one by one as they are generated,\n    regardless of whether there are a finite or infinite number of\n    responses.","4.4.3":"4.4.3  \n    Is Logic Programming Mathematical Logic?","4.4.3#p1":"\n    The means of combination used in the query language may at first seem\n    identical to the operations and,\n    or, and not of\n    mathematical logic, and the application of query-language rules is in\n    fact accomplished through a legitimate method of\n    \n    inference. This identification of the query language with\n    mathematical logic is not really valid, though, because the query language\n    provides a \n    control structure that interprets the logical statements\n    procedurally.  We can often take advantage of this control structure.\n    For example, to find all of the supervisors of programmers we could\n    formulate a query in either of two logically equivalent forms:\n    (and (job ?x (computer programmer))\n     (supervisor ?x ?y))\n    or\n    (and (supervisor ?x ?y)\n     (job ?x (computer programmer)))\n    If a company has\n    \n    many more supervisors than programmers,\n    it is better to use the first form rather than the second,\n    because the data base must be scanned for each intermediate result\n    (frame) produced by the first clause of the and.\n  ","4.4.3#footnote-link-1":"1","4.4.3#p2":"\n    The aim of logic programming is to provide the programmer with\n    techniques for decomposing a computational problem into two separate\n    problems:\n    \"what\" is to be computed, and \"how\" this\n    should be computed.  This is accomplished by selecting a subset of the\n    statements of mathematical logic that is powerful enough to be able to\n    describe anything one might want to compute, yet weak enough to have a\n    controllable procedural interpretation.  The intention here is that,\n    on the one hand, a program specified in a logic programming language\n    should be an effective program that can be carried out by a computer.\n    Control (\"how\" to compute) is effected by using the order of\n    evaluation of the language.  We should be able to arrange the order of\n    clauses and the order of subgoals within each clause so that the\n    computation is done in an order deemed to be effective and efficient.\n    At the same time, we should be able to view the result of the\n    computation (\"what\" to compute) as a simple consequence of the\n    laws of logic.\n  ","4.4.3#p3":"\n    Our query language can be regarded as just such a procedurally\n    interpretable subset of mathematical logic.  An assertion represents a\n    simple fact (an atomic proposition).  A rule represents the\n    implication that the rule conclusion holds for those cases where the\n    rule body holds.  A rule has a natural procedural interpretation: To\n    establish the conclusion of the rule, establish the body of the rule.\n    Rules, therefore, specify computations.  However, because rules can\n    also be regarded as statements of mathematical logic, we can justify any\n    \"inference\" accomplished by a logic program by asserting that\n    the same result could be obtained by working entirely within\n    mathematical logic.","4.4.3#footnote-link-2":"2","4.4.3#h1":"Infinite loops","4.4.3#p4":"\n    A consequence of the procedural interpretation of logic programs is\n    that it is possible to construct hopelessly inefficient programs for\n    solving certain problems.  An extreme case of inefficiency occurs when\n    the system falls into infinite loops in making deductions.  As a\n    simple example, suppose we are setting up a data base of famous\n    marriages, including\n    (assert! (married Minnie Mickey))\n    If we now ask\n    (married Mickey ?who)\n    we will get no response, because the system doesn't know that if\n    $A$ is married to $B$,\n    then $B$ is married to\n    $A$.  So we assert the rule\n    (assert! (rule (married ?x ?y)\n         (married ?y ?x)))\n    and again query\n    (married Mickey ?who)\n    Unfortunately, this will drive the system into an infinite loop, as\n    follows:\n    \n\t    The system finds that the married rule is\n\t    applicable; that is, the rule conclusion\n\t    (married ?x ?y)\n\t    successfully unifies with the query pattern\n\t    (married Mickey ?who)\n\t    to produce a frame in which\n\t  ?x\n\tis bound to\n\tMickey\n\tand\n\t?y      \n\tis bound to\n\t?who.\n\tSo the interpreter proceeds to evaluate the rule body\n\t(married ?y ?x)\n\tin this frame—in effect, to process the query\n\t(married ?who Mickey).\n\t    One answer appears directly as an assertion in the data\n\t    base:\n\t    (married Minnie Mickey).\n\t  \n\tThe married rule is also applicable, so the\n\tinterpreter again evaluates the rule body, which this time is equivalent\n\tto\n\t(married Mickey ?who).\n    The system is now in an infinite loop.  Indeed, whether the system\n    will find the simple answer\n    (married Minnie Mickey)\n    before it goes into the loop depends on implementation details concerning the\n    order in which the system checks the items in the data base.  This is a very\n    simple example of the kinds of loops that can occur. Collections of\n    interrelated rules can lead to loops that are much harder to anticipate, and\n    the appearance of a loop can depend on the order of clauses in an\n    and (see\n    exercise 4.65) or on low-level details\n    concerning the order in which the system processes queries.","4.4.3#footnote-link-3":"3","4.4.3#h2":"Problems with not","4.4.3#p5":"\n    Another quirk in the query system concerns\n    not.\n    Given the data base of\n    section 4.4.1, consider the\n    following two queries:\n    (and (supervisor ?x ?y)\n     (not (job ?x (computer programmer))))\n\n(and (not (job ?x (computer programmer)))\n     (supervisor ?x ?y))\n    These two queries do not produce the same result.  The first query\n    begins by finding all entries in the data base that match\n    (supervisor ?x ?y),\n    and then filters the resulting frames by removing the ones in which the\n    value of\n    ?x\n    satisfies\n    (job ?x (computer programmer)).\n      \n    The second query begins by filtering the \n    incoming frames to remove those that can satisfy\n    (job ?x (computer programmer)).\n      \n    Since the only incoming frame is empty, it checks the data base\n    \n        to see if there are any \n      \n    patterns that satisfy\n    (job ?x (computer programmer)).\n      \n    Since there generally are entries of this form, the\n    not clause filters out the empty frame and\n    returns an empty stream of frames.  Consequently, the entire compound query\n    returns an empty stream.\n  ","4.4.3#p6":"\n    The trouble is that our implementation of not\n    really is meant to serve as a filter on values for the variables.  If a\n    not clause is processed with a frame in which\n    some of the variables remain unbound (as does\n    ?x\n    in the example above), the system will produce unexpected results. Similar\n    problems occur with the use of\n    lisp-value—the\n\tLisp\n\tpredicate can't work if some of its arguments are unbound.\n      \n    See exercise 4.78.\n  ","4.4.3#p7":"\n    There is also a much more serious way in which the\n    not of the query language differs from the\n    not of mathematical logic.  In logic, we\n    interpret the statement \"not $P$\" to\n    mean that $P$ is not true.  In the query system,\n    however, \"not $P$\" means that\n    $P$ is not deducible from the knowledge in the\n    data base.  For example, given the personnel data base of\n    section 4.4.1, the system would\n    happily deduce all sorts of not statements,\n    such as that Ben Bitdiddle is not a baseball fan, that it is not raining\n    outside, and that $2 + 2$\n    is not 4. In other\n    words, the not of logic programming languages\n    reflects the so-called \n    closed world assumption that all relevant information has been\n    included in the data base.","4.4.3#footnote-link-4":"4","4.4.3#footnote-link-5":"5","4.4.3#ex-4.65":"\n    Louis Reasoner mistakenly deletes the\n    outranked-by\n    rule (section 4.4.1) from the\n    data base.  When he realizes this, he quickly reinstalls it.  Unfortunately,\n    he makes a slight change in the rule, and types it in as\n    (rule (outranked-by ?staff-person ?boss)\n      (or (supervisor ?staff-person ?boss)\n          (and (outranked-by ?middle-manager ?boss)\n               (supervisor ?staff-person ?middle-manager))))\n    Just after Louis types this information into the system, DeWitt\n    Aull comes by to find out who outranks Ben Bitdiddle. He issues\n    the query\n    (outranked-by (Bitdiddle Ben) ?who)\n\n    After answering, the system goes into an infinite loop.  Explain why.\n    ","4.4.3#ex-4.66":"\n    Cy D. Fect, looking forward to the day when he will rise in the\n    organization, gives a query to find all the wheels (using the\n    wheel rule of\n    section 4.4.1):\n    (wheel ?who)\n    To his surprise, the system responds\n    \n    Why is Oliver Warbucks listed four times?\n    ","4.4.3#ex-4.67":"\n    Ben has been\n    \n    generalizing the query system to provide statistics about the\n    company.  For example, to find the total salaries of all the computer\n    programmers one will be able to say\n    (sum ?amount\n     (and (job ?x (computer programmer))\n          (salary ?x ?amount)))\n    In general, Ben's new system allows expressions of the form\n    \n(accumulation-function variable\n                       $\\langle query$ $pattern\\rangle$)\n      \n    where\n    accumulation-function\n    can be things like sum,\n    average, or\n    maximum.\n    Ben reasons that it should be a cinch to implement this.  He will simply\n    feed the query pattern to\n    qeval.\n    This will produce a stream of frames.  He will then pass this stream through\n    a mapping function that extracts the value of the designated variable from\n    each frame in the stream and feed the resulting stream of values to the\n    accumulation function.  Just as Ben completes the implementation and is\n    about to try it out, Cy walks by, still puzzling over the\n    wheel query result in\n    exercise 4.66.  When Cy shows Ben the\n    system's response, Ben groans, \"Oh, no, my simple accumulation\n    scheme won't work!\"\n    What has Ben just realized?  Outline a method he can use to salvage the\n    situation.\n    ","4.4.3#ex-4.68":"\n    Devise a\n    \n    way to install a loop detector in the query system so as to\n    avoid the kinds of simple loops illustrated in the text and in\n    exercise 4.65.  The general idea is\n    that the system should maintain some sort of history of its current chain of\n    deductions and should not begin processing a query that it is already\n    working on.  Describe what kind of information (patterns and frames)\n    is included in this history, and how the check should be made.  (After\n    you study the details of the query-system implementation in\n    section 4.4.4, you may\n    want to modify the system to include your loop detector.)\n    ","4.4.3#ex-4.69":"\n    Define rules to implement the\n    reverse operation\n    of exercise 2.18, which returns a list containing\n    the same elements as a given list\n\t  in reverse order.\n    (Hint: Use\n    append-to-form.)\n      \n    Can your rules answer both\n    (reverse (1 2 3) ?x)\n    and (reverse ?x (1 2 3))?","4.4.3#ex-4.70":"\n\tBeginning with the data base and the rules you formulated in\n\texercise 4.64, devise a rule for adding\n\t\"greats\" to a grandson relationship. This should enable the\n\tsystem to deduce that Irad is the great-grandson of Adam, or that Jabal\n\tand Jubal are the great-great-great-great-great-grandsons of Adam.\n\t(Hint: Represent the fact about Irad, for example, as\n\t((great grandson) Adam Irad).\n\tWrite rules that determine if a list ends in the word\n\tgrandson.\n\tUse this to express a rule that allows one to derive the relationship\n\t((great .  ?rel) ?x ?y),\n\twhere ?rel is a list ending in\n\tgrandson.)\tCheck your rules on queries such\n\tas ((great grandson) ?g ?ggs) and\n\t(?relationship Adam Irad).\n      ","4.4.3#footnote-1":"That a particular method of inference is\n    legitimate is not a trivial assertion.  One must prove that if one\n    starts with true premises, only true conclusions can be derived.  The\n    method of inference represented by rule applications is \n    modus ponens,\n    the familiar method of inference that says that if A is\n    true and A implies B is true, then we may conclude that B\n    is true.","4.4.3#footnote-2":"We must qualify this statement by\n    agreeing that, in speaking of the \"inference\" accomplished\n    by a logic program, we assume that the computation terminates.\n    Unfortunately, even this qualified statement is false for our\n    implementation of the query language (and also false for programs in\n    Prolog and most other current logic programming languages) because of\n    our use of not and\n    lisp-value.\n      \n    As we will describe below, the not implemented\n    in the query language is not always consistent with the\n    not of mathematical logic, and\n    lisp-value\n    introduces additional complications.  We could implement a language\n    consistent with mathematical logic by simply removing\n    not and\n    lisp-value\n    from the language and agreeing to write programs using only simple queries,\n    and, and or.\n    However, this would greatly restrict the expressive power of the language.\n    One of the major concerns of research in logic programming was to find ways\n    to achieve more consistency with mathematical logic without unduly\n    sacrificing expressive power.","4.4.3#footnote-3":"This is\n    not a problem of the logic but one of the procedural interpretation of the\n    logic provided by our interpreter. We could write an interpreter that would\n    not fall into a loop here. For example, we could enumerate all the proofs\n    derivable from our assertions and our rules in a breadth-first rather than a\n    depth-first order.  However, such a system makes it more difficult to take\n    advantage of the order of deductions in our programs.  One attempt to\n    build sophisticated control into such a program is described in\n    de Kleer et al. 1977.  \n    Another technique, which does not lead to such serious control problems, is\n    to put in special knowledge, such as detectors for particular kinds of loops\n    (exercise 4.68).  However, there can\n    be no general scheme for reliably preventing a system from going down\n    infinite paths in performing deductions.  Imagine a diabolical rule of\n    the form \"To show $P(x)$ is true, show that\n    $P(f(x))$ is true,\" for some suitably\n    chosen function $f$.","4.4.3#footnote-4":"Consider the query\n    (not (baseball-fan (Bitdiddle Ben))).\n      \n    The system finds that\n    (baseball-fan (Bitdiddle Ben))\n    is not in the data base, so the empty frame does not satisfy the pattern and\n    is not filtered out of the initial stream of frames.  The result of the\n    query is thus the empty frame, which is used to instantiate the input query\n    to produce\n    (not (baseball-fan (Bitdiddle Ben))).","4.4.3#footnote-5":"A discussion and justification of this\n    treatment of not can be found in the article\n    \"Negation as Failure\" by\n    \n    Clark (1978).","4.4.4":"4.4.4.8   Frames and Bindings","4.4.4#p1":"\n    Section 4.4.2 described how the query\n    system works. Now we fill in the details by presenting a complete\n    implementation of the system.\n  ","4.4.4#p2":"\n      The\n      \n      driver loop for the query system repeatedly reads input expressions.\n      If the expression is a rule or assertion to be added to\n      the data base, then the information is added.  Otherwise the\n      expression is assumed to be a query.  The driver passes this query to\n      the evaluator qeval\n      together with an initial frame stream consisting of a single empty frame.\n      The result of the evaluation is a stream of frames generated by satisfying\n      the query with variable values found in the data base.  These frames are\n      used to form a new stream consisting of copies of the original query in\n      which the variables are instantiated with values supplied by the stream of\n      frames, and this final stream is\n      printed at the terminal:(define input-prompt \";;; Query input:\")\n(define output-prompt \";;; Query results:\")\n\n(define (query-driver-loop)\n  (prompt-for-input input-prompt)\n  (let ((q (query-syntax-process (read))))\n    (cond ((assertion-to-be-added? q)\n           (add-rule-or-assertion! (add-assertion-body q))\n           (newline)\n           (display \"Assertion added to data base.\")\n           (query-driver-loop))\n          (else\n           (newline)\n           (display output-prompt)\n           (display-stream\n            (stream-map\n             (lambda (frame)\n              (instantiate q\n               frame\n               (lambda (v f)\n                 (contract-question-mark v))))\n             (qeval q (singleton-stream '()))))\n  (query-driver-loop))))) \n\t  Here, as in the other evaluators in this chapter,\n\t  we use an\n\t  \n\t  abstract syntax for expressions of the query language.\n\t  The implementation of the\n\t  expression syntax, including the predicate\n\t  assertion-to-be-added?\n\t  and the selector\n\t  add-assertion-body,\n\t  is given in section 4.4.4.7.\n\t  Add-rule-or-assertion!\n\t  is defined in section 4.4.4.5.\n\t","4.4.4#p3":"\n\t    Before doing any processing on an input expression, the driver loop\n\t    transforms it syntactically into a form that makes the processing more\n\t    efficient. This involves changing the \n\t    \n\t    representation of pattern variables.  When the query is instantiated, any\n\t    variables that remain unbound are transformed back to the input\n\t    representation before being printed.  These transformations are performed\n\t    by the two procedures\n\t    query-syntax-process\n\t    and\n\t    contract-question-mark\n\t    (section 4.4.4.7).\n\t  ","4.4.4#p4":"\n          To\n          \n          instantiate an expression, we copy it, replacing any variables in\n\t  the expression by their values in a given frame.  The values are\n\t  themselves instantiated, since they could contain variables (for\n\t  example, if\n\t  ?x\n\t  in exp is bound to\n\t  ?y\n\t  as the result of unification and\n\t  ?y\n\t  is in turn bound to 5).\n\t  The action to take if a variable cannot be\n\t  instantiated is given by a\n\t  procedural\n\t  argument to\n\t  instantiate.\n\t  (define (instantiate exp frame unbound-var-handler)\n(define (copy exp)\n  (cond ((var? exp)\n         (let ((binding (binding-in-frame exp frame)))\n           (if binding\n               (copy (binding-value binding))\n               (unbound-var-handler exp frame))))\n        ((pair? exp)\n         (cons (copy (car exp)) (copy (cdr exp))))\n        (else exp)))\n  (copy exp)) \n\t  The procedures that manipulate bindings are defined in\n\t  section 4.4.4.8.\n\t  ","4.4.4#p5":"\n      The\n      qevalprocedure,\n      called by the\n      query-driver-loop,\n\t\n      is the basic evaluator of the query system.  It takes as inputs a query\n      and a stream of frames, and it returns a stream of extended frames.\n      It identifies\n      special\n      forms by a \n      \n      data-directed dispatch using get and\n      put, just as we did in implementing generic\n      operations in chapter 2.  Any query that is not identified as a\n      special\n      form is assumed to be a simple query, to be processed by\n      simple-query.\n\t(define (qeval query frame-stream)\n  (let ((qproc (get (type query) 'qeval)))\n    (if qproc\n        (qproc (contents query) frame-stream)\n        (simple-query query frame-stream)))) Type\n      and contents, defined in\n      section 4.4.4.7, implement \n      \n\t  the abstract syntax of the special forms.\n\t","4.4.4#h3":"Simple queries","4.4.4#p6":"\n      The\n      simple-queryprocedure\n      handles simple queries.  It takes as arguments a simple query (a pattern)\n      together with a stream of frames, and it returns the stream formed by\n      extending each frame by all data-base matches of the query.\n      (define (simple-query query-pattern frame-stream)\n  (stream-flatmap\n   (lambda (frame)\n     (stream-append-delayed\n      (find-assertions query-pattern frame)\n      (delay (apply-rules query-pattern frame))))\n   frame-stream)) ","4.4.4#p7":"\n      For each frame in the input stream, we use\n      find-assertions\n      (section 4.4.4.3) to match the pattern\n      against all assertions in the data base, producing a stream of extended\n      frames, and we use\n      apply-rules\n      (section 4.4.4.4) to apply\n      all possible rules, producing another stream of extended frames.\n      These two streams are combined (using\n      stream-append-delayed,\n\t\n      section 4.4.4.6) to make a stream of all\n      the ways that the given pattern can be satisfied consistent with the\n      original frame (see exercise 4.72).\n      The streams for the individual input frames are combined using\n      stream-flatmap\n      (section 4.4.4.6) to form one large stream\n      of all the ways that any of the frames in the original input stream can be\n      extended to produce a match with the given pattern.\n    ","4.4.4#h4":"Compound queries","4.4.4#p8":"And\n\t  queries are handled as illustrated in\n\t  figure \n\t  by the\n\tconjoinprocedure. Conjoin\n      takes as inputs the conjuncts and the frame stream and returns the stream\n      of extended frames.  First, conjoin processes\n      the stream of frames to find the stream of all possible frame extensions\n      that satisfy the first query in the conjunction.  Then, using this as the\n      new frame stream, it recursively applies\n      conjoin to the rest of the queries.\n      (define (conjoin conjuncts frame-stream)\n  (if (empty-conjunction? conjuncts)\n      frame-stream\n      (conjoin (rest-conjuncts conjuncts)\n               (qeval (first-conjunct conjuncts)\n                      frame-stream)))) The\n      expression(put 'and 'qeval conjoin) \n      sets up\n      qeval\n      to dispatch to conjoin when an\n      andform\n      is encountered.\n    ","4.4.4#p9":"Or\n\t  queries are handled\n\t\n      similarly, as shown in\n      \n      figure .\n\t\n      The output streams for the various disjuncts of the\n      or are computed separately and merged using\n      the\n      interleave-delayedprocedure\n      from section 4.4.4.6.\n      (See exercises 4.72\n      and 4.73.)\n      (define (disjoin disjuncts frame-stream)\n  (if (empty-disjunction? disjuncts)\n      the-empty-stream\n      (interleave-delayed\n       (qeval (first-disjunct disjuncts) frame-stream)\n       (delay (disjoin (rest-disjuncts disjuncts)\n                       frame-stream)))))\n\n(put 'or 'qeval disjoin) ","4.4.4#p10":"\n      The predicates and selectors for the\n      syntax\n      of conjuncts and disjuncts\n      are given in section 4.4.4.7.\n    ","4.4.4#h5":"Filters","4.4.4#p11":"Not\n\t  is\n\t\n      handled by the method outlined in\n      section 4.4.2.  We attempt to extend\n      each frame in the input stream to satisfy the query being negated, and we\n      include a given frame in the output stream only if it cannot be extended.\n      (define (negate operands frame-stream)\n  (stream-flatmap\n   (lambda (frame)\n     (if (stream-null? (qeval (negated-query operands)\n                              (singleton-stream frame)))\n         (singleton-stream frame)\n         the-empty-stream))\n   frame-stream))\n\n(put 'not 'qeval negate) ","4.4.4#p12":"Lisp-value\n      is a filter similar to not.\n      \n\t  Each frame in\n\t  the stream is used to instantiate the variables in the pattern, the\n\t  indicated predicate is applied, and the frames for which the predicate\n\t  returns false are filtered out of the input stream.  An error results\n\t  if there are unbound pattern variables.\n\t(define (lisp-value call frame-stream)\n  (stream-flatmap\n   (lambda (frame)\n     (if (execute\n          (instantiate\n           call\n           frame\n           (lambda (v f)\n             (error \"Unknown pat var - - LISP-VALUE\" v))))\n              (singleton-stream frame)\n              the-empty-stream))\n        frame-stream))\n\n        (put 'lisp-value 'qeval lisp-value) ","4.4.4#p13":"Execute,\n\t  which applies the predicate to the arguments, must\n\t  eval\n\t  the predicate expression to get the procedure\n\t  to apply. However, it must not evaluate the arguments, since they are\n\t  already the actual arguments, not expressions whose evaluation\n\t  (in Lisp) will produce the arguments.\n\t  Note that\n\t  execute is implemented using\n\t  eval\n\t  and apply from the\n\t  underlying Lisp system.\n\t(define (execute exp)\n  (apply (eval (predicate exp) user-initial-environment)\n         (args exp))) ","4.4.4#p14":"\n      The\n      always-true special form\n      provides for a query that is always satisfied.  It ignores its contents\n      (normally empty) and simply passes through all the frames in the input\n      stream.\n      Always-true\n\t  is used by the\n\t  rule-body\n\t  selector (section 4.4.4.7)\n\t\n      to provide bodies for rules that were defined without bodies (that is,\n      rules whose bodies are always satisfied).\n      (define (always-true ignore frame-stream) frame-stream)\n\n(put 'always-true 'qeval always-true) \n      The selectors that define the syntax of\n      not\n      and\n      lisp-value\n      are given in section 4.4.4.7.\n    ","4.4.4#p15":"Find-assertions,\n\t\n      called by\n      simple-query\n      (section 4.4.4.2), takes as input a pattern\n      and a frame. It returns a stream of frames, each extending the given one\n      by a data-base match of the given pattern.  It uses\n      fetch-assertions\n      (section 4.4.4.5) to get a stream of all the\n      assertions in the data base that should be checked for a match against the\n      pattern and the frame.  The reason for\n      fetch-assertions\n      here is that we can often apply simple tests that will eliminate many of\n      the entries in the data base from the pool of candidates for a successful\n      match. The system would still work if we eliminated\n      fetch-assertions\n      and simply checked a stream of all assertions in the data base, but\n      the computation would be less efficient because we would need to make\n      many more calls to the matcher.\n      (define (find-assertions pattern frame)\n  (stream-flatmap (lambda (datum)\n                    (check-an-assertion datum pattern frame))\n                  (fetch-assertions pattern frame))) ","4.4.4#p16":"Check-an-assertion\n      takes as arguments a data object\n      \n\t  (assertion),       \n\t\n      a pattern, and a frame and\n      returns either a one-element stream containing the extended frame or\n      the-empty-stream\n      if the match fails.\n      (define (check-an-assertion assertion query-pat query-frame)\n  (let ((match-result\n         (pattern-match query-pat assertion query-frame)))\n    (if (eq? match-result 'failed)\n        the-empty-stream\n        (singleton-stream match-result)))) \n      The basic pattern matcher returns either the\n      symbol failed\n      or an extension of the given frame.  The basic idea of the matcher is to\n      check the pattern against the data, element by element, accumulating\n      bindings for the pattern variables.  If the pattern and the data\n      object are the same, the match succeeds and we return the frame of\n      bindings accumulated so far.  Otherwise, if the pattern is a variable\n      \n      we extend the current frame by binding the variable to the data, so\n      long as this is consistent with the bindings already in the frame.  If\n      the pattern and the data are both pairs, we (recursively) match the\n      car\n      of the pattern against the\n      car\n      of the data to produce a frame; in this frame we then match the\n      cdr\n      of the pattern against the\n      cdr\n      of the data.  If none of these cases are applicable, the match fails and\n      we return the\n      symbol failed.(define (pattern-match pat dat frame)\n  (cond ((eq? frame 'failed) 'failed)\n        ((equal? pat dat) frame)\n        ((var? pat) (extend-if-consistent pat dat frame))\n        ((and (pair? pat) (pair? dat))\n         (pattern-match (cdr pat)\n                        (cdr dat)\n                        (pattern-match (car pat)\n                                       (car dat)\n                                       frame)))\n        (else 'failed))) ","4.4.4#p17":"\n      Here is the\n      procedure\n      that extends a frame by adding a new binding, if this is consistent with\n      the bindings already in the frame:\n      (define (extend-if-consistent var dat frame)\n  (let ((binding (binding-in-frame var frame)))\n    (if binding\n        (pattern-match (binding-value binding) dat frame)\n        (extend var dat frame)))) \n      If there is no binding for the variable in the frame, we simply add\n      the binding of the variable to the data.  Otherwise we match, in the\n      frame, the data against the value of the variable in the frame.  If\n      the stored value contains only constants, as it must if it was stored\n      during pattern matching by\n      extend-if-consistent,\n      then the match simply tests whether the stored and new values are the\n      same.  If so, it returns the unmodified frame; if not, it returns a failure\n      indication.  The stored value may, however, contain pattern variables\n      if it was stored during unification (see\n      section 4.4.4.4). The recursive match of the\n      stored pattern against the new data will add or check bindings for the\n      variables in this pattern.  For example, suppose we have a frame in which\n      ?x\n      is bound to\n      (f ?y)\n      and\n      ?y\n      is unbound, and we wish to augment this frame by a binding of\n      ?x\n      to\n      (f b).\n      We look up\n      ?x\n      and find that it is bound to\n      (f ?y).\n      This leads us to match\n      (f ?y)\n      against the proposed new value\n      (f b)\n      in the same frame.  Eventually this match extends the frame by adding a\n      binding of\n      ?y\n      to\n      b.?X\n      remains bound to\n      (f ?y).\n      We never modify a stored binding and we never store more than one binding\n      for a given variable.\n    ","4.4.4#p18":"\n      The\n      procedures\n      used by\n      extend-if-consistent\n      to manipulate bindings are defined in\n      section 4.4.4.8.\n    ","4.4.4#h7":"Patterns with dotted tails","4.4.4#p19":"\n\t  If a pattern contains a dot followed by a pattern variable, the\n\t  pattern variable matches the rest of the data list (rather than the\n\t  next element of the data list), just as one would expect with the\n\t  \n\t  dotted-tail notation described in\n\t  exercise 2.20. Although the pattern\n\t  matcher we have just implemented doesn't look for dots, it does\n\t  behave as we want.  This is because the Lisp\n\t  read primitive, which is used by\n\t  query-driver-loop to read the query\n\t  and represent it as a list structure, treats dots in a special way.\n\t","4.4.4#p20":"\n\t  When read sees a\n\t  \n\t  dot, instead of making\n\t  the next item be the\n\t  next element of a list (the car of a\n\t  cons whose\n\t  cdr will be the rest of the list) it\n\t  makes the next item be the cdr of the\n\t  list structure.  For example, the list structure produced by\n\t  read for the pattern\n\t  (computer ?type) could be constructed\n\t  by evaluating the expression\n\t  (cons 'computer (cons '?type '())),\n\t  and that for (computer ?type) could be\n\t  constructed by evaluating the expression\n\t  (cons 'computer '?type).\n\t","4.4.4#p21":"\n\t  Thus, as pattern-match recursively\n\t  compares cars and\n\t  cdrs of a data list and a pattern that\n\t  had a dot, it eventually matches the variable after the dot (which is\n\t  a cdr of the pattern) against a sublist\n\t  of the data list, binding the variable to that list.  For example,\n\t  matching the pattern (computer ?type)\n\t  against (computer programmer trainee)\n\t  will match ?type against the list\n\t  (programmer trainee).\n\t","4.4.4#p22":"Apply-rules\n      is the rule analog of\n      find-assertions\n      (section 4.4.4.3).  It takes as input a\n      pattern and a frame, and it forms a stream of extension frames by applying\n      rules from the data base.\n      Stream-flatmap\n      maps\n      apply-a-rule\n      down the stream of possibly applicable rules (selected by\n      fetch-rules,\n      section 4.4.4.5) and combines the resulting\n      streams of frames.\n      (define (apply-rules pattern frame)\n  (stream-flatmap (lambda (rule)\n                    (apply-a-rule rule pattern frame))\n                  (fetch-rules pattern frame))) ","4.4.4#p23":"Apply-a-rule\n\t  applies rules\n\t\n      using the method outlined in\n      section 4.4.2.  It first augments its\n      argument frame by unifying the rule conclusion with the pattern in the\n      given frame.  If this succeeds, it evaluates the rule body in this new\n      frame.\n    ","4.4.4#p24":"\n      Before any of this happens, however, the program renames all the variables\n      in the rule with unique new names.  The reason for this is to prevent the\n      variables for different rule applications from becoming confused with each\n      other.  For instance, if two rules both use a variable \n      named ?x,\n      then each one may add a binding for\n      ?x\n      to the frame when it is applied.  These two\n      ?x's\n      have nothing to do with each other, and we should not be fooled into\n      thinking that the two bindings must be consistent.  Rather than rename\n      variables, we could devise a more clever environment structure; however,\n      the renaming approach we have chosen here is the most straightforward,\n      even if not the most efficient.  (See\n      exercise 4.80.)  Here is the\n      apply-a-ruleprocedure:(define (apply-a-rule rule query-pattern query-frame)\n (let ((clean-rule (rename-variables-in rule)))\n       (let ((unify-result\n             (unify-match query-pattern\n                          (conclusion clean-rule)\n                          query-frame)))\n   (if (eq? unify-result 'failed)\n       the-empty-stream\n       (qeval (rule-body clean-rule)\n              (singleton-stream unify-result)))))) \n      The selectors\n      rule-body\n      and conclusion that extract parts\n      of a rule are defined in section 4.4.4.7.\n    ","4.4.4#p25":"\n      We generate unique variable names by associating a unique identifier\n      (such as a number) with each rule application and combining this\n      identifier with the original variable names.  For example, if the\n      rule-application identifier is 7, we might change each\n      ?x\n      in the rule to\n      ?x-7\n      and each\n      ?y\n      in the rule to\n      ?y-7.(Make-new-variable\n      and\n      new-rule-application-id\n      are included with the syntax\n      procedures\n      in section 4.4.4.7.)\n      (define (rename-variables-in rule)\n  (let ((rule-application-id (new-rule-application-id)))\n    (define (tree-walk exp)\n      (cond ((var? exp)\n             (make-new-variable exp rule-application-id))\n            ((pair? exp)\n             (cons (tree-walk (car exp))\n                   (tree-walk (cdr exp))))\n            (else exp)))\n    (tree-walk rule))) ","4.4.4#p26":"\n      The\n      \n      unification algorithm is implemented as a\n      procedure\n      that takes as inputs two patterns and a frame and returns either the\n      extended frame or the\n      symbol failed.\n      The unifier is like the pattern matcher except that it is\n      symmetrical—variables are allowed on both sides of the match.\n      Unify-match\n      is basically the same as\n      pattern-match,\n      except that there is\n      \n\t  extra code\n\t\n      (marked\n      \"***\" below) to handle\n      the case where the object on the right side of the match is a variable.\n      (define (unify-match p1 p2 frame)\n  (cond ((eq? frame 'failed) 'failed)\n        ((equal? p1 p2) frame)\n        ((var? p1) (extend-if-possible p1 p2 frame))\n        ((var? p2) (extend-if-possible p2 p1 frame))  \n        ((and (pair? p1) (pair? p2))\n         (unify-match (cdr p1)\n                      (cdr p2)\n                      (unify-match (car p1)\n                                   (car p2)\n                                   frame)))\n        (else 'failed))) ","4.4.4#p27":"\n      In unification, as in one-sided pattern matching, we want to accept a\n      proposed extension of the frame only if it is consistent with existing\n      bindings.  The\n      procedureextend-if-possible\n      used in unification is the same as the\n      extend-if-consistent\n      used in pattern matching except for two special checks, marked\n      \"***\" in the program below.  In\n      the first case, if the variable we are trying to match is not bound, but\n      the value we are trying to match it with is itself a (different) variable,\n      it is necessary to check to see if the value is bound, and if so, to match\n      its value.  If both parties to the match are unbound, we may bind either\n      to the other.\n    ","4.4.4#p28":"\n      The second check deals with attempts to bind a variable to a pattern\n      that includes that variable.  Such a situation can occur whenever a\n      variable is repeated in both patterns.  Consider, for example,\n      unifying the two patterns\n      (?x ?x)\n      and\n      (?y $\\langle expression$ $involving$?y$\\rangle$)\n      in a frame where both\n      ?x\n      and\n      ?y\n      are unbound.  First\n      ?x\n      is matched against\n      ?y,\n      making a binding of\n      ?x\n      to\n      ?y.\n      Next, the same\n      ?x\n      is matched against the given\n      expression involving\t  \n      ?y.\n\t\n      Since\n      ?x\n      is already bound to\n      ?y,\n      this results in matching\n      ?y\n      against the\n      \n\t  expression.\n\t\n      If we think of the unifier as finding a set of\n      values for the pattern variables that make the patterns the same, then\n      these patterns imply instructions to find a\n      ?y\n      such that\n      ?y\n      is equal to the expression involving\n      ?y.\n\t  There is no general method for solving such equations, so we\n\t\n      reject such\n      bindings; these cases are recognized by the predicate \n      depends-on?.\n      On the other hand, we do not want to reject attempts\n      to bind a variable to itself.  For example, consider unifying\n      (?x ?x)\n      and\n      (?y ?y).\n      The second attempt to bind\n      ?x\n      to\n      ?y\n      matches\n      ?y(the stored value of ?x)\n      against\n      ?y(the new value of ?x).).\n      \n      This is taken care of by the\n      equal?\n      clause of\n      unify-match.(define (extend-if-possible var val frame)\n  (let ((binding (binding-in-frame var frame)))\n    (cond (binding\n           (unify-match\n            (binding-value binding) val frame))\n          ((var? val)                      \n           (let ((binding (binding-in-frame val frame)))\n             (if binding\n                 (unify-match\n                  var (binding-value binding) frame)\n                 (extend var val frame))))\n          ((depends-on? val var frame)     \n           'failed)\n          (else (extend var val frame))))) ","4.4.4#footnote-link-1":"1","4.4.4#p29":"Depends-on?\n      is a predicate that tests whether an expression proposed to be the value\n      of a pattern variable depends on the variable. This must be done relative\n      to the current frame because the expression may contain occurrences of a\n      variable that already has a value that depends on our test variable.\n      The structure of\n      depends-on?\n      is a simple recursive tree walk in which we substitute for the values of\n      variables whenever necessary.\n      (define (depends-on? exp var frame)\n  (define (tree-walk e)\n    (cond ((var? e)\n           (if (equal? var e)\n                true\n                (let ((b (binding-in-frame e frame)))\n                  (if b\n                      (tree-walk (binding-value b))\n                      false))))\n          ((pair? e)\n           (or (tree-walk (car e))\n               (tree-walk (cdr e))))\n          (else false)))\n        (tree-walk exp)) ","4.4.4#p30":"\n      One important problem in designing logic programming languages is that\n      of arranging things so that as few irrelevant\n      \n      data-base entries as\n      possible will be examined in checking a given pattern. For this purpose,\n      we will represent an assertion as a list whose head is a string that\n      represents the kind of information of the assertion.\n      \n\t  Then, \n\t  in addition to storing all assertions in one big stream,\n\t  we store all assertions whose\n\t  cars\n\t  are constant symbols in separate streams, in a table indexed by the\n\t  symbol. To fetch an assertion that may match a pattern, we first\n\t  check to see if the\n\t  car\n\t  of the pattern is a constant symbol. If so, we return (to be tested\n\t  using the matcher) all the stored assertions that have the same\n\t  car.\n\t  If the pattern's\n\t  car\n\t  is not a constant symbol, we return all the stored assertions.\n\t  Cleverer methods could also take advantage of information in the\n\t  frame, or try also to optimize the case where the\n\t  car\n\t  of the pattern is not a constant symbol.\n\t  We avoid building our criteria for indexing (using the\n\t  car,\n\t  handling only the case of constant symbols) into the program;\n\t  instead we call on predicates and selectors that embody our\n\t  criteria.\n\t(define THE-ASSERTIONS the-empty-stream)\n\n(define (fetch-assertions pattern frame)\n  (if (use-index? pattern)\n      (get-indexed-assertions pattern)\n      (get-all-assertions)))\n\n(define (get-all-assertions) THE-ASSERTIONS)\n\n(define (get-indexed-assertions pattern)\n  (get-stream (index-key-of pattern) 'assertion-stream)) Get-stream\n      looks up a stream in the table and returns an empty stream if nothing is\n      stored there.\n      (define (get-stream key1 key2)\n  (let ((s (get key1 key2)))\n    (if s s the-empty-stream))) ","4.4.4#p31":"\n\t  Rules are stored similarly, using the\n\t  car\n\t  of the rule conclusion.  Rule conclusions are arbitrary patterns, however,\n\t  so they differ from assertions in that they can contain variables.\n\t  A pattern whose\n\t  car\n\t  is a constant symbol can match rules whose conclusions start with a\n\t  variable as well as rules whose conclusions have the same\n\t  car.\n\t  Thus, when fetching rules that might match a pattern whose\n\t  car is a\n\t  constant symbol we fetch all rules whose conclusions start with a\n\t  variable as well as those whose conclusions have the same\n\t  car\n\t  as the pattern.  For this purpose we store all rules whose conclusions\n\t  start with a variable in a separate stream in our table, indexed by the\n\t  symbol ?.\n\t(define THE-RULES the-empty-stream)\n\n(define (fetch-rules pattern frame)\n  (if (use-index? pattern)\n      (get-indexed-rules pattern)\n      (get-all-rules)))\n\n(define (get-all-rules) THE-RULES)\n\n(define (get-indexed-rules pattern)\n  (stream-append\n   (get-stream (index-key-of pattern) 'rule-stream)\n   (get-stream '? 'rule-stream))) ","4.4.4#p32":"Add-rule-or-assertion!\n      is used by\n      query-driver-loop\n      to add assertions and rules to the data base.  Each item is stored in the\n      index.\n      (define (add-rule-or-assertion! assertion)\n  (if (rule? assertion)\n      (add-rule! assertion)\n      (add-assertion! assertion)))\n\n(define (add-assertion! assertion)\n  (store-assertion-in-index assertion)\n  (let ((old-assertions THE-ASSERTIONS))\n    (set! THE-ASSERTIONS\n          (cons-stream assertion old-assertions))\n          'ok))\n\n(define (add-rule! rule)\n  (store-rule-in-index rule)\n  (let ((old-rules THE-RULES))\n    (set! THE-RULES (cons-stream rule old-rules))\n    'ok)) ","4.4.4#p33":"  \n      To actually store an assertion or a rule, we store it in the appropriate stream.\n      (define (store-assertion-in-index assertion)\n  (if (indexable? assertion)\n      (let ((key (index-key-of assertion)))\n        (let ((current-assertion-stream\n          (get-stream key 'assertion-stream)))\n      (put key\n           'assertion-stream\n           (cons-stream assertion\n                        current-assertion-stream))))))\n\n(define (store-rule-in-index rule)\n  (let ((pattern (conclusion rule)))\n    (if (indexable? pattern)\n        (let ((key (index-key-of pattern)))\n          (let ((current-rule-stream\n            (get-stream key 'rule-stream)))\n        (put key\n             'rule-stream\n             (cons-stream rule\n                          current-rule-stream))))))) ","4.4.4#p34":"\n\t  The following\n\t  procedures\n\t  define how the data-base index is used.  A pattern (an assertion or a rule\n\t  conclusion) will be stored in the table if it starts with a variable or a\n\t  constant symbol.\n\t  (define (indexable? pat)\n\t      (or (constant-symbol? (car pat))\n\t      (var? (car pat)))) \n\t  The key under which a pattern is stored in the table is either\n\t  ?\n\t  (if it starts with a variable) or the\n\t  constant symbol\n\t  with which it starts.\n\t(define (index-key-of pat)\n  (let ((key (car pat)))\n    (if (var? key) '? key))) \n\t  The index will be used to retrieve items that might match a pattern if\n\t  the pattern starts with a\n\tconstant symbol.\n\t(define (use-index? pat)\n  (constant-symbol? (car pat))) ","4.4.4#ex-4.71":"\n      What is the purpose of the\n      let bindings\n      in the\n      procedures\n      add-assertion!\n      and\n      add-rule!?\n      What would be wrong with the following implementation of\n      add-assertion!?\n      Hint: Recall the definition of the infinite stream of ones in\n      section 3.5.2:\n\t  (define ones (cons-stream 1 ones)).\n      (define (add-assertion! assertion)\n  (store-assertion-in-index assertion)\n  (set! THE-ASSERTIONS\n        (cons-stream assertion THE-ASSERTIONS))\n  'ok) ","4.4.4#p35":"\n      The query system uses a few stream operations that were not presented\n      in chapter 3.\n    ","4.4.4#p36":"Stream-append-delayed\n      and\n      interleave-delayed\n      are just like\n      stream-append\n      and\n      interleave\n      (section 3.5.3),\n      except that they take a delayed argument (like the\n      integralprocedure\n      in section 3.5.4).\n      This postpones looping in some cases (see\n      exercise 4.72).\n      (define (stream-append-delayed s1 delayed-s2)\n  (if (stream-null? s1)\n      (force delayed-s2)\n      (cons-stream\n       (stream-car s1)\n       (stream-append-delayed (stream-cdr s1) delayed-s2))))\n\n(define (interleave-delayed s1 delayed-s2)\n  (if (stream-null? s1)\n      (force delayed-s2)\n      (cons-stream\n       (stream-car s1)\n       (interleave-delayed (force delayed-s2)\n                           (delay (stream-cdr s1)))))) ","4.4.4#p37":"Stream-flatmap,\n      which is used throughout the query evaluator to map a\n      procedure\n      over a stream of frames and combine the resulting streams of frames,\n      is the stream analog of the flatmapprocedure\n      introduced for ordinary lists in\n      section 2.2.3. Unlike ordinary\n      flatmap, however, we accumulate the streams\n      with an interleaving process, rather than simply appending them (see\n      exercises 4.73\n      and 4.74).\n      (define (stream-flatmap proc s)\n  (flatten-stream (stream-map proc s)))\n\n(define (flatten-stream stream)\n  (if (stream-null? stream)\n      the-empty-stream\n      (interleave-delayed\n       (stream-car stream)\n       (delay (flatten-stream (stream-cdr stream)))))) ","4.4.4#p38":"\n      The evaluator also uses the following simple\n      procedure\n      to generate a stream consisting of a single element:\n      (define (singleton-stream x)\n  (cons-stream x the-empty-stream)) ","4.4.4#p39":"Type\n      and contents, used by\n      qeval\n      (section 4.4.4.2), specify that a\n      special form\n      is identified by\n      the symbol in its car.\n      They are the same as the\n      type-tag\n      and contents\n      procedures\n      in section 2.4.2, except for the\n      error message.\n      (define (type exp)\n  (if (pair? exp)\n      (car exp)\n      (error \"Unknown expression TYPE\" exp)))\n\n(define (contents exp)\n  (if (pair? exp)\n      (cdr exp)\n      (error \"Unknown expression CONTENTS\" exp))) ","4.4.4#p40":"\n      The following\n      procedures,\n      used by\n      query-driver-loop\n      (in section 4.4.4.1), specify\n      that rules and assertions are added to the data base by expressions of\n      the form\n      (assert! rule-or-assertion):\n      (define (assertion-to-be-added? exp)\n  (eq? (type exp) 'assert!))\n\n(define (add-assertion-body exp)\n  (car (contents exp))) ","4.4.4#p41":"\n      Here are the syntax definitions for the and,\n      or, not, and\n      lisp-value special forms\n      (section 4.4.4.2):\n      (define (empty-conjunction? exps) (null? exps))\n(define (first-conjunct exps) (car exps))\n(define (rest-conjuncts exps) (cdr exps))\n\n(define (empty-disjunction? exps) (null? exps))\n(define (first-disjunct exps) (car exps))\n(define (rest-disjuncts exps) (cdr exps))\n\n(define (negated-query exps) (car exps))\n\n(define (predicate exps) (car exps))\n(define (args exps) (cdr exps)) ","4.4.4#p42":"\n      The following three\n      procedures\n      define the syntax of rules:\n      (define (rule? statement)\n  (tagged-list? statement 'rule))\n\n(define (conclusion rule) (cadr rule))\n\n(define (rule-body rule)\n  (if (null? (cddr rule))\n      '(always-true)\n      (caddr rule))) ","4.4.4#p43":"Query-driver-loop\n\t  (section 4.4.4.1) calls\n\t  query-syntax-process\n\t  to transform pattern variables in the expression, which\n\t  have the form ?symbol,\n\t  into the internal format (? symbol).\n\t  That is to say, a pattern such as\n\t  (job ?x ?y) is actually represented\n\t  internally by the system as\n\t  (job (? x) (? y)). This increases the\n\t  efficiency of query processing, since it means that the system can\n\t  check to see if an expression is a pattern variable by checking\n\t  whether the car of the expression is the\n\t  symbol ?, rather than having to extract\n\t  characters from the symbol. The syntax transformation is accomplished\n\t  by the following procedure:(define (query-syntax-process exp)\n  (map-over-symbols expand-question-mark exp))\n\n(define (map-over-symbols proc exp)\n  (cond ((pair? exp)\n         (cons (map-over-symbols proc (car exp))\n               (map-over-symbols proc (cdr exp))))\n        ((symbol? exp) (proc exp))\n        (else exp)))\n\n(define (expand-question-mark symbol)\n  (let ((chars (symbol->string symbol)))\n    (if (string=? (substring chars 0 1) \"?\")\n        (list '?\n              (string->symbol\n               (substring chars 1 (string-length chars))))\n        symbol))) ","4.4.4#footnote-link-2":"2","4.4.4#p44":"\n\t  Once the variables are transformed in this way, the variables in a\n\t  pattern are lists starting with\n\t  ?,\n\t  and the constant symbols\n\t  (which need to be recognized for\n\t  data-base indexing, section 4.4.4.5) are\n\t  just the symbols.\n\t  (define (var? exp)\n\t      (tagged-list? exp '?))\n\n\t      (define (constant-symbol? exp) (symbol? exp)) ","4.4.4#p45":"\n\t  Unique variables are constructed during rule application\n\t  (in section 4.4.4.4) by means of\n\t  the following procedures.\n\t  The unique identifier for a rule application is a number, which is\n\t  incremented each time a rule is applied.\n\t  (define rule-counter 0)\n\n\t      (define (new-rule-application-id)\n\t      (set! rule-counter (+ 1 rule-counter))\n\t      rule-counter)\n\n\t      (define (make-new-variable var rule-application-id)\n\t      (cons '? (cons rule-application-id (cdr var)))) ","4.4.4#p46":"\n\t  When query-driver-loop instantiates the\n\t  query to print the answer, it converts any unbound pattern variables\n\t  back to the right form for printing, using\n\t  (define (contract-question-mark variable)\n  (string->symbol\n   (string-append \"?\" \n     (if (number? (cadr variable))\n         (string-append (symbol->string (caddr variable))\n                                     \"-\"\n                                     (number->string (cadr variable)))\n         (symbol->string (cadr variable)))))) ","4.4.4#p47":"\n      Frames are represented as lists of bindings, which are\n      variable-value pairs:\n      (define (make-binding variable value)\n  (cons variable value))\n\n(define (binding-variable binding)\n  (car binding))\n\n(define (binding-value binding)\n  (cdr binding))\n\n(define (binding-in-frame variable frame)\n  (assoc variable frame))\n\n(define (extend variable value frame)\n  (cons (make-binding variable value) frame)) ","4.4.4#ex-4.72":"\n      Louis Reasoner wonders why the\n      simple-query\n      and disjoinprocedures\n      (section 4.4.4.2) are implemented using\n      \n\t  explicit delay\n\t  operations, \n\t\n      rather than being defined as follows:\n      (define (simple-query query-pattern frame-stream)\n  (stream-flatmap\n   (lambda (frame)\n     (stream-append (find-assertions query-pattern frame)\n                    (apply-rules query-pattern frame)))\n   frame-stream))\n\n(define (disjoin disjuncts frame-stream)\n  (if (empty-disjunction? disjuncts)\n      the-empty-stream\n      (interleave\n       (qeval (first-disjunct disjuncts) frame-stream)\n       (disjoin (rest-disjuncts disjuncts) frame-stream))))\n      Can you give examples of queries where these simpler definitions would\n      lead to undesirable behavior?\n      ","4.4.4#ex-4.73":"\n      Why do disjoin and\n      stream-flatmap\n      interleave the streams rather than simply append them?  Give examples that\n      illustrate why interleaving works better.  (Hint: Why did we use\n      interleave in\n      section 3.5.3?)\n      ","4.4.4#ex-4.74":"\n    Why does\n    flatten-stream\n    use\n    delay explicitly?\n    What would be wrong with defining it as follows:\n    (define (flatten-stream stream)\n  (if (stream-null? stream)\n      the-empty-stream\n      (interleave\n       (stream-car stream)\n       (flatten-stream (stream-cdr stream)))))","4.4.4#ex-4.75":"\n      Alyssa P. Hacker proposes to use a simpler version of\n      stream-flatmap\n      in negate,\n      lisp-value,\n      and\n      find-assertions.\n      She observes that the\n      procedure\n      that is mapped over the frame stream in these cases always produces either\n      the empty stream or a singleton stream, so no interleaving is needed when\n      combining these streams.\n      \n          Fill in the missing expressions in Alyssa's program.\n          \n(define (simple-stream-flatmap proc s)\n  (simple-flatten (stream-map proc s)))\n\n(define (simple-flatten stream)\n  (stream-map ??\n              (stream-filter ?? stream)))\n\t    \n          Does the query system's behavior change if we change it in this\n\t  way?\n\t","4.4.4#ex-4.76":"\n      Implement for the query language a\n      new special form called\n\tunique.Unique\n      should succeed if there is precisely one item in the data base satisfying\n      a specified query.  For example,\n      (unique (job ?x (computer wizard)))\n      should print the one-item stream\n      (unique (job (Bitdiddle Ben) (computer wizard)))\n      since Ben is the only computer wizard, and\n      (unique (job ?x (computer programmer)))\n      should print the empty stream, since there is more than one computer\n      programmer.  Moreover,(and (job ?x ?j) (unique (job ?anyone ?j)))\n      should list all the jobs that are filled by only one person, and the\n      people who fill them.\n      \n      There are two parts to implementing unique.\n      The first is to write a\n      procedure\n      that handles this\n      special\n      form, and the second is to make\n      qeval\n      dispatch to that\n      procedure.\n      The second part is trivial, since\n      qeval\n      does its dispatching in a data-directed way.  If your\n      procedure\n      is called\n      uniquely-asserted,\n      all you need to do is\n      (put 'unique 'qeval uniquely-asserted)\n      and\n      qeval\n      will dispatch to this\n      procedure\n      for every query whose\n      type(car)\n      is the\n      symbolunique.\n      The real problem is to write the\n      procedureuniquely-asserted.\n      This should take as input the contents(cdr)\n      of the unique query, together with a stream\n      of frames.  For each frame in the stream, it should use\n      qeval\n      to find the stream of all extensions to the frame that satisfy the given\n      query.  Any stream that does not have exactly one item in it should be\n      eliminated.  The remaining streams should be passed back to be accumulated\n      into one big stream that is the result of the\n      unique query.  This is similar to the\n      implementation of the notspecial\n      form.\n      \n      Test your implementation by forming a query that lists all people who\n      supervise precisely one person.\n      ","4.4.4#ex-4.77":"\n      Our implementation of and as a series\n      combination of queries\n      \n\t  (figure )\n\t\n      is\n      elegant, but it is inefficient because in processing the second query of\n      the and we must scan the data base for each\n      frame produced by the first query.  If the data base has\n      $N$ elements, and a typical query produces a\n      number of output frames proportional to $N$\n      (say $N/k$), then scanning the data base for\n      each frame produced by the first query will require\n      $N^{2}/k$ calls to the pattern matcher.\n      Another approach would be to process the two clauses of the\n      and separately, then look for all pairs of\n      output frames that are compatible.  If each query produces\n      $N/k$ output frames, then this means that we\n      must perform $N^{2}/k^{2}$ compatibility\n      checks—a factor of $k$ fewer than the\n      number of matches required in our current  method.\n      \n      Devise an implementation of and that uses\n      this strategy.  You  must implement a\n      procedure\n      that takes two frames as inputs, checks whether the bindings in the\n      frames are compatible, and, if so, produces a frame that merges the two\n      sets of bindings.  This operation is similar to unification.\n      ","4.4.4#ex-4.78":"\n      In section 4.4.3 we saw that\n      not and\n      lisp-value\n      can cause the query language to give \"wrong\" answers if\n      these filtering operations are applied to frames in which variables\n      are unbound.  Devise a way to fix this shortcoming.  One idea is to\n      perform the filtering in a \"delayed\" manner by appending to\n      the frame a \"promise\" to filter that is fulfilled only when\n      enough variables have been bound to make the operation possible.  We could\n      wait to perform filtering until all other operations have been performed.\n      However, for efficiency's sake, we would like to perform filtering\n      as soon as possible so as to cut down on the number of intermediate frames\n      generated.\n      ","4.4.4#ex-4.79":"\n      Redesign the query language as a\n      \n      nondeterministic program to be\n      implemented using the evaluator of\n      section 4.3, rather than as a stream\n      process.  In this approach, each query will produce a single answer\n      (rather than the stream of all answers) and the user can type\n      try-again\n      to see more answers.  You should find that much of the mechanism we built\n      in this section is subsumed by nondeterministic search and backtracking.\n      You will probably also find, however, that your new query language has\n      subtle differences in behavior from the one implemented here.  Can you\n      find examples that illustrate this difference?\n      ","4.4.4#ex-4.80":"\n      When we implemented the\n      Lisp\n      evaluator in section 4.1, we saw how to use\n      local environments to avoid\n      \n      name conflicts between the parameters of\n      procedures.\n      For example, in evaluating\n      (define (square x)\n  (* x x))\n\n(define (sum-of-squares x y)\n  (+ (square x) (square y)))\n\n(sum-of-squares 3 4) \n      there is no confusion between the x in\n      square and the x\n      in\n      sum-of-squares,\n      because we evaluate the body of each\n      procedure\n      in an environment that is specially constructed to contain\n      bindings for the local\n      variables.\n      In the query system, we used a\n      different strategy to avoid name conflicts in applying rules.  Each\n      time we apply a rule we rename the variables with new names that are\n      guaranteed to be unique.  The analogous strategy for the\n      Lisp\n      evaluator would be to do away with local environments and simply\n      rename the variables in the body of a\n      procedure\n      each time we apply the\n      procedure.\n      Implement for the query language a rule-application method that uses\n      environments rather than renaming.  See if you can build on your\n      environment structure to create constructs in the query language for\n      dealing with large systems, such as the rule analog of\n      \n      block-structured\n      procedures.\n      Can you relate any of this to the problem of making deductions in a\n      context (e.g., \"If I supposed that $P$\n      were true, then I would be able to deduce $A$\n      and $B$.\") as a method of problem\n      solving?  (This problem is open-ended.)\n      ","4.4.4#footnote-1":"In general,\n      unifying\n      ?y\n\t  with an expression involving\n\t?y\n      would require our being able to find a\n      \n      fixed point of the equation\n      ?y$ = \\textit{expression involving}$?y.\n\t\n\t      It is sometimes possible to syntactically form an expression that appears\n\t      to be the solution.  For example,\n\t      ?y$=$(f ?y)\n\t  seems to have the fixed point\n\t  (f (f (f …))),\n\t    \n\t  which we can produce by beginning with the expression\n\t  (f ?y)\n\t  and repeatedly substituting\n\t  (f ?y)\n\t  for\n\t  ?y.\n\t  Unfortunately, not every such equation has a meaningful fixed point.  The\n\t  issues that arise here are similar to the issues of manipulating \n\t  \n\t  infinite series in mathematics.  For example, we know that 2 is the\n\t  solution to the equation $y = 1 + y/2$.\n\t  Beginning with the expression $1 + y/2$ and\n\t  repeatedly substituting $1 + y/2$ for\n\t  $y$ gives\n\t  \n            \\[\n\t    2 \\ = \\ y \\ = \\ 1 + y/2 \\ = \\ 1 + (1+y/2)/2 \\ = \\ 1 + 1/2 + y/4 \\ = \\ \\cdots,\n\t    \\]\n\t  \n\t  which leads to\n\t  \n            \\[ 2 \\ = \\  1 + 1/2 + 1/4 + 1/8 +\\cdots. \\]\n\t  \n\t  However, if we try the same manipulation beginning with the\n\t  observation that $-1$ is the solution to the\n\t  equation $y \\ = \\  1 + 2y$, we obtain\n\t  \n            \\[ -1 \\ = \\  y \\ = \\  1 + 2y \\ = \\  1 + 2(1 + 2y) \\ = \\  1 + 2 + 4y \\ = \\  \\cdots, \\]\n\t  \n\t  which leads to \n\t  \n            \\[ -1 \\ = \\  1 + 2 + 4 + 8 +\\cdots. \\]\n\t  \n\t  Although the formal manipulations used in deriving these two equations\n\t  are identical, the first result is a valid assertion about infinite\n\t  series but the second is not.  Similarly, for our unification results,\n\t  reasoning with an arbitrary syntactically constructed expression may\n\t  lead to errors.\n        ","4.4.4#footnote-2":"Most Lisp systems give the user\n\t  the ability to modify the ordinary read\n\t  procedure to perform such transformations by defining \n\t  reader macro characters.  Quoted\n\t  expressions are already handled in this way: The reader automatically\n\t  translates 'expression into\n\t  (quote expression) before the\n\t  evaluator sees it.  We could arrange for\n\t  ?expression to be transformed into\n\t  (? expression) in the same way; however,\n\t  for the sake of clarity we have included the transformation\n\t  procedure here explicitly.\n\t  Expand-question-mark and\n\t  contract-question-mark use several\n\t  procedures with string in their names.\n\t  These are Scheme primitives.","5#p1":"\n    We began this book by studying processes and by describing processes\n    in terms of\n    procedures\n    written in\n    Lisp.\n    To explain the meanings of these\n    procedures,\n    we used a succession of models of evaluation: the\n    substitution model of chapter 1, the environment model of\n    chapter 3, and the metacircular evaluator of chapter 4.  Our\n    examination of the metacircular evaluator, in particular, dispelled much of\n    the mystery of how\n    Lisp-like languages are interpreted.\n    But even the metacircular evaluator leaves important questions\n    unanswered, because it fails to elucidate the mechanisms of control in a\n    Lisp\n    system.  For instance, the evaluator does not explain how the\n    evaluation of a subexpression manages to return a value to the\n    expression that uses this value, nor does\n    the evaluator explain how some recursive procedures generate\n    iterative processes (that is, are evaluated using constant space)\n    whereas other recursive procedures generate recursive\n    processes.\n    \n        These questions remain unanswered because the metacircular\n        evaluator is itself a Lisp program and hence inherits the\n        control structure of the underlying Lisp system. In order to\n        provide a more complete description of the control structure\n        of the Lisp evaluator, we must work at a more primitive level\n        than Lisp itself.\n      ","5#p2":"\n        In this chapter we \n      \n    will describe processes in terms of the step-by-step\n    operation of a traditional computer.  Such a computer, or \n    register machine, sequentially executes \n    instructions that\n    manipulate the contents of a fixed set of storage elements called \n    registers.  A typical register-machine instruction applies a\n    primitive operation to the contents of some registers and assigns the\n    result to another register.  Our descriptions of processes executed by\n    register machines will look very much like \"machine-language\"\n    programs for traditional computers.  However, instead of focusing on\n    the machine language of any particular computer, we will examine\n    several\n    Lisp procedures\n    and design a specific register machine to\n    execute each\n    procedure.\n    Thus, we will approach our task from the\n    perspective of a hardware architect rather than that of a\n    machine-language computer programmer.  In designing register machines,\n    we will develop mechanisms for implementing important programming\n    constructs such as recursion.  We will also present a language for\n    describing designs for register machines.  In\n    section 5.2 we will\n    implement a\n    Lisp\n    program that uses these descriptions to simulate the machines we design.\n  ","5#p3":"\n    Most of the primitive operations of our register machines are very\n    simple.  For example, an operation might add the numbers fetched from\n    two registers, producing a result to be stored into a third register.\n    Such an operation can be performed by easily described hardware.  In\n    order to deal with list structure, however, we will also use the\n    memory operations\n    car,cdr,\n    and\n    cons,\n    which require an elaborate storage-allocation mechanism.  In\n    section 5.3 we study their\n    implementation in terms of more elementary operations.\n  ","5#p4":"\n    In section 5.4, after we have accumulated\n    experience formulating simple\n    procedures\n    as register machines, we will design a\n    machine that carries out the algorithm described by the metacircular\n    evaluator of section 4.1.  This will fill in\n    the gap in our understanding of how\n    Scheme expressions\n    are interpreted, by providing an explicit model for the mechanisms of\n    control in the evaluator.\n    In section 5.5 we will study a simple\n    compiler that translates\n    Scheme\n    programs into sequences of instructions that can be executed directly with\n    the registers and operations of the evaluator register machine.\n  ","5.1":"\n    Design a register machine to compute\n    \n    factorials using the iterative\n    algorithm specified by the following\n    procedure.\n    Draw data-path and\n    controller diagrams for this machine.\n    (define (factorial n)\n  (define (iter product counter)\n    (if (> counter n)\n        product\n        (iter (* counter product)\n              (+ counter 1))))\n  (iter 1 1)) ","5.1#p1":"\n    To design a register machine, we must design its data paths\n    (registers and operations) and the controller that sequences\n    these operations.  To illustrate the design of a simple register\n    machine, let us examine Euclid's Algorithm, which is used to compute\n    \n    the greatest common divisor (GCD) of two integers.  As we saw in\n    section 1.2.5,\n    \n    Euclid's Algorithm can be\n    carried out by an iterative process, as specified by the following\n    procedure:(define (gcd a b)\n  (if (= b 0)\n      a\n      (gcd b (remainder a b)))) ","5.1#p2":"\n    A machine to carry out this algorithm must keep track of two numbers,\n    $a$ and $b$, so let us\n    assume that these numbers are stored in two registers with those names. The\n    basic operations required are testing whether the contents of register\n    b is zero and computing the remainder of the\n    contents of register a divided by the contents\n    of register b.\n\n    The remainder operation is a complex process, but assume for the moment that\n    we have a primitive device that computes remainders. On each cycle of the\n    GCD algorithm, the contents of register a must\n    be replaced by the contents of register b, and\n    the contents of b must be replaced by the\n    remainder of the old contents of a divided by\n    the old contents of b. It would be convenient\n    if these replacements could be done simultaneously, but in our model of\n    register machines we will assume that only one register can be assigned a\n    new value at each step. To accomplish the replacements, our machine will use\n    a third \"temporary\" register, which we call\n    t. (First the remainder will be placed in\n    t, then the contents of\n    b will be placed in\n    a, and finally the remainder stored in\n    t will be placed in\n    b.)\n  ","5.1#p3":"\n    We can illustrate the registers and operations required for this\n    machine by using the\n    \n    data-path diagram shown in\n    figure 5.1.  In this\n    diagram, the registers (a,\n    b, and t) are\n    represented by rectangles.  Each way to assign a value to a register is\n    indicated by an arrow with an X behind the \n    head, pointing from the source of data to the register.  \n    We can think of the X as a button that, when pushed, allows\n    the value at the source to \"flow\" into the designated register.\n    The label next to each button is the name we will use to refer to the\n    button.  The names are arbitrary, and can be chosen to have mnemonic value\n    (for example, a<-b denotes pushing the\n    button that assigns the contents of register b\n    to register a).  The source of data for a\n    register can be another register (as in the\n    a<-b assignment), an operation result (as in\n    the t<-r assignment), or a constant\n    (a built-in value that cannot be changed, represented in a data-path\n    diagram by a triangle containing the constant).\n  ","5.1#p4":"\n    An operation that computes a value from constants and the contents\n    of registers is represented in a data-path diagram by a trapezoid\n    containing a name for the operation.  For example, the box marked\n    rem in\n    figure 5.1 represents an operation that\n    computes the remainder of the contents of the registers\n    a and b to which\n    it is attached.  Arrows (without buttons) point from the input registers and\n    constants to the box, and arrows connect the operation's output value\n    to registers. A test is represented by a circle containing a name for the\n    test.  For  example, our GCD machine has an operation that tests whether the\n    contents of register b is zero.  A\n    \n    test also has arrows from its input\n    registers and constants, but it has no output\n    arrows; its value is used by the controller rather than by the data\n    paths.  Overall, the data-path diagram shows the registers and\n    operations that are required for the machine and how they must be\n    connected.  If we view the arrows as wires and the\n    X buttons as switches, the data-path diagram\n    is very like the wiring diagram for a machine that could be constructed\n    from electrical components.\n    ","5.1#p5":"\n    In order for the data paths to actually compute GCDs, the buttons must\n    be pushed in the correct sequence.  We will describe this sequence in\n    terms of a\n    \n    controller diagram, as illustrated in\n    figure 5.2.  The elements of the\n    controller diagram indicate how the data-path components should be operated.\n    The rectangular boxes in the controller diagram identify data-path buttons\n    to be pushed, and the arrows describe the sequencing from one step to the\n    next.  The diamond in the diagram represents a decision.  One of the two\n    sequencing arrows will be followed, depending on the value of the data-path\n    test identified in the diamond.  We can interpret the controller in terms\n    of a physical analogy: Think of the diagram as a maze in which a marble is\n    rolling.  When the marble rolls into a box, it pushes the data-path button\n    that is named by the box.  When the marble rolls into a decision node (such\n    as the test for\n    b$\\, =0$), it leaves\n    the node on the path determined by the result of the indicated test.\n    Taken together, the data paths and the controller completely describe\n    a machine for computing GCDs.  We start the controller (the rolling\n    marble) at the place marked start, after\n    placing numbers in registers a and\n    b.  When the controller reaches\n    done, we will find the value of the GCD in\n    register a.\n    ","5.1#fig-5.2":"","5.1.1":"5.1.1  \n    A Language for Describing Register Machines","5.1.1#p1":"\n    Data-path and controller diagrams are adequate for representing simple\n    machines such as GCD, but they are unwieldy for describing large\n    machines such as a\n    Lisp\n    interpreter.  To make it possible to deal with complex machines, we will\n    create a language that presents, in textual form, all the\n    information given by the data-path and controller\n    diagrams. We will start with a notation that directly\n    mirrors the diagrams.\n  ","5.1.1#p2":"\n    We define the data paths of a machine by describing the registers and\n    the operations.  To describe a register, we give it a name\n    and specify the buttons that control assignment to it.  We give each\n    of these buttons a name and specify the source of the data that enters\n    the register under the button's control.  (The source is a register,\n    a constant, or an operation.) To describe an operation, we give\n    it a name and specify its inputs (registers or constants).\n  ","5.1.1#p3":"\n    We define the controller of a machine as a sequence of \n    instructions together with \n      labels that identify\n      entry points in the sequence. An instruction is one of the following:\n      \n\t  The name of a data-path button to push to assign a value to\n\t  a register.  (This corresponds to a box in the controller diagram.)\n\t\n\t  A\n\t  test\n\t  instruction, which performs a\n\t  specified test.\n\t\n\t  A\n\t  \n\t  conditional branch (branch instruction)\n\t  to a\tlocation indicated by a controller label, based on the result of\n\t  the previous test.  (The test and branch together correspond to a\n\t  diamond in the controller diagram.)  If the test is false, the\n\t  controller should continue with the next instruction in the sequence.\n\t  Otherwise, the controller should continue with the instruction after\n\t  the label.\n\t\n\t  An\t  \n\t  \n\t  unconditional branch\n\t  (goto\n\t  instruction) naming a controller label at which to continue execution.\n\t\n\n    The machine starts at the beginning of the controller instruction\n    sequence and stops when execution reaches the end of the sequence.\n    Except when a branch changes the flow of control, instructions are\n    executed in the order in which they are listed.\n    ","5.1.1#p4":"\n    Figure 5.3 shows the GCD machine\n    described in this way.  This example only hints at the generality of these\n    descriptions, since the GCD machine is a very simple case: Each register has\n    only one button, and each button and test is used only once in the\n    controller.\n  ","5.1.1#p5":"\n    Unfortunately, it is difficult to read such a description.  In order\n    to understand the controller instructions we must constantly refer\n    back to the definitions of the button names and the operation names,\n    and to understand what the buttons do we may have to refer to the\n    definitions of the operation names.  We will thus transform our\n    notation to combine the information from the data-path and controller\n    descriptions so that we see it all together.\n  ","5.1.1#p6":"\n    To obtain this form of description, we will replace the arbitrary\n    button and operation names by the definitions of their behavior.  That\n    is, instead of saying (in the controller) \"Push button\n    t<-r\" and separately saying (in the\n    data paths) \"Button t<-r assigns the\n    value of the rem operation to register\n    t\" and \"The\n    rem operation's inputs are the contents\n    of registers\n    a and b,\"\n    we will say (in the controller) \"Push the button that assigns to\n    register t the value of the\n    rem operation on the contents of registers\n    a and b.\"\n    Similarly, instead of saying (in the controller) \"Perform the\n    = test\" and separately saying (in the\n    data paths) \"The = test operates on the\n    contents of register b and the\n    constant 0,\" we will say \"Perform the\n    = test on the\n    \n    contents of register b and the\n    constant 0.\"  We will omit the data-path description, leaving only\n    the controller sequence.  Thus, the GCD machine is described as follows:\n    (controller\n test-b\n   (test (op =) (reg b) (const 0))\n   (branch (label gcd-done))\n   (assign t (op rem) (reg a) (reg b))\n   (assign a (reg b))\n   (assign b (reg t))\n   (goto (label test-b))\n gcd-done) ","5.1.1#p7":"\n    This form of description is easier to read than the kind illustrated\n    in figure 5.3, but it also has disadvantages:\n    It is more verbose for large machines,\n      because complete descriptions of the data-path elements are repeated\n      whenever the elements are mentioned in the controller instruction\n      sequence.  (This is not a problem in the GCD example, because each\n      operation and button is used only once.)  Moreover, repeating the\n      data-path descriptions obscures the actual data-path structure of the\n      machine; it is not obvious for a large machine how many registers,\n      operations, and buttons there are and how they are interconnected.\n      \n\tBecause the controller instructions in a machine definition look like\n\tLisp\n\texpressions, it is easy to forget that they are\n\tnot arbitrary\n\tLisp\n\texpressions.  They can notate only legal machine operations.  For\n\texample, operations can operate directly only on constants and the\n\tcontents of registers, not on the results of other operations.\n      \n\n    In spite of these disadvantages, we will use this register-machine\n    language throughout this chapter, because we will be more concerned with\n    understanding controllers than with understanding the elements and\n    connections in data paths.  We should keep in mind,\n    however, that data-path design is crucial in designing real machines.\n  ","5.1.1#ex-5.2":"\n    Use the register-machine language to describe the\n    \n    iterative factorial\n    machine of exercise 5.1.\n    ","5.1.1#h1":"Actions","5.1.1#p8":"\n    Let us modify the GCD machine so that we can type in the numbers\n    whose GCD we want and get the answer\n    printed at our terminal.\n    We will not discuss how to make a machine that can read and print,\n    but will assume (as we do when we use\n    read\n    and display in\n    Scheme)\n    that they are available as primitive\n    operations.","5.1.1#footnote-link-1":"1","5.1.1#p9":"Read\n    is like the operations we have been using in that it produces a value that\n    can be stored in a register.  But\n    read\n    does not take inputs from any registers; its value depends on\n    something that happens outside the parts of the machine we are\n    designing.  We will allow our machine's operations to have such\n    behavior, and thus will draw and notate the use of\n    read\n    just as we do any other operation that computes a value.\n  ","5.1.1#p10":"Print,\n    on the other hand, differs from the operations we have\n    been using in a fundamental way: It does not produce an output value\n    to be stored in a register.  Though it has an effect, this effect is\n    not on a part of the machine we are designing.  We will refer to this\n    kind of operation as an action.  We will represent an action in\n    a data-path diagram just as we represent an operation that computes a\n    value—as a trapezoid that contains the name of the action.\n    Arrows point to the action box from any inputs (registers or\n    constants).  We also associate a button with the action.  Pushing the\n    button makes the action happen.  To make a controller push an action\n    button we use a new kind of instruction called\n    perform.  Thus,\n    the action of printing\n    the contents of register\n    a is represented\n    in a controller sequence by the instruction\n    (perform (op print) (reg a))","5.1.1#p11":"\n\tFigure \n    shows the data paths and controller for\n    the new GCD machine.  Instead of having the machine stop after printing\n    the answer, we have made it start over, so that it repeatedly\n    reads a pair of numbers, computes their GCD, and prints\n    the result.\n    This structure is like the driver loops we used in the interpreters of\n    chapter 4.\n    ","5.1.1#fig-":"","5.1.1#footnote-1":"This assumption glosses over a\n    great deal of complexity.  Usually a large portion of the implementation of\n    a Lisp system is dedicated to making reading\n    and printing work.","5.1.2":"5.1.2  \n    Abstraction in Machine Design","5.1.2#p1":"\n    We will often define a machine to include \"primitive\"\n    operations that are actually very complex.  For example, in\n    sections 5.4 and 5.5\n    we will treat\n    Scheme's\n    environment manipulations as primitive.  Such abstraction is valuable\n    because it allows us to ignore the details of parts of a machine so that we\n    can concentrate on other aspects of the design.  The fact that we have\n    swept a lot of complexity under the rug, however, does not mean that a\n    machine design is unrealistic.  We can always replace the complex\n    \"primitives\" by simpler primitive operations.\n  ","5.1.2#p2":"\n    Consider the GCD machine. The machine has an instruction that computes\n    the remainder of the contents of registers a\n    and b and assigns the result to register\n    t.  If we want to construct the GCD machine\n    without using a primitive remainder operation, we must specify how to\n    compute remainders in terms of simpler operations, such as subtraction.\n    Indeed, we can write a \n    Scheme procedure\n    that finds remainders in this way:\n    (define (remainder n d)\n  (if (< n d)\n      n\n      (remainder (- n d) d))) \n\n    We can thus replace the remainder operation in the GCD machine's\n    data paths with a subtraction operation and a comparison test.\n    Figure 5.5 shows the data paths and\n    controller for the elaborated machine. The instruction\n    (assign t (op rem) (reg a) (reg b))\n    in the GCD controller definition is replaced by a sequence of\n    instructions that contains a loop, as shown in\n    figure 5.6.\n    ","5.1.2#fig-5.5":"","5.1.2#ex-5.3":"\n    Design a machine to compute\n    \n    square roots using Newton's method, as\n    described in section 1.1.7 and implemented with the following code in section 1.1.8:\n    (define (sqrt x)\n  (define (good-enough? guess)\n    (< (abs (- (square guess) x)) 0.001))\n  (define (improve guess)\n    (average guess (/ x guess)))\n  (define (sqrt-iter guess)\n    (if (good-enough? guess)\n        guess\n        (sqrt-iter (improve guess))))\n  (sqrt-iter 1.0)) \n    Begin by assuming that\n    good-enough?\n    and improve operations are available as\n    primitives.  Then show how to expand these in terms of arithmetic\n    operations.  Describe each version of the sqrt\n    machine design by drawing a data-path diagram and writing a controller\n    definition in the register-machine language.\n    ","5.1.3":"5.1.3  \n    Subroutines","5.1.3#p1":"When designing a machine to perform a computation, we would often\n    prefer to arrange for components to be shared by different parts of\n    the computation rather than duplicate the components.  Consider a\n    machine that includes two GCD computations—one that finds the GCD of\n    the contents of registers a and\n    b and one that finds the\n    GCD of the contents of registers c and\n    d.  We might start\n    by assuming we have a primitive gcd operation,\n    then expand the two instances of gcd in terms\n    of more primitive operations.\n    \n\tFigure \n    shows just the GCD portions of the resulting machine's data paths,\n    without showing how they connect to the rest of the machine.  The figure\n    also shows the corresponding portions of the machine's controller\n    sequence.\n    ","5.1.3#fig-":"","5.1.3#p2":"\n    This machine has two remainder operation boxes and two boxes for\n    testing equality.  If the duplicated components are complicated, as is the\n    remainder box, this will not be an economical way to build the\n    machine.  We can avoid duplicating the data-path components by using\n    the same components for both GCD computations, provided that doing so\n    will not affect the rest of the larger machine's computation.  If the\n    values in registers a and\n    b are not needed by the time the\n    controller gets to gcd-2 (or if these values\n    can be moved to other registers for safekeeping), we can change the machine\n    so that it uses registers a and\n    b, rather than registers\n    c and d, in\n    computing the second GCD as well as the first.  If we do this, we obtain the\n    controller sequence shown in\n    figure 5.8.\n  ","5.1.3#p3":"\n    We have removed the duplicate data-path components (so that the data paths\n    are again as in figure 5.1), but the\n    controller now has two GCD sequences that differ only in their entry-point\n    labels.  It would be better to replace these two sequences by branches to a\n    single sequence—a gcdsubroutine—at the end of which we branch back to the\n    correct place in the main instruction sequence.  We can accomplish this as\n    follows: Before branching to gcd, we place a\n    distinguishing value (such as 0 or 1) into a special register, \n    continue.  At the end of the\n    gcd subroutine we return either to\n    after-gcd-1 or to after-gcd-2, depending\n    on the value of the continue register.\n    Figure 5.9 shows the relevant portion\n    of the resulting controller sequence, which includes only a single copy of\n    the gcd instructions.\n    ","5.1.3#p4":"\n    This is a reasonable approach for handling small problems, but it would be\n    awkward if there were many instances of GCD computations in the controller\n    sequence.  To decide where to continue executing after the\n    gcd subroutine, we would need tests in the data\n    paths and branch instructions in the controller for all the places that use\n    gcd. A more powerful method for implementing\n    subroutines is to have the continue register\n    hold the label of the entry point in the controller sequence at which\n    execution should continue when the subroutine is finished. Implementing this\n    strategy requires a new kind of connection between the data paths and the\n    controller of a register machine: There must be a way to assign to a\n    register a label in the controller sequence in such a way that this value\n    can be fetched from the register and used to continue execution at the\n    designated entry point.\n  ","5.1.3#p5":"\n    To reflect this ability, we will extend the\n    assign\n    instruction of the register-machine language to allow a register to be\n    assigned as value a label from the controller sequence (as a special\n    kind of constant).  We will also extend the\n    goto\n    instruction to allow execution to continue at the entry point described by\n    the contents of a register rather than only at an entry point described by\n    a constant label.  Using these new constructs we can terminate the\n    gcd subroutine with a branch to the location\n    stored in the continue register.  This leads\n    to the controller sequence shown in\n    figure 5.10.\n  ","5.1.3#p6":"\n    A machine with more than one subroutine could use multiple\n    continuation registers (e.g., gcd-continue,\n    factorial-continue) or we could have all\n    subroutines share a single\n    continue register.  Sharing is more economical,\n    but we must be careful if we have a subroutine\n    (sub1) that calls another subroutine\n    (sub2).  Unless\n    sub1 saves the contents of\n    continue in some other register before setting\n    up continue for the call to\n    sub2, sub1 will\n    not know where to go when it is finished.  The mechanism developed in the\n    next section to handle recursion also provides a better solution to this\n    problem of nested subroutine calls.\n  ","5.1.4":"5.1.4  \n    Using a Stack to Implement Recursion","5.1.4#p1":"\n    With the ideas illustrated so far, we can implement any\n    \n    iterative\n    process by specifying a register machine that has a register\n    corresponding to each state variable of the process.  The machine\n    repeatedly executes a controller loop, changing the contents\n    of the registers, until some termination condition is satisfied.  At\n    each point in the controller sequence, the state of the machine\n    (representing the state of the iterative process) is completely\n    determined by the contents of the registers (the values of the state\n    variables).\n  ","5.1.4#p2":"\n    Implementing\n    \n    recursive processes, however, requires an additional\n    mechanism.  Consider the following recursive method for computing\n    factorials, which we first examined in\n    section 1.2.1:\n    (define (factorial n)\n  (if (= n 1)\n      1\n      (* (factorial (- n 1)) n))) \n\n    As we see from the\n    procedure,\n    computing $n!$ requires computing\n    $(n-1)!$.  Our GCD machine, modeled on the\n    procedure(define (gcd a b)\n  (if (= b 0)\n      a\n      (gcd b (remainder a b)))) \n\n    similarly had to compute another GCD.  But there is an important\n    difference between the gcdprocedure,\n    which reduces the original computation to a new GCD computation, and\n    factorial, which requires computing another\n    factorial as a subproblem.  In GCD, the answer to the new GCD computation is\n    the answer to the original problem.  To compute the next GCD, we simply\n    place the new arguments in the input registers of the GCD machine and reuse\n    the machine's data paths by executing the same controller sequence.\n    When the machine is finished solving the final GCD problem, it has completed\n    the entire computation.\n  ","5.1.4#p3":"\n    In the case of factorial (or any recursive process) the answer to the\n    new factorial subproblem is not the answer to the original problem.\n    The value obtained for $(n-1)!$ must be\n    multiplied by $n$ to get the final answer.  If\n    we try to imitate the GCD design, and solve the factorial subproblem by\n    decrementing the n register and rerunning the\n    factorial machine, we will no longer have available the old value of\n    n by which to multiply the result.  We thus\n    need a second factorial machine to work on the subproblem.  This second\n    factorial computation itself has a factorial subproblem, which\n    requires a third factorial machine, and so on.  Since each factorial\n    machine contains another factorial machine within it, the total\n    machine contains an infinite nest of similar machines and hence cannot\n    be constructed from a fixed, finite number of parts.\n  ","5.1.4#p4":"\n    Nevertheless, we can implement the factorial process as a register\n    machine if we can arrange to use the same components for each nested\n    instance of the machine.  Specifically, the machine that computes\n    $n!$\n    should use the same components to work on the subproblem of computing\n    $(n-1)!$, on the subproblem for\n    $(n-2)!$, and so on.  This is\n    plausible because, although the factorial process dictates that an\n    unbounded number of copies of the same machine are needed to perform a\n    computation, only one of these copies needs to be active at any given\n    time.  When the machine encounters a recursive subproblem, it can\n    suspend work on the main problem, reuse the same physical parts to\n    work on the subproblem, then continue the suspended computation.\n  ","5.1.4#p5":"\n    In the subproblem, the contents of the registers will be different\n    than they were in the main problem. (In this case the\n    n register is decremented.)  In order to be\n    able to continue the suspended computation, the machine must save the\n    contents of any registers that will be needed after the subproblem is\n    solved so that these can be restored to continue the suspended computation.\n    In the case of factorial, we will save the old value of\n    n, to be restored when we are finished\n    computing the factorial of the decremented n\n    register.","5.1.4#footnote-link-1":"1","5.1.4#p6":"\n    Since there is no a priori limit on the depth of nested\n    recursive calls, we may need to save an arbitrary number of register\n    values.  These values must be restored in the reverse of the order in\n    which they were saved, since in a nest of recursions the last\n    subproblem to be entered is the first to be finished.  This dictates\n    the use of a stack, or \"last in, first out\" data\n    structure, to save register values.  We can extend the register-machine\n    language to include a stack by adding two kinds of instructions: Values are\n    placed\n    on the stack using a\n    save instruction and\n    restored from the stack using a\n    restore\n    instruction.  After a sequence of values has been\n    saved on the stack, a sequence of\n    restores will retrieve these values in reverse\n    order.","5.1.4#footnote-link-2":"2","5.1.4#p7":"\n    With the aid of the stack, we can reuse a single copy of the factorial\n    machine's data paths for each factorial subproblem.  There is a\n    similar design issue in reusing the controller sequence that operates\n    the data paths.  To reexecute the factorial computation, the\n    controller cannot simply loop back to the beginning, as with\n    an iterative process, because after solving the\n    $(n-1)!$ subproblem\n    the machine must still multiply the result by\n    $n$.  The controller\n    must suspend its computation of $n!$, solve the\n    $(n-1)!$ subproblem,\n    then continue its computation of $n!$.  This\n    view of the factorial computation suggests the use of the subroutine\n    mechanism described in section 5.1.3, which\n    has the controller use a\n    continue register to transfer to the part of\n    the sequence that solves a subproblem and then continue where it left off on\n    the main problem. We can thus make a factorial subroutine that returns to\n    the entry point stored in the continue\n    register. Around each subroutine call, we save and restore\n    continue just as we do the\n    n register, since each \"level\" of\n    the factorial computation will use the same\n    continue register. That is, the factorial\n    subroutine must put a new value in continue\n    when it calls itself for a subproblem, but it will need the old value in\n    order to return to the place that called it to solve a subproblem.\n  ","5.1.4#fig-5.11":"","5.1.4#p8":"\n\tFigure \n    shows the data paths and controller for\n    a machine that implements the recursive\n    factorialprocedure.\n    The machine has a stack and three registers, called\n    n, val, and\n    continue.  To simplify the data-path diagram,\n    we have not named the register-assignment buttons, only the stack-operation\n    buttons (sc and sn\n    to save registers, rc and\n    rn to restore registers).  To operate the\n    machine, we put in register n the number whose\n    factorial we wish to compute and start the machine.  When the machine\n    reaches fact-done, the computation is finished\n    and the answer will be found in the val\n    register.  In the controller sequence, n and\n    continue are saved before each recursive call\n    and restored upon return from the call.  Returning from a call is\n    accomplished by branching to the location stored in\n    continue. \n    Continue\n    is initialized when the machine starts so that the last return will go to\n    fact-done.  The\n    val\n    register, which holds the result of the factorial computation, is not\n    saved before the recursive call, because the old contents of\n    val is not useful after the subroutine returns.\n    Only the new value, which is the value produced by the subcomputation, is\n    needed.\n  ","5.1.4#p9":"\n    Although in principle the factorial computation requires an infinite\n    machine, the machine in\n    \n\tfigure \n    is actually finite except for the stack, which is potentially unbounded.  Any\n    particular physical implementation of a stack, however, will be of finite\n    size, and this will limit the depth of recursive calls that can be handled\n    by the machine.  This implementation of factorial illustrates the general\n    strategy for realizing recursive algorithms as ordinary register machines\n    augmented by stacks.  When a recursive subproblem is encountered, we save on\n    the stack the registers whose current values will be required after the\n    subproblem is solved, solve the recursive subproblem, then restore the saved\n    registers and continue execution on the main problem.  The\n    continue register must always be saved.\n    Whether there are other registers that need to be saved depends on the\n    particular machine, since not all recursive computations need the original\n    values of registers that are modified during solution of the subproblem\n    (see exercise 5.4).\n  ","5.1.4#h1":"A double recursion","5.1.4#p10":"\n    Let us examine a more complex recursive process, the tree-recursive\n    computation of the\n    \n    Fibonacci numbers, which we introduced in\n    section 1.2.2:\n    (define (fib n)\n  (if (< n 2)\n      n\n      (+ (fib (- n 1)) (fib (- n 2))))) \n\n    Just as with factorial, we can implement the recursive Fibonacci\n    computation as a register machine with registers\n    n, val,\n    and continue.  The machine is more complex than\n    the one for factorial, because there are two places in the controller\n    sequence where we need to perform recursive calls—once to compute\n    Fib$(n-1)$ and once to compute\n    Fib$(n-2)$.  To set up for each of these calls,\n    we save the registers whose values will be needed later, set the\n    n\n    register to the number whose Fib we need to compute recursively\n    ($n-1$ or $n-2$), and\n    assign to continue the entry point in the main\n    sequence to which to return (afterfib-n-1 or\n    afterfib-n-2, respectively).  We then go to\n    fib-loop.  When we return from the\n    recursive call, the answer is in val.\n    Figure 5.12 shows the controller sequence\n    for this machine.\n    ","5.1.4#fig-":"","5.1.4#ex-5.4":"\n    Specify register machines that implement each of the following\n    procedures.\n    For each machine, write a controller instruction sequence\n    and draw a diagram showing the data paths.\n    \n\tRecursive exponentiation:\n\t(define (expt b n)\n  (if (= n 0)\n      1\n      (* b (expt b (- n 1))))) \n\tIterative exponentiation:\n\t(define (expt b n)\n  (define (expt-iter counter product)\n    (if (= counter 0)\n        product\n        (expt-iter (- counter 1) (* b product))))\n  (expt-iter n 1)) ","5.1.4#ex-5.5":"\n    Hand-simulate the factorial and Fibonacci machines, using some\n    nontrivial input (requiring execution of at least one recursive call).\n    Show the contents of the stack at each significant point in the\n    execution.\n    ","5.1.4#ex-5.6":"\n    Ben Bitdiddle observes that the Fibonacci machine's controller sequence\n    has an extra save and an extra\n    restore, which can be removed to make a faster\n    machine.  Where are these instructions?\n    ","5.1.4#footnote-1":"One might argue that we don't need to save the old\n    n; after we decrement it and solve the\n    subproblem, we could simply increment it to recover the old value.  Although\n    this strategy works for factorial, it cannot work in general, since the old\n    value of a register cannot always be computed from the new one.","5.1.4#footnote-2":"In section 5.3 we\n    will see how to implement a stack in terms of more primitive\n    operations.","5.1.5":"5.1.5  \n    Instruction Summary","5.1.5#p1":"\n    A controller instruction in our register-machine language\n    has one of the following forms, where each\n    input$_i$ is\n    \n\teither\n      (reg register-name)\n    or\n    (const constant-value).","5.1.5#p2":"\n    These instructions were introduced in\n    section 5.1.1:\n    \n(assign $register-name$ (reg $register-name$))\n\n(assign $register-name$ (const $constant-value$))\n\n(assign $register-name$ (op $operation-name$) $input_{1}$ $\\ldots$ $input_{n}$)\n\n(perform (op $operation-name$) $input_{1}$ $\\ldots$ $input_{n}$)\n\n(test (op $operation-name$) $input_{1}$ $\\ldots$ $input_{n}$)\n\n(branch (label $label-name$))\n\n(goto (label $label-name$))\n      ","5.1.5#p3":"\n    The use of registers to hold labels was introduced in\n    section 5.1.3:\n    \n(assign $register-name$ (label $label-name$))\n\n(goto (reg $register-name$))\n      ","5.1.5#p4":"\n    Instructions to use the stack were introduced in\n    section 5.1.4:\n    \n(save $register-name$)\n\n(restore $register-name$)\n      ","5.1.5#p5":"\n    The only kind of\n    $\\langle constant-value \\rangle$\n    we have seen so far is a number, but later we will \n    use strings, symbols,\n    and lists.\n    \n\tFor example,\n\t(const \"abc\") is the string\n\t\"abc\",\n\t(const abc) is the symbol\n\tabc,\n\t(const (a b c))\n\tis the list\n\t(a b c),\n\tand\n\t(const ())\n\tis the empty list.\n      ","5.2":"5.2  A Register-Machine Simulator","5.2#p1":"In order to gain a good understanding of the design of register machines,\n    we must test the machines we design to see if they perform as expected.\n    One way to test a design is to hand-simulate the operation of the\n    controller, as in exercise 5.5.  But this is\n    extremely tedious for all but the simplest machines.  In this section we\n    construct a simulator for machines described in the register-machine\n    language.  The simulator is a\n    Scheme\n    program with\n    four interface\n    procedures.\n    The first uses a description of a register\n    machine to construct a model of the machine (a data structure whose\n    parts correspond to the parts of the machine to be simulated), and the\n    other three allow us to simulate the machine by manipulating the\n    model:\n    (make-machine register-names operations controller)\n\tconstructs and returns a model of the machine with the given\n\tregisters, operations, and controller.\n      (set_register_contents machine-model register-name value)\n\tstores a value in a simulated register in the given machine.\n      (get-register-contents machine-model, register-name)\n\treturns the contents of a simulated register in the given machine.\n      (start machine-model)\n\tsimulates the execution of the given machine, starting from the\n\tbeginning of the controller sequence and stopping when it reaches the\n\tend of the sequence.\n      ","5.2#p2":"\n    As an example of how these\n    procedures\n    are used, we can define\n    gcd-machine\n    to be a model of the GCD machine\n    of section 5.1.1 as follows:\n    (define gcd-machine\n  (make-machine\n   '(a b t)\n   (list (list 'rem remainder) (list '= =))\n   '(test-b\n       (test (op =) (reg b) (const 0))\n       (branch (label gcd-done))\n       (assign t (op rem) (reg a) (reg b))\n       (assign a (reg b))\n       (assign b (reg t))\n       (goto (label test-b))\n     gcd-done))) \n    The first argument to\n    make-machine\n    is a list of register names. The next argument is a table (a list of\n    two-element lists) that pairs each operation name with a \n    Scheme procedure\n    that implements the operation (that is, produces the same output value\n    given the same input values). The last argument specifies the controller\n    as a list of labels and machine instructions, as in\n    section 5.1.\n  ","5.2#p3":"\n    To compute GCDs with this machine, we set the input registers, start the\n    machine, and examine the result when the simulation terminates:\n    (set-register-contents! gcd-machine 'a 206) (set-register-contents! gcd-machine 'b 40) (start gcd-machine) (get-register-contents gcd-machine 'a) \n\n    This computation will run much more slowly than a\n    gcdprocedure\n    written in\n    Scheme,\n      because we will simulate low-level machine instructions, such as\n      assign, by much more complex operations.\n  ","5.2#ex-5.7":"\n    Use the simulator to test the machines you designed in\n    exercise 5.4.\n    ","5.2.1":"5.2.1  \n    The Machine Model(define (make-new-machine)\n  (let ((pc (make-register 'pc))\n        (flag (make-register 'flag))\n        (stack (make-stack))\n        (the-instruction-sequence '()))\n    (let ((the-ops\n           (list (list 'initialize-stack\n                       (lambda () (stack 'initialize)))))\n          (register-table\n           (list (list 'pc pc) (list 'flag flag))))\n      (define (allocate-register name)\n        (if (assoc name register-table)\n            (error \"Multiply defined register: \" name)\n            (set! register-table\n                  (cons (list name (make-register name))\n                        register-table)))\n        'register-allocated)\n      (define (lookup-register name)\n        (let ((val (assoc name register-table)))\n          (if val\n              (cadr val)\n              (error \"Unknown register:\" name))))\n      (define (execute)\n        (let ((insts (get-contents pc)))\n          (if (null? insts)\n              'done\n              (begin\n                ((instruction-execution-proc (car insts)))\n                (execute)))))\n      (define (dispatch message)\n        (cond ((eq? message 'start)\n               (set-contents! pc the-instruction-sequence)\n               (execute))\n              ((eq? message 'install-instruction-sequence)\n               (lambda (seq) (set! the-instruction-sequence seq)))\n              ((eq? message 'allocate-register) allocate-register)\n              ((eq? message 'get-register) lookup-register)\n              ((eq? message 'install-operations)\n               (lambda (ops) (set! the-ops (append the-ops ops))))\n              ((eq? message 'stack) stack)\n              ((eq? message 'operations) the-ops)\n              (else (error \"Unknown request - - MACHINE\" message))))\n      dispatch)))","5.2.1#p1":"\n    The machine model generated by\n    make-machine\n    is represented as a\n    procedure\n    with local state using the message-passing techniques\n    developed in chapter 3.  To build this model,\n    make-machine\n    begins by calling the\n    proceduremake-new-machine\n    to construct\n    the parts of the machine model that are common to all register\n    machines.  This basic machine model constructed by\n    make-new-machine\n    is essentially a container for some registers and a stack, together with an\n    execution mechanism that processes the controller instructions one by one.\n  ","5.2.1#p2":"Make-machine\n    then extends this basic model (by sending it\n    messages) to include the registers, operations, and controller of the\n    particular machine being defined.  First it allocates a register in\n    the new machine for each of the supplied register names and installs\n    the designated operations in the machine.  Then it uses an \n    assembler (described below in\n    section 5.2.2) to transform the controller list\n    into instructions for the new machine and installs these as the\n    machine's instruction sequence.\n    Make-machine\n    returns as its value the modified machine model.\n    (define (make-machine register-names ops controller-text)\n  (let ((machine (make-new-machine)))\n    (for-each (lambda (register-name)\n                ((machine 'allocate-register) register-name))\n              register-names)\n    ((machine 'install-operations) ops)    \n    ((machine 'install-instruction-sequence)\n    (assemble controller-text machine))\n    machine)) ","5.2.1#h1":"Registers","5.2.1#p3":"\n    We will represent a register as a\n    procedure\n    with local state, as in\n    chapter 3.  The\n    proceduremake-register creates a register that\n    holds a value that can be accessed or changed:\n    (define (make-register name)\n  (let ((contents '*unassigned*))\n    (define (dispatch message)\n      (cond ((eq? message 'get) contents)\n            ((eq? message 'set)\n             (lambda (value) (set! contents value)))\n            (else\n             (error \"Unknown request - - REGISTER\" message))))\n    dispatch)) \n\n    The following\n    procedures\n    are used to access registers:\n    (define (get-contents register)\n  (register 'get))\n\n(define (set-contents! register value)\n  ((register 'set) value)) ","5.2.1#h2":"The stack","5.2.1#p4":"\n    We can also represent a stack as a\n    procedure\n    with local state.  The\n    proceduremake-stack\n    creates a stack whose local state consists\n    of a list of the items on the stack.  A stack accepts requests to\n    push an item onto the stack, to\n    pop the top item off the stack\n    and return it, and to\n    initialize the stack to empty.\n    (define (make-stack)\n  (let ((s '()))\n    (define (push x)\n      (set! s (cons x s)))\n    (define (pop)\n      (if (null? s)\n          (error \"Empty stack - - POP\")\n          (let ((top (car s)))\n            (set! s (cdr s))\n            top)))\n    (define (initialize)\n      (set! s '())\n      'done)\n    (define (dispatch message)\n      (cond ((eq? message 'push) push)\n            ((eq? message 'pop) (pop))\n            ((eq? message 'initialize) (initialize))\n            (else (error \"Unknown request - - STACK\"\n                         message))))\n    dispatch)) \n\n    The following\n    procedures\n    are used to access stacks:\n    (define (pop stack)\n  (stack 'pop))\n\n(define (push stack value)\n  ((stack 'push) value)) ","5.2.1#h3":"The basic machine","5.2.1#p5":"\n    The\n    make-new-machineprocedure,\n    shown in figure 5.13, constructs an\n    object whose local state consists of a stack, an initially empty instruction\n    sequence, a list of operations that initially contains an operation to \n    \n    initialize the stack, and a \n    register table that initially contains two\n    registers, named\n    flag and\n    pc\n    (for \"program counter\"). The internal\n    procedureallocate-register\n    adds new entries to the register table, and the internal\n    procedurelookup-register\n    looks up registers in the table.\n  ","5.2.1#p6":"\n    The flag register is used to control branching\n    in the simulated machine.\n    Test\n    instructions set the contents of\n    flag to the result of the test (true or false).\n    Branch\n    instructions decide whether or not to branch by examining the contents of\n    flag.\n  ","5.2.1#p7":"\n    The pc register determines the sequencing of\n    instructions as the machine runs.  This sequencing is implemented by the\n    internal\n    procedureexecute.\n    In the simulation model, each machine instruction is a data structure\n    that includes a\n    procedure\n    of no arguments, called the \n    instruction execution procedure,\n    such that calling this\n    procedure\n    simulates executing the instruction.  As the simulation runs,\n    pc points to the place in the instruction\n    sequence beginning with the next instruction to be executed.  \n    Execute\n    gets that instruction, executes it by calling the instruction execution\n    procedure,\n    and repeats this cycle until there are no more instructions to execute\n    (i.e., until pc points to the end of the\n    instruction sequence).\n  ","5.2.1#p8":"\n    As part of its operation, each instruction execution\n    procedure\n    modifies\n    pc to indicate the next instruction to be\n    executed.\n    Branch and\n\tgoto instructions\n      \n    change pc to point to the new destination.\n    All other instructions simply advance pc,\n    making it point to the next instruction in the sequence.  Observe that\n    each call to execute calls\n    execute again, but this does not produce an\n    infinite loop because running the instruction execution\n    procedure\n    changes the contents of pc.\n  ","5.2.1#p9":"Make-new-machine\n    returns a\n    dispatch procedure\n    that implements message-passing access to the internal state.  Notice that\n    starting the machine is accomplished by setting\n    pc to the beginning of the instruction sequence\n    and calling execute.\n  ","5.2.1#p10":"\n    For convenience, we provide an alternate\n    procedural\n    interface to a machine's\n    start operation, as well as\n    procedures\n    to set and examine register contents, as specified at the beginning of\n    section 5.2:\n    (define (start machine)\n  (machine 'start))\n\n(define (get-register-contents machine register-name)\n  (get-contents (get-register machine register-name)))\n\n(define (set-register-contents! machine register-name value)\n  (set-contents! (get-register machine register-name) value)\n  'done) \n\n    These\n    procedures\n    (and many\n    procedures\n    in sections 5.2.2 and 5.2.3)\n    use the following to look up the register with a given name in a given\n    machine:\n    (define (get-register machine reg-name)\n  ((machine 'get-register) reg-name)) ","5.2.2":"5.2.2  \n    The Assembler\n\n  The assembler transforms the sequence of controller\n  expressions\n  for a machine into a corresponding list of machine instructions, each with its\n  execution\n  procedure.\n  Overall, the assembler is much like the evaluators we studied in\n  chapter 4—there is an input language (in this case, the\n  register-machine language) and we must perform an appropriate action for each\n  type of component in the language.\n   \n  ","5.2.2#p1":"\n    The technique of producing an execution\n    procedure\n    for each instruction is just what we used in\n    section 4.1.7 to speed\n    up the evaluator by separating analysis from runtime execution.  As we\n    saw in chapter 4, much useful\n    \n    analysis of\n    Scheme\n    expressions could\n    be performed without knowing the actual values of\n    variables.\n    Here, analogously, much useful analysis of register-machine-language\n    expressions can be performed without knowing the actual contents of\n    machine registers.  For example, we can replace references to\n    registers by pointers to the register objects, and we can\n    replace references to labels by pointers to the place in the\n    instruction sequence that the label designates.\n  ","5.2.2#p2":"\n    Before it can generate the instruction execution\n    procedures,\n    the assembler must know what all the labels refer to, so it begins by\n    scanning the controller text to separate the labels from the\n    instructions.  As it scans the text, it constructs both a list of\n    instructions and a table that associates each label with a pointer\n    into that list.  Then the assembler augments the instruction list by\n    inserting the execution\n    procedure\n    for each instruction.\n  ","5.2.2#p3":"\n    The assembleprocedure\n    is the main entry to the assembler. It takes the controller\n    text and the\n    machine model as arguments and returns the instruction sequence to be stored\n    in the model.\n    Assemble\n    calls\n    extract-labels\n    to build the initial instruction list and label table from the supplied\n    controller text.  The second argument\n    to\n    extract-labels\n    is a\n    procedure\n    to be called to process these results: This\n    procedure\n    uses\n    update-insts!\n    to generate the instruction execution\n    procedures\n    and insert them into the instruction list, and returns the modified list.\n    (define (assemble controller-text machine)\n  (extract-labels controller-text\n                  (lambda (insts labels)\n                    (update-insts! insts labels machine)\n                    insts))) ","5.2.2#p4":"Extract-labels takes as arguments a list\n        text (the sequence of controller instruction\n        expressions) and a receive procedure.\n        Receive will be called with two values: (1) a\n        list insts of instruction data structures,\n        each containing an instruction from text; and\n        (2) a table called labels, which associates\n        each label from text with the position in the\n        list insts that the label designates.\n      (define (extract-labels text receive)\n   (if (null? text)\n       (receive '() '())\n       (extract-labels (cdr text)\n        (lambda (insts labels)\n          (let ((next-inst (car text)))\n            (if (symbol? next-inst)\n                (receive insts\n                         (cons (make-label-entry next-inst\n                                                 insts)\n                               labels))\n                (receive (cons (make-instruction next-inst)\n                               insts)\n                         labels))))))) Extract-labels\n    works by sequentially scanning the elements of the\n    text\n     and accumulating the\n    insts and the\n    labels. If an element is a\n    symbol\n    (and thus a label) an appropriate entry is added to the\n    labels table.  Otherwise the element is\n    accumulated onto the insts\n    list.","5.2.2#footnote-link-1":"1","5.2.2#p5":"Update-insts!\n    modifies the instruction list, which initially contains only \n    the text of the instructions,\n    to include the corresponding execution\n    procedures:(define (update-insts! insts labels machine)\n  (let ((pc (get-register machine 'pc))\n        (flag (get-register machine 'flag))\n        (stack (machine 'stack))\n        (ops (machine 'operations)))\n    (for-each\n     (lambda (inst)\n       (set-instruction-execution-proc! \n        inst\n        (make-execution-procedure\n         (instruction-text inst) labels machine\n         pc flag stack ops)))\n     insts))) ","5.2.2#p6":"\n    The machine instruction data structure simply pairs the\n    \n    instruction text with the corresponding execution\n    procedure.\n    The execution\n    procedure\n    is not yet available when\n    extract-labels\n    constructs the instruction, and is inserted later by\n    update-insts!.(define (make-instruction text)\n  (cons text '()))\n\n(define (instruction-text inst)\n  (car inst))\n\n(define (instruction-execution-proc inst)\n  (cdr inst))\n\n(define (set-instruction-execution-proc! inst proc)\n  (set-cdr! inst proc)) \n\n    The\n    instruction text\n    is not used by our simulator, but is handy to keep\n    around for debugging (see\n    exercise 5.16).\n  ","5.2.2#p7":"\n    Elements of the label table are pairs:\n    (define (make-label-entry label-name insts)\n  (cons label-name insts)) \n    Entries will be looked up in the table with\n    (define (lookup-label labels label-name)\n  (let ((val (assoc label-name labels)))\n    (if val\n        (cdr val)\n        (error \"Undefined label - - ASSEMBLE\" label-name)))) ","5.2.2#ex-5.8":"\n    The following register-machine code is ambiguous, because the label\n    here is defined more than once:\n    start\n  (goto (label here))\nhere\n  (assign a (const 3))\n  (goto (label there))\nhere\n  (assign a (const 4))\n  (goto (label there))\nthere\n    With the simulator as written, what will the contents of register\n    a be when control reaches\n    there?  Modify the\n    extract-labelsprocedure\n    so that the assembler will signal an error if the same label\n    name is used to indicate two different locations.\n    ","5.2.2#footnote-1":"\n    Using the\n    receiveprocedure\n    here is a way to get\n    extract-labels\n    to effectively return two\n    values—labels and\n    insts—without explicitly making a\n    compound data structure to hold them.  An alternative implementation, which\n    returns an explicit pair of values, is\n    (define (extract-labels text)\n  (if (null? text)\n      (cons '() '())\n      (let ((result (extract-labels (cdr text))))\n        (let ((insts (car result)) (labels (cdr result)))\n          (let ((next-inst (car text)))\n            (if (symbol? next-inst)\n                (cons insts\n                      (cons (make-label-entry next-inst insts) labels))\n                (cons (cons (make-instruction next-inst) insts)\n                      labels)))))))\n    which would be called by assemble as follows:\n    (define (assemble controller-text machine)\n  (let ((result (extract-labels controller-text)))\n    (let ((insts (car result)) (labels (cdr result)))\n      (update-insts! insts labels machine)\n      insts)))\n\n    You can consider our use of receive as\n    demonstrating an elegant way to\n    \n    return multiple values, or simply an excuse\n    to show off a programming trick.  An argument like\n    receive that is the next\n    procedure\n    to be invoked is called a\n    \"continuation.\"  Recall that we\n    also used continuations to implement the backtracking control\n    structure in the amb evaluator in\n    section 4.3.3.","5.2.3":"5.2.3","5.2.3#p1":"\n    The assembler calls\n    make-execution-procedure\n    to generate the execution\n    procedure\n    for an instruction.\n    Like the analyzeprocedure\n    in the evaluator of section 4.1.7,\n    this dispatches on the type of instruction to generate the appropriate\n    execution\n    procedure.(define (make-execution-procedure inst labels machine\n                                  pc flag stack ops)\n  (cond ((eq? (car inst) 'assign)\n         (make-assign inst machine labels ops pc))\n        ((eq? (car inst) 'test)\n         (make-test inst machine labels ops flag pc))\n        ((eq? (car inst) 'branch)\n         (make-branch inst machine labels flag pc))\n        ((eq? (car inst) 'goto)\n         (make-goto inst machine labels pc))\n        ((eq? (car inst) 'save)\n         (make-save inst machine stack pc))\n        ((eq? (car inst) 'restore)\n         (make-restore inst machine stack pc))\n        ((eq? (car inst) 'perform)\n         (make-perform inst machine labels ops pc))\n        (else (error \"Unknown instruction type - - ASSEMBLE\"\n                     inst)))) \n        For each type of instruction in the register-machine language,\n        there is a generator that builds an appropriate execution\n        procedure. The details of these procedures determine the\n        meaning of the individual instructions in the register-machine\n        language. We use data abstraction to isolate the detailed\n        syntax of register-machine expressions from the general\n        execution mechanism, as we did for evaluators in\n        section 4.1.2, by\n        using syntax procedures to extract and classify the parts of\n        an instruction.\n      ","5.2.3#h1":"Assign instructions","5.2.3#p2":"\n    The\n    make-assign procedure handles assign\n    instructions:\n    (define (make-assign inst machine labels operations pc)\n  (let ((target\n         (get-register machine (assign-reg-name inst)))\n        (value-exp (assign-value-exp inst)))\n    (let ((value-proc\n           (if (operation-exp? value-exp)\n               (make-operation-exp\n                value-exp machine labels operations)\n               (make-primitive-exp\n                (car value-exp) machine labels))))\n      (lambda ()     ; execution procedure for assign\n        (set-contents! target (value-proc))\n        (advance-pc pc))))) Make-assign\n\textracts the target register name (the second element of the instruction)\n\tand the value expression (the rest of the list that forms the instruction)\n\tfrom the assign instruction using the selectors\n\t(define (assign-reg-name assign-instruction)\n  (cadr assign-instruction))\n\n(define (assign-value-exp assign-instruction)\n  (cddr assign-instruction)) \n        The register name is looked up \n      \n    with\n    get-register\n    to produce the target register object.  The value expression is passed to\n    make-operation-exp\n    if the value is the result of an operation, and\n    \n    to\n    make-primitive-exp\n    otherwise.  These\n    procedures\n    (shown below)\n    parse\n    the value expression and produce an execution\n    procedure\n    for the value.  This is a\n    procedure\n    of no arguments, called \n    value-proc,\n    which will be evaluated during the simulation to produce the actual\n    value to be assigned to the register.  Notice that the work of looking\n    up the register name and\n    parsing\n    the value expression is performed\n    just once, at assembly time, not every time the instruction is\n    simulated.  This saving of work is the reason we use execution\n    procedures,\n    and corresponds directly to the saving in work we obtained by separating\n    program analysis from execution in the evaluator of\n    section 4.1.7.\n  ","5.2.3#p3":"\n    The result returned by\n    make-assign\n    is the execution\n    procedure\n    for the assign instruction.  When this\n    procedure\n    is called (by the machine model's executeprocedure),\n    it sets the contents of the target register to the result obtained by\n    executing\n    value-proc.\n    Then it advances the pc to the next instruction\n    by running the\n    procedure(define (advance-pc pc)\n  (set-contents! pc (cdr (get-contents pc)))) Advance-pc\n    is the normal termination for all instructions except\n    branch and\n    goto.\n      ","5.2.3#h2":"Test,\n\t  branch, and\n\t  goto\n\t  instructions\n\t","5.2.3#p4":"Make-test\n    handles test instructions in a similar way.\n    It extracts the expression that specifies the condition to be tested and\n    generates an execution\n    procedure\n    for it.  At simulation time, the\n    procedure\n    for the condition is called, the result is assigned to the\n    flag register, and the\n    pc is advanced:\n    (define (make-test inst machine labels operations flag pc)\n  (let ((condition (test-condition inst)))\n    (if (operation-exp? condition)\n        (let ((condition-proc\n               (make-operation-exp\n                condition machine labels operations)))\n          (lambda ()\n            (set-contents! flag (condition-proc))\n            (advance-pc pc)))\n        (error \"Bad TEST instruction - - ASSEMBLE\" inst))))\n\n(define (test-condition test-instruction)\n  (cdr test-instruction)) ","5.2.3#p5":"\n    The execution\n    procedure\n    for a branch instruction checks the contents of\n    the flag register and either sets the contents\n    of the pc to the branch destination (if the\n    branch is taken) or else just advances the pc\n    (if the branch is not taken).  Notice that the indicated destination in a\n    branch instruction must be a label, and the\n    make-branchprocedure\n    enforces this.  Notice also that the label is looked up at assembly time,\n    not each time the branch instruction is\n    simulated.\n    (define (make-branch inst machine labels flag pc)\n  (let ((dest (branch-dest inst)))\n    (if (label-exp? dest)\n        (let ((insts\n               (lookup-label labels (label-exp-label dest))))\n          (lambda ()\n            (if (get-contents flag)\n                (set-contents! pc insts)\n                (advance-pc pc))))\n        (error \"Bad BRANCH instruction - - ASSEMBLE\" inst))))\n\n(define (branch-dest branch-instruction)\n  (cadr branch-instruction)) ","5.2.3#p6":"\n    A\n    goto\n    instruction is similar to a branch, except that the destination may be\n    specified either as a label or as a register, and there is no condition to\n    check—the pc is always set to the\n    new destination.\n    (define (make-goto inst machine labels pc)\n  (let ((dest (goto-dest inst)))\n    (cond ((label-exp? dest)\n           (let ((insts\n                  (lookup-label labels\n                                (label-exp-label dest))))\n             (lambda () (set-contents! pc insts))))\n          ((register-exp? dest)\n           (let ((reg\n                  (get-register machine\n                                (register-exp-reg dest))))\n             (lambda ()\n               (set-contents! pc (get-contents reg)))))\n          (else (error \"Bad GOTO instruction - - ASSEMBLE\"\n            inst)))))\n\n(define (goto-dest goto-instruction)\n  (cadr goto-instruction)) ","5.2.3#h3":"Other instructions","5.2.3#p7":"\n    The stack instructions\n    save and restore\n    simply use the stack with the designated register and advance the\n    pc:\n    (define (make-save inst machine stack pc)\n  (let ((reg (get-register machine\n              (stack-inst-reg-name inst))))\n    (lambda ()\n      (push stack (get-contents reg))\n      (advance-pc pc))))\n\n(define (make-restore inst machine stack pc)\n  (let ((reg (get-register machine\n              (stack-inst-reg-name inst))))\n    (lambda ()\n      (set-contents! reg (pop stack))    \n      (advance-pc pc))))\n\n(define (stack-inst-reg-name stack-instruction)\n  (cadr stack-instruction)) ","5.2.3#p8":"\n    The final instruction type, handled by\n    make-perform,\n    generates an execution\n    procedure\n    for the action to be performed.  At simulation time, the action\n    procedure\n    is executed and the pc advanced.\n    (define (make-perform inst machine labels operations pc)\n  (let ((action (perform-action inst)))\n    (if (operation-exp? action)\n        (let ((action-proc\n               (make-operation-exp\n                action machine labels operations)))\n          (lambda ()\n            (action-proc)\n            (advance-pc pc)))\n        (error \"Bad PERFORM instruction - - ASSEMBLE\" inst))))\n\n(define (perform-action inst) (cdr inst)) ","5.2.3#h4":"Execution\n    procedures\n    for subexpressions","5.2.3#p9":"\n    The value of a\n    reg,\n    label, or\n    const\n    expression may be needed for assignment to a register\n    (make-assign)\n    or for input to an operation\n    (make-operation-exp,\n    below).  The following\n    procedure\n    generates execution\n    procedures\n    to produce values for these expressions during the simulation:\n    (define (make-primitive-exp exp machine labels)\n  (cond ((constant-exp? exp)\n         (let ((c (constant-exp-value exp)))\n           (lambda () c)))\n        ((label-exp? exp)\n         (let ((insts\n                (lookup-label labels\n                              (label-exp-label exp))))\n           (lambda () insts)))\n        ((register-exp? exp)\n         (let ((r (get-register machine\n                   (register-exp-reg exp))))\n           (lambda () (get-contents r))))\n        (else\n         (error \"Unknown expression type - - ASSEMBLE\" exp)))) \n        The syntax of reg,\n        label, and const\n        expressions is determined by\n        (define (register-exp? exp) (tagged-list? exp 'reg))\n\n(define (register-exp-reg exp) (cadr exp))\n\n(define (constant-exp? exp) (tagged-list? exp 'const))\n\n(define (constant-exp-value exp) (cadr exp))\n\n(define (label-exp? exp) (tagged-list? exp 'label))\n\n(define (label-exp-label exp) (cadr exp)) ","5.2.3#p10":"Assign,\n\tperform, and\n\ttest\n\tinstructions\n      \n    may include the application of a machine operation (specified by an\n    op expression) to some operands (specified\n    by reg and\n    const\n    expressions). The following\n    procedure\n    produces an execution\n    procedure\n    for an \"operation expression\"—a list containing the\n    operation and operand expressions from the instruction:\n    (define (make-operation-exp exp machine labels operations)\n  (let ((op (lookup-prim (operation-exp-op exp) operations))\n        (aprocs\n         (map (lambda (e)\n                (make-primitive-exp e machine labels))\n              (operation-exp-operands exp))))\n    (lambda ()\n      (apply op (map (lambda (p) (p)) aprocs))))) \n\tThe syntax of operation expressions is determined by\n\t(define (operation-exp? exp)\n  (and (pair? exp) (tagged-list? (car exp) 'op)))\n\n(define (operation-exp-op operation-exp)\n  (cadr (car operation-exp)))\n\n(define (operation-exp-operands operation-exp)\n  (cdr operation-exp)) \n\n    Observe that the treatment of operation expressions is very much like\n    the treatment of\n    procedure\n    applications by the\n    analyze-applicationprocedure\n    in the evaluator of section 4.1.7 in\n    that we generate an execution\n    procedure\n    for each operand.  \n    \n\tAt simulation time, we call the operand\t\n\tprocedures\n\tand apply the Scheme procedure\t\n      \n    that simulates the operation to the resulting values. \n\n    ","5.2.3#p11":"\n  The simulation\n    procedure\n    is found by looking up the operation name in the operation table for the\n    machine:\n    (define (lookup-prim symbol operations)\n  (let ((val (assoc symbol operations)))\n    (if val\n        (cadr val)\n        (error \"Unknown operation - - ASSEMBLE\" symbol)))) ","5.2.3#ex-5.9":"\n    The treatment of machine operations above permits them to operate\n    on labels as well as on constants and the contents of registers.\n    Modify the expression-processing\n    procedures\n    to enforce the condition that operations can be used only with registers\n    and constants.\n    ","5.2.3#ex-5.10":"\n\tDesign a new syntax for register-machine instructions and modify the\n\tsimulator to use your new syntax.  Can you implement your new\n\tsyntax without changing any part of the simulator except the\n\tsyntax procedures in this section?","5.2.3#ex-5.11":"\n    When we introduced\n    save and\n    restore in\n    section 5.1.4, we didn't specify\n    what would happen if you tried to restore a register that was not the last\n    one saved, as in the sequence\n    (save y)\n(save x)\n(restore y)There are several reasonable possibilities for the meaning of\n    restore:\n    (restore y)\n\tputs into y the last value saved on the\n\tstack, regardless of what register that value came from.  This is the\n\tway our simulator behaves.  Show how to take advantage of this\n\tbehavior to eliminate one instruction from the Fibonacci machine of\n\tsection 5.1.4\n\t(figure 5.12).\n      (restore y)\n\tputs into y the last value saved on the\n\tstack, but only if that value was saved from\n\ty; otherwise, it signals an error.  Modify\n\tthe simulator to behave this way.  You will have to change\n\tsave to put the register name on the stack\n\talong with the value.\n      (restore y)\n\tputs into y the last value saved from\n\ty regardless of what other registers were\n\tsaved after y and not restored.  Modify the\n\tsimulator to behave this way.  You will have to associate a separate\n\tstack with each register.  You should make the\n\tinitialize-stack\n\toperation initialize all the register stacks.\n      ","5.2.3#ex-5.12":"\n    The simulator can be used to help determine the data paths required\n    for implementing a machine with a given controller.  Extend\n    the assembler to store the following information in the machine model:\n    \n\ta list of all instructions, with duplicates removed, sorted by\n\tinstruction type\n\t(assign,\n\tgoto,\n\tand so on);\n      \n\ta list (without duplicates) of the registers used to hold entry points\n\t(these are the registers referenced by\n\tgoto\n\tinstructions);\n      \n\ta list (without duplicates) of the registers that are\n\tsaved\n\tor restored;\n      \n\tfor each register, a list (without duplicates) of the sources from\n\twhich it is assigned (for example, the sources for register\n\tval in the factorial machine of\n\tfigure 5.11 are\n\t(const 1)\n\tand\n\t((op *) (reg n) (reg val))).\n\t  \n    Extend the message-passing interface to the machine to provide access to\n    this new information.  To test your analyzer, define the Fibonacci machine\n    from figure 5.12 and examine the lists you\n    constructed.\n    ","5.2.3#ex-5.13":"\n    Modify the simulator so that it uses the controller sequence to determine\n    what registers the machine has rather than requiring a list of registers as\n    an argument to\n    make-machine.\n    Instead of preallocating the registers in\n    make-machine,\n    you can allocate them one at a time when they are first seen during assembly\n    of the instructions.\n    ","5.2.4":"5.2.4  \n    Monitoring Machine Performance","5.2.4#p1":"\n    Simulation is useful not only for verifying the correctness of a\n    proposed machine design but also for measuring the machine's\n    \n    performance.  For example, we can install in our simulation program a\n    \"meter\" that measures the number of stack operations used in a\n    computation.  To do this, we modify our simulated stack to keep track\n    of the number of times registers are saved on the stack and the\n    maximum depth reached by the stack, and add a message to the stack's\n    interface that prints the statistics, as shown below.\n    We also add an operation to the basic machine model to print the\n    stack statistics, by initializing\n    the-ops\n    in\n    make-new-machine\n    to\n    (list (list 'initialize-stack\n            (lambda () (stack 'initialize)))\n      (list 'print-stack-statistics\n            (lambda () (stack 'print-statistics))))\n\n    Here is the new version ofmake-stack:(define (make-stack)\n  (let ((s '())\n        (number-pushes 0)\n        (max-depth 0)\n        (current-depth 0))\n    (define (push x)\n      (set! s (cons x s))\n      (set! number-pushes (+ 1 number-pushes))\n      (set! current-depth (+ 1 current-depth))\n      (set! max-depth (max current-depth max-depth)))\n    (define (pop)\n      (if (null? s)\n          (error \"Empty stack - - POP\")\n          (let ((top (car s)))\n            (set! s (cdr s))\n            (set! current-depth (- current-depth 1))\n            top)))    \n    (define (initialize)\n      (set! s '())\n      (set! number-pushes 0)\n      (set! max-depth 0)\n      (set! current-depth 0)\n      'done)\n    (define (print-statistics)\n      (newline)\n      (display (list 'total-pushes  '= number-pushes\n                     'maximum-depth '= max-depth)))\n    (define (dispatch message)\n      (cond ((eq? message 'push) push)\n            ((eq? message 'pop) (pop))\n            ((eq? message 'initialize) (initialize))\n            ((eq? message 'print-statistics)\n             (print-statistics))\n            (else\n             (error \"Unknown request - - STACK\" message))))\n    dispatch)) ","5.2.4#p2":"\n    Exercises 5.15\n    through 5.19\n    describe other useful monitoring and debugging features that can be\n    added to the register-machine simulator.\n  ","5.2.4#ex-5.14":"\n    Measure the number of pushes and the maximum stack depth required to\n    compute\n    $n!$ for various small values of\n    $n$ using the factorial\n    machine shown in figure 5.11.  From your\n    data determine formulas in terms of $n$ for the\n    total number of push operations and the maximum stack depth used in\n    computing $n!$ for any\n    $n > 1$. Note that each of these is a linear\n    function of $n$ and is thus determined by two\n    constants.  In order to get the statistics printed, you will have to augment\n    the factorial machine with instructions to initialize the stack and print\n    the statistics. You may want to also modify the machine so that it\n    repeatedly reads a value for $n$, computes the\n    factorial, and prints\n    the result (as we did for the GCD machine in\n    figure 5.4), so that you will not have to\n    repeatedly invoke\n    get-register-contents,set-register-contents!,\n    and\n    start.","5.2.4#ex-5.15":"\n    Add \n    instruction counting\n    to the register machine simulation.\n    That is, have the machine model keep track of the number of\n    instructions executed.  Extend the machine model's interface to\n    accept a new message that prints the value of the instruction count and\n    resets the count to zero.\n    ","5.2.4#ex-5.16":"\n    Augment the simulator to provide for \n    instruction tracing.\n    That is, before each instruction is executed, the simulator should print\n    the text of the instruction.  Make the machine model accept\n    trace-on\n    and\n    trace-off\n    messages to turn tracing on and off.\n    ","5.2.4#ex-5.17":"\n    Extend the instruction tracing of\n    exercise 5.16 so that\n    before printing an instruction, the simulator prints any labels that\n    immediately precede that instruction in the controller sequence.  Be\n    careful to do this in a way that does not interfere with instruction\n    counting (exercise 5.15).\n    You will have to make the simulator retain the necessary label information.\n    ","5.2.4#ex-5.18":"\n    Modify the\n    make-registerprocedure\n    of section 5.2.1 so that registers can be\n    \n    traced. Registers should accept messages that turn tracing on and off.  When\n    a register is traced, assigning a value to the register should print the\n    name of the register, the old contents of the register, and the new contents\n    being assigned.  Extend the interface to the machine model to permit you to\n    turn tracing on and off for designated machine registers.\n    ","5.2.4#ex-5.19":"\n    Alyssa P. Hacker wants a \n    breakpoint feature in the simulator to help her debug her machine\n    designs.  You have been hired to install this feature for her.  She wants to\n    be able to specify a place in the controller sequence where the simulator\n    will stop and allow her to examine the state of the machine.  You are to\n    implement a\n    procedure\n(set-breakpoint $machine$ $label$ $n$)\n      \n    that sets a breakpoint just before the $n$th\n    instruction after the given label.  For example,\n    (set-breakpoint gcd-machine 'test-b 4)\n    installs a breakpoint in\n    gcd-machine\n    just before the assignment to register a.\n    When the simulator reaches the breakpoint it should print the label and the\n    offset of the breakpoint and stop executing instructions.  Alyssa can then\n    use\n    get-register-contents\n    and\n    set-register-contents!\n    to manipulate the state of the simulated machine.  She should then be able\n    to continue execution by saying\n    \n(proceed-machine $machine$)\n      \n    She should also be able to remove a specific breakpoint by means of\n    \n(cancel-breakpoint $machine$ $label$ $n$)\n      \n    or to remove all breakpoints by means of\n    \n(cancel-all-breakpoints $machine$)\n      ","5.3":"5.3  Storage Allocation and Garbage Collection","5.3#p1":"\n    In section 5.4, we will show how to implement a\n    Lisp      \n    evaluator as a register machine.  In order to simplify the discussion, we\n    will assume that our register machines can be equipped with a\n    list-structured memory, in which the basic operations for\n    manipulating list-structured data are primitive.  Postulating the existence\n    of such a memory is a useful abstraction when one is focusing on the\n    mechanisms of control in\n    a Lisp\n    interpreter, but this does not reflect a realistic view of the actual\n    primitive data operations of contemporary computers.  To obtain a more\n    complete picture of how \n    a Lisp system operates,\n    we must investigate how list structure can be represented in a way that is\n    compatible with conventional computer memories.\n  ","5.3#p2":"\n    There are two considerations in implementing list structure.  The first is\n    purely an issue of representation: how to represent the\n    \"box-and-pointer\" structure of\n    Lisp\n    pairs, using only the storage and addressing capabilities of typical computer\n    memories.  The second issue concerns the management of memory as a\n    computation proceeds. The operation of a\n    Lisp\n    system depends crucially on the ability to\n    continually create new data objects.  These include objects that are\n    explicitly created by the\n    Lisp procedures\n    being interpreted as well as structures created by the interpreter itself,\n    such as environments and argument lists.  Although the constant creation of\n    new data objects would pose no problem on a computer with an infinite amount\n    of rapidly addressable memory, computer memories are available only in\n    finite sizes (more's the pity).\n    Lisp\n    thus provide an \n    automatic storage allocation facility to\n    support the illusion of an infinite memory.  When a data object is no longer\n    needed, the memory allocated to it is automatically recycled and used to\n    construct new data objects.  There are various techniques for providing such\n    automatic storage allocation.  The method we shall discuss in this section\n    is called garbage collection.\n  ","5.3.1":"5.3.1  \n    Memory as Vectors","5.3.1#p1":"\n    A conventional computer memory can be thought of as an array of\n    cubbyholes, each of which can contain a piece of information.  Each\n    cubbyhole has a unique name, called its \n    address or \n    location.  Typical memory systems provide two primitive operations:\n    one that fetches the data stored in a specified location and one that\n    assigns new data to a specified location.  Memory addresses can be\n    incremented to support sequential access to some set of the\n    cubbyholes.  More generally, many important data operations require\n    that memory addresses be treated as data, which can be stored in\n    memory locations and manipulated in machine registers.  The\n    representation of list structure is one application of such \n    address arithmetic.\n  ","5.3.1#p2":"\n    To model computer memory, we use a new kind of data structure called a \n    vector.  Abstractly, a vector is a compound data object whose\n    individual elements can be accessed by means of an integer index in an\n    amount of time that is independent of the index. In order to describe memory operations, we use two\n    primitive Scheme procedures\n    for manipulating vectors:(vector-ref vector n)\n\treturns the nth element of the vector.\n      (vector-set! vector n value)\n\tsets the $n$th element of the vector to the\n\tdesignated value.\n      \n\n    For example, if v is a vector, then\n    (vector-ref v 5)\n    gets the fifth entry in the vector v and\n    (vector-set! v 5 7)\n    changes the value of the fifth entry of the vector\n    v\n    to 7. For computer memory, this access can be implemented\n    through the use of address arithmetic to combine a base address\n    that specifies the beginning location of a vector in memory with an\n    index that specifies the offset of a particular element of the\n    vector.\n  ","5.3.1#footnote-link-1":"1","5.3.1#footnote-link-2":"2","5.3.1#h1":"Representing\n    Lisp\n    data","5.3.1#p3":"\n    We can use vectors to implement the basic pair structures required for a\n    list-structured memory.  Let us imagine that computer memory is divided into\n    two vectors: \n    the-cars\n    and \n    the-cdrs.\n    We will represent list structure as follows: A pointer to a pair is an index\n    into the two vectors.  The \n    car\n    of the pair is the entry in\n    the-cars\n    with the designated index, and the \n    cdr\n    of the pair is the entry in\n    the-cdrs\n    with the designated index.  We also need a representation for objects other\n    than pairs (such as numbers and\n    symbols)\n    and a way to distinguish one kind of data from another. There are many\n    methods of accomplishing this, but they all reduce to using \n    typed pointers, that is, to extending the notion of\n    \"pointer\" to include information on data type. The data type enables the system to\n    distinguish a pointer to a pair (which consists of the \"pair\"\n    data type and an index into the memory vectors) from pointers to other\n    kinds of data (which consist of some other data type and whatever is\n    being used to represent data of that type).  Two data objects are\n    \n    considered to be the same\n    (eq?\n    if their pointers are identical.\n    \n\tFigure \n    illustrates the use of this method to represent \n    ((1 2) 3 4),\n    whose box-and-pointer diagram is also shown.  We use letter prefixes to\n    denote the data-type information.  Thus, a pointer to the pair with\n    index 5 is denoted p5, the empty list\n    is denoted by the pointer e0, and a pointer to\n    the number 4 is denoted n4.  In the\n    box-and-pointer diagram, we have indicated at the lower left of each pair\n    the vector index that specifies where the\n    car\n    and\n    cdr\n    of the pair are stored.  The blank locations in\n    the-cars\n    and\n    the-cdrs\n    may contain parts of other list structures (not of interest here).\n    ","5.3.1#footnote-link-3":"3","5.3.1#fig-":"","5.3.1#p4":"\n    A pointer to a number, such as n4,\n    might consist of a type indicating numeric data together with the\n    actual representation of the number 4.\n    To deal with numbers that are too large to be represented in the fixed\n    amount of space allocated for a single pointer, we could use a distinct \n    bignum data type, for which the pointer designates a list in which\n    the parts of the number are stored.","5.3.1#footnote-link-4":"4","5.3.1#footnote-link-5":"5","5.3.1#p5":"\n\tA\n\t\n\tsymbol might be represented as a typed pointer that designates a\n\tsequence of the characters that form the symbol's\n\tprinted representation. This sequence is constructed by the\n\tLisp reader\n\twhen the character string is initially encountered in input.  Since\n\twe want two instances of a symbol\n\tto be recognized as the \"same\" symbol by\n\teq?\n\tand we want\n\teq?\n\tto be a simple test for equality of pointers, we must ensure that if the\n\treader sees the same character string twice, it will use the same pointer\n\t(to the same sequence of characters) to represent both occurrences.\n\tTo accomplish this, the reader maintains a table, traditionally called the\n\tobarray, of all the symbols it has ever encountered.  When the\n        reader encounters a character string and is about to construct a\n        symbol, it checks the obarray to see if it\n\thas ever before seen the same character string.  If it has not, it\n\tuses the characters to construct a new symbol (a typed pointer to a\n\tnew character sequence) and enters this pointer in the obarray.\n\tIf the reader has seen the string before, it returns the\n        symbol pointer stored in the obarray.\n\tThis process of replacing character strings\n\tby unique pointers is called \n\tinterning symbols.\n      ","5.3.1#h2":"Implementing the primitive list operations","5.3.1#p6":"\n    Given the above representation scheme, we can replace each\n    \"primitive\" list operation of a register machine with one or\n    more primitive vector operations.  We will use two registers,\n    the-cars\n    and\n    the-cdrs,\n    to identify the memory vectors, and will\n    assume that\n    vector-ref\n    and\n    vector-set!\n    are available as primitive operations.  We also assume that numeric\n    operations on pointers (such as incrementing a pointer, using a pair pointer\n    to index a vector, or adding two numbers) use only the index portion of\n    the typed pointer.\n  ","5.3.1#p7":"\n    For example, we can make a register machine support the instructions\n    \n(assign reg$_{1}$ (op car) (reg reg$_{2}$))\n\n(assign reg$_{1}$ (op cdr) (reg reg$_{2}$))\n      \n    if we implement these, respectively, as\n    \n(assign reg$_{1}$ (op vector-ref) (reg the-cars) (reg reg$_{2}$))\n\n(assign reg$_{1}$ (op vector-ref) (reg the-cdrs) (reg reg$_{2}$))\n      \n\n    The instructions\n    \n(perform (op set-car!) (reg reg$_{1}$) (reg reg$_{2}$))\n\n(perform (op set-cdr!) (reg reg$_{1}$) (reg reg$_{2}$))\n      \n    are implemented as\n    \n(perform\n (op vector-set!) (reg the-cars) (reg reg$_{1}$) (reg reg$_{2}$))\n\n (perform\n (op vector-set!) (reg the-cdrs) (reg reg$_{1}$) (reg reg$_{2}$))\n      ","5.3.1#p8":"Cons\n    is performed by allocating an unused index and storing the arguments to\n    cons\n    in\n    the-cars\n    and\n    the-cdrs\n    at that indexed vector position.  We presume that there is a special\n    register,\n    free, that always holds a pair pointer\n    containing the next available index, and that we can increment the index\n    part of that pointer to find the next free location.\n    For example, the instruction\n    \n(assign reg$_{1}$ (op cons) (reg reg$_{2}$) (reg reg$_{3}$))\n      \n    is implemented as the following sequence of vector\n    operations:\n(perform\n (op vector-set!) (reg the-cars) (reg free) (reg reg$_{2}$))\n(perform\n (op vector-set!) (reg the-cdrs) (reg free) (reg reg$_{3}$))\n      (assign reg$_{1}$ (reg free))\n      (assign free (op +) (reg free) (const 1))\n      ","5.3.1#footnote-link-6":"6","5.3.1#footnote-link-7":"7","5.3.1#p9":"\n    The\n    eq?\n    operation\n    \n(op eq?) (reg reg$_{1}$) (reg reg$_{2}$)\n      \n    simply tests the equality of all fields in the registers, and\n    predicates such as\n    pair?,null?,symbol?,\n    and\n    number?,\n    need only check the type field.\n  ","5.3.1#h3":"Implementing stacks","5.3.1#p10":"\n    Although our register machines use stacks, we need do nothing special\n    here, since stacks can be modeled in terms of lists.  The stack can be\n    \n    a list of the saved values, pointed to by a special register\n    the-stack.\n    Thus,\n    (save reg)\n    can be implemented as\n    \n(assign the-stack (op cons) (reg reg) (reg the-stack))\n      \n    Similarly,\n    (restore reg)\n    can be implemented as\n    \n(assign reg (op car) (reg the-stack))\n(assign the-stack (op cdr) (reg the-stack))\n      \n    and\n    (perform (op initialize-stack))\n    can be implemented as\n    \n(assign the-stack (const ()))\n      \n\n    These operations can be further expanded in terms of the vector\n    operations given above.  In conventional computer architectures,\n    however, it is usually advantageous to allocate the stack as a\n    separate vector.  Then pushing and popping the stack can be\n    accomplished by incrementing or decrementing an index into that\n    vector.\n  ","5.3.1#ex-5.20":"\n    Draw the box-and-pointer representation and the memory-vector representation\n    \n\t(as in figure )\n      \n    of the list structure produced by\n\n    (define x (cons 1 2))\n(define y (list x x))\n    with the free pointer initially\n    p1.  What is the final value of\n    free$\\,$?  What\n    pointers represent the values of x and\n    y?\n    ","5.3.1#ex-5.21":"\n    Implement register machines for the following\n    procedures.\n    Assume that the list-structure memory operations are available as\n    machine primitives.\n    \n\tRecursive\n\tcount-leaves:(define (count-leaves tree)\n  (cond ((null? tree) 0)\n        ((not (pair? tree)) 1)\n        (else (+ (count-leaves (car tree))\n                 (count-leaves (cdr tree)))))) \n\tRecursive\n\tcount-leaves\n\twith explicit counter:\n\t(define (count-leaves tree)\n  (define (count-iter tree n)\n    (cond ((null? tree) n)\n          ((not (pair? tree)) (+ n 1))\n          (else (count-iter (cdr tree)\n                            (count-iter (car tree) n)))))\n  (count-iter tree 0)) ","5.3.1#ex-5.22":"\n    Exercise 3.12 of\n    section 3.3.1\n    presented an\n    appendprocedure\n    that appends two lists to form a new list and an\n    append!\n      procedure\n      \n    that splices two lists together.  Design a register machine to\n    \n    implement\n    each of these\n    procedures.\n    Assume that the list-structure memory operations are\n    available as primitive operations.\n    ","5.3.1#footnote-1":"We could represent\n    memory as lists of items. However, the access time would then not be\n    independent of the index, since accessing the\n    $n$th element of a list requires\n    $n-1$cdr\n    operations.","5.3.1#footnote-2":"For completeness, we should specify a\n    make-vector\n    operation that constructs vectors.  However, in the present application we\n    will use vectors only to model fixed divisions of the computer\n    memory.","5.3.1#footnote-3":"This is\n    precisely the same \n    \"tagged data\" idea we introduced in chapter 2 for\n    dealing with generic operations.  Here, however, the data types are\n    included at the primitive machine level rather than constructed\n    through the use of lists.\n    ","5.3.1#p11":"\n      Type information may be encoded in\n      a variety of ways, depending on the details of the machine on which the\n      Lisp\n      system is to be implemented.  The execution efficiency of\n      Lisp\n      programs will be strongly dependent on how cleverly this choice is made, but\n      it is difficult to formulate general design rules for good choices.  The\n      most straightforward way to implement typed pointers is to allocate a fixed\n      set of bits in each pointer to be a\n      type field that encodes the data type.  Important questions to be\n      addressed in designing such a representation include the following:\n      How many type bits are required?  How large must the vector indices\n      be?  How efficiently can the primitive machine instructions be used to\n      manipulate the type fields of pointers?  Machines that include special\n      hardware for the efficient handling of type fields are said to have\n      tagged architectures.\n    ","5.3.1#footnote-4":"This decision on the\n    representation of numbers determines whether\n    eq?,\n    which tests equality of pointers, can be used to test for equality of\n    numbers.  If the pointer contains the number itself, then equal numbers will\n    have the same pointer.  But if the pointer contains the index of a location\n    where the number is stored, equal numbers will be guaranteed to have\n    equal pointers only if we are careful never to store the same number\n    in more than one location.","5.3.1#footnote-5":"This is just like writing a\n    number as a sequence of digits, except that each \"digit\" is a\n    number between 0 and the largest number that can be stored in a single\n    pointer.","5.3.1#footnote-6":"There are\n    other ways of finding free storage.  For example, we could link together\n    all the unused pairs into a \n    free list.  Our free locations are consecutive (and hence can be\n    accessed by incrementing a pointer) because we are using a compacting\n    garbage collector, as we will see in\n    section 5.3.2.","5.3.1#footnote-7":"This is essentially the implementation of\n    cons\n    in terms of\n    set-car!\n    and\n    set-cdr!,\n    as described in section 3.3.1.\n    The operation\n    get-new-pair\n    used in that implementation is realized here by the\n    free pointer.","5.3.2":"5.3.2  \n    Maintaining the Illusion of Infinite Memory","5.3.2#p1":"\n    The representation method outlined in\n    section 5.3.1 solves the problem of\n    implementing list structure, provided that we have an infinite amount of\n    memory. With a real computer we will eventually run out of free space in\n    which to construct new pairs. However, most of the pairs\n    generated in a typical computation are used only to hold intermediate\n    results.  After these\n    results are accessed, the pairs are no longer needed—they are \n    garbage.  For instance, the computation\n    (accumulate + 0 (filter odd? (enumerate-interval 0 n)))\n    constructs two lists: the enumeration and the result of filtering\n    the enumeration.  When the accumulation is complete, these lists are\n    no longer needed, and the allocated memory can be reclaimed.  If we\n    can arrange to collect all the garbage periodically, and if this turns\n    out to recycle memory at about the same rate at which we construct new\n    pairs, we will have preserved the illusion that there is an infinite\n    amount of memory.\n  ","5.3.2#footnote-link-1":"1","5.3.2#p2":"\n    In order to recycle pairs, we must have a way to determine which\n    allocated pairs are not needed (in the sense that their contents can\n    no longer influence the future of the computation).  The method we\n    shall examine for accomplishing this is known as garbage\n    collection.  Garbage collection is based on the observation that, at\n    any moment in \n    a Lisp interpretation,\n    the only objects that can\n    affect the future of the computation are those that can be reached by\n    some succession of\n    car\n    and\n    cdr\n    operations starting from the pointers that are currently in the machine\n    registers.  Any memory cell that is not so accessible may be\n    recycled.\n  ","5.3.2#footnote-link-2":"2","5.3.2#p3":"\n    There are many ways to perform garbage collection.  The method we\n    shall examine here is called \n    stop-and-copy.  The basic idea is to divide memory into two\n    halves: \"working memory\" and \"free memory.\"  When\n    cons\n    constructs pairs, it allocates these in working memory.  When working memory\n    is full, we perform garbage collection by locating all the useful pairs in\n    working memory and copying these into consecutive locations in free memory.\n    (The useful pairs are located by tracing all the\n    car\n     and\n    cdr\n    pointers, starting with the machine registers.)  Since we do not copy the\n    garbage, there will presumably be additional free memory that we can\n    use to allocate new pairs.  In addition, nothing in the working memory\n    is needed, since all the useful pairs in it have been copied.  Thus,\n    if we interchange the roles of working memory and free memory, we can\n    continue processing; new pairs will be allocated in the new working\n    memory (which was the old free memory).  When this is full, we can\n    copy the useful pairs into the new free memory (which was the old\n    working memory).","5.3.2#footnote-link-3":"3","5.3.2#h1":"Implementation of a stop-and-copy garbage collector","5.3.2#p4":"\n    We now use our register-machine language to describe the stop-and-copy\n    algorithm in more detail.  We will assume that there is a register\n    called \n    root that contains a pointer to a structure\n    that eventually points at all accessible data.  This can be arranged by\n    storing the contents of all the machine registers in a preallocated list\n    pointed at by root just before starting\n    garbage collection.\n    We also assume that, in addition to the current working memory, there is\n    free memory available into which we can copy the useful data.  The current\n    working memory consists of vectors whose base addresses are in \n    registers called\n    the-cars\n    and\n    the-cdrs,\n    and the free memory is in registers called \n    new-cars\n    and\n    new-cdrs.","5.3.2#footnote-link-4":"4","5.3.2#p5":"\n    Garbage collection is triggered when we exhaust the free cells in the\n    current working memory, that is, when a\n    cons\n    operation attempts to increment the free\n    pointer beyond the end of the memory vector.  When the garbage-collection\n    process is complete, the root pointer will\n    point into the new memory, all objects accessible from the\n    root will have been moved to the new memory,\n    and the free pointer will indicate the next\n    place in the new memory where a new pair can be allocated.  In addition,\n    the roles of working memory and new memory will have been\n    interchanged—new pairs will be constructed in the new memory,\n    beginning at the place indicated by free, and\n    the (previous) working memory will be available as the new memory for the\n    next garbage collection.\n    Figure \n    shows the arrangement of memory just before and just after garbage\n    collection.\n    ","5.3.2#fig-":"","5.3.2#p6":"\n    The state of the garbage-collection process is controlled by\n    maintaining two pointers:\n    free and\n    scan.  These are initialized to point to the\n    beginning of the new memory.  The algorithm begins by relocating the pair\n    pointed at by root to the beginning of the new\n    memory.  The pair is copied, the root pointer\n    is adjusted to point to the new location, and the\n    free pointer is incremented.  In addition, the\n    old location of the pair is marked to show that its contents have been\n    moved.  This marking is done as follows: In the\n    car\n    position, we place a special tag that signals that this is an already-moved\n    object.  (Such an object is traditionally called a \n    broken heart.)\n    In the\n    cdr\n    position we place a \n    forwarding address that points at the location to which the object\n    has been moved.\n  ","5.3.2#footnote-link-5":"5","5.3.2#p7":"\n    After relocating the root, the garbage collector enters its basic\n    cycle.  At each step in the algorithm, the\n    scan pointer\n    (initially pointing at the relocated root) points at a pair that has\n    been moved to the new memory but whose\n    car\n    and\n    cdr\n    pointers still refer to objects in the old memory.  These objects are each\n    relocated, and the scan pointer is incremented.\n    To relocate an object (for example, the object indicated by the\n    car\n    pointer of the pair we are scanning) we check to see if the object has\n    already been moved (as indicated by the presence of a broken-heart tag\n    in the\n    car\n    position of the object).  If the object has not\n    already been moved, we copy it to the place indicated by\n    free,\n    update free, set up a broken heart at the\n    object's old location, and update the pointer to the object (in this\n    example, the\n    car\n    pointer of the pair we are scanning) to point\n    to the new location.  If the object has already been moved, its\n    forwarding address (found in the\n    cdr\n    position of the broken heart) is substituted for the pointer in the pair\n    being scanned. Eventually, all accessible objects will have been moved and\n    scanned, at which point the scan pointer will\n    overtake the free pointer and the process will\n    terminate.\n  ","5.3.2#p8":"\n    We can specify the stop-and-copy algorithm as a sequence of instructions for\n    a register machine.  The basic step of relocating an object is accomplished\n    by a subroutine called\n    relocate-old-result-in-new.\n    This subroutine gets its argument, a pointer to the object to be relocated,\n    from a register named \n    old.  It relocates the designated object\n    (incrementing free in the process),\n    puts a pointer to the relocated object into a register called \n    new, and returns by branching to the entry\n    point stored in the register\n    relocate-continue.\n    To begin garbage collection, we invoke this subroutine to relocate the\n    root pointer, after initializing\n    free and scan.\n    When the relocation of root has been\n    accomplished, we install the new pointer as the new\n    root and enter the main loop of the garbage\n    collector.\n    begin-garbage-collection\n  (assign free (const 0))\n  (assign scan (const 0))\n  (assign old (reg root))\n  (assign relocate-continue (label reassign-root))\n  (goto (label relocate-old-result-in-new))\nreassign-root\n  (assign root (reg new))\n  (goto (label gc-loop))","5.3.2#p9":"\n    In the main loop of the garbage collector we must determine whether\n    there are any more objects to be scanned.  We do this by testing\n    whether the scan pointer is coincident with\n    the free pointer.  If the pointers are equal,\n    then all accessible objects have been relocated, and we branch to\n    gc-flip,\n    which cleans things up so that we can continue the interrupted computation.\n    If there are still pairs to be scanned, we call the relocate subroutine to\n    relocate the\n    car\n    of the next pair (by placing the\n    car\n    pointer in old).  The\n    relocate-continue\n    register is set up so that the subroutine will return to update the\n    car\n    pointer.\n    gc-loop\n  (test (op =) (reg scan) (reg free))\n  (branch (label gc-flip))\n  (assign old (op vector-ref) (reg new-cars) (reg scan))\n  (assign relocate-continue (label update-car))\n  (goto (label relocate-old-result-in-new))","5.3.2#p10":"\n    At\n    update-car,\n    we modify the\n    car\n    pointer of the pair being scanned, then proceed to relocate the\n    cdr\n    of the pair.  We return to\n    update-cdr\n    when that relocation has been accomplished. After relocating and updating\n    the\n    cdr,\n    we are finished scanning that pair, so we continue with the main loop.\n    update-car\n  (perform\n  (op vector-set!) (reg new-cars) (reg scan) (reg new))\n  (assign old (op vector-ref) (reg new-cdrs) (reg scan))\n  (assign relocate-continue (label update-cdr))\n  (goto (label relocate-old-result-in-new))\n\nupdate-cdr\n  (perform\n  (op vector-set!) (reg new-cdrs) (reg scan) (reg new))\n  (assign scan (op +) (reg scan) (const 1))\n  (goto (label gc-loop))","5.3.2#p11":"\n    The subroutine\n    relocate-old-result-in-new\n    relocates objects as follows: If the object to be relocated (pointed at by\n    old) is not a pair, then we return the same\n    pointer to the object unchanged (in new).\n    (For example, we may be scanning a pair whose\n    car\n    is the number 4.  If we represent the\n    car\n    by n4, as described in\n    section 5.3.1, then we want the\n    \"relocated\"car\n    pointer to still be n4.)  Otherwise, we\n    must perform the relocation.  If the\n    car\n    position of the pair to be relocated contains a broken-heart tag, then the\n    pair has in fact already been moved, so we retrieve the forwarding address\n    (from the\n    cdr\n    position of the broken heart) and return this in\n    new. If the pointer in\n    old points at a yet-unmoved pair, then we move\n    the pair to the first free cell in new memory (pointed at by\n    free) and set up the broken heart by storing a\n    broken-heart tag and forwarding address at the old location.\n    Relocate-old-result-in-new\n    uses a register \n    oldcr\n    to hold the\n    car\n    or the\n    cdr\n    of the object pointed at by old.\nrelocate-old-result-in-new\n  (test (op pointer-to-pair?) (reg old))\n  (branch (label pair))\n  (assign new (reg old))\n  (goto (reg relocate-continue))\npair\n  (assign oldcr (op vector-ref) (reg the-cars) (reg old))\n  (test (op broken-heart?) (reg oldcr))\n  (branch (label already-moved))\n  (assign new (reg free)) \n  \n  (assign free (op +) (reg free) (const 1))\n  \n  (perform (op vector-set!)\n  (reg new-cars) (reg new) (reg oldcr))\n  (assign oldcr (op vector-ref) (reg the-cdrs) (reg old))\n  (perform (op vector-set!)\n  (reg new-cdrs) (reg new) (reg oldcr))\n  \n  (perform (op vector-set!)\n  (reg the-cars) (reg old) (const broken-heart))\n  (perform\n  (op vector-set!) (reg the-cdrs) (reg old) (reg new))\n  (goto (reg relocate-continue))\nalready-moved\n  (assign new (op vector-ref) (reg the-cdrs) (reg old))\n  (goto (reg relocate-continue))\n      ","5.3.2#footnote-link-6":"6","5.3.2#p12":"\n    At the very end of the garbage collection process, we interchange the\n    role of old and new memories by interchanging pointers: interchanging\n    the-cars\n    with\n    new-cars,\n    and\n    the-cdrs\n    with\n    new-cdrs.\n    We will then be ready to perform another garbage\n    collection the next time memory runs out.\n    gc-flip\n      (assign temp (reg the-cdrs))\n      (assign the-cdrs (reg new-cdrs))\n      (assign new-cdrs (reg temp))\n      (assign temp (reg the-cars))\n      (assign the-cars (reg new-cars))\n      (assign new-cars (reg temp)) ","5.3.2#footnote-1":"This may not be true eventually,\n    because memories may get large enough so that it would be impossible\n    to run out of free memory in the lifetime of the computer.  For\n    example, there are about\n    $3\\times 10^{13}$ microseconds\n    in a year, so if we were to\n    cons\n    once per\n    microsecond\n    we would need about\n    $10^{15}$\n    cells of memory to build a machine\n    that could operate for 30 years without running out of memory.  That much\n    memory seems absurdly large by today's standards, but it is not\n    physically impossible.  On the other hand, processors are getting faster and \n    \n        a future computer may have \n      \n    large numbers of processors operating in\n    parallel on a single memory, so it may be possible to use up memory much\n    faster than we have postulated.","5.3.2#footnote-2":"We assume here that the stack is represented as a list\n    as described in section 5.3.1, so that\n    items on the stack are accessible via the pointer in the stack\n    register.","5.3.2#footnote-3":"This idea was invented and first implemented\n    by\n    \n    Minsky, as part of the implementation of \n    \n    Lisp for the PDP-1 at the\n    \n    MIT Research Laboratory of Electronics.  It was further developed by\n    \n    Fenichel and Yochelson (1969) for use in the Lisp implementation for the\n    \n    Multics time-sharing system.  Later, \n    \n    Baker (1978) developed a \"real-time\" version of the method,\n    which does not require the computation to stop during garbage collection.\n    Baker's idea was extended by \n    \n    Hewitt,\n    \n    Lieberman, and\n    \n    Moon (see Lieberman and Hewitt 1983) to take\n    advantage of the fact that some structure is more volatile\n    and other structure is more permanent.  \n    \n    An alternative commonly used garbage-collection technique is the \n    mark-sweep method. This consists of tracing all the structure\n    accessible from the machine registers and marking each pair we reach.\n    We then scan all of memory, and any location that is unmarked is\n    \"swept up\" as garbage and made available for reuse.  A full\n    discussion of the mark-sweep method can be found in \n    Allen 1978.\n    \n    The Minsky-Fenichel-Yochelson algorithm is the dominant algorithm in\n    use for large-memory systems because it examines only the useful part\n    of memory.  This is in contrast to mark-sweep, in which the sweep\n    phase must check all of memory.  A second advantage of stop-and-copy\n    is that it is a \n    compacting garbage collector.  That is, at the\n    end of the garbage-collection phase the useful data will have been\n    moved to consecutive memory locations, with all garbage pairs\n    compressed out.  This can be an extremely important performance\n    consideration in machines with virtual memory, in which accesses to\n    widely separated memory addresses may require extra paging\n    operations.","5.3.2#footnote-4":"This list of\n    registers does not\n    include\n    the registers used by the storage-allocation\n    \n\tsystem—root,\n      the-cars,the-cdrs,\n    and the other registers that will be introduced in this section.","5.3.2#footnote-5":"The term \n    broken heart was coined by\n    \n    David Cressey, who wrote a garbage collector\n    for \n    \n    MDL, a dialect of Lisp developed at MIT during the early 1970s.","5.3.2#footnote-6":"The\n    garbage collector uses the low-level predicate\n    pointer-to-pair?\n    instead of the list-structure\n    pair?\n    operation because in a real system there might be various things\n    that are treated as pairs for garbage-collection purposes.\n    For example,\n    \n    in a Scheme system that conforms to the IEEE standard\n      \n    a\n    \n\tprocedure\n      \n    object may be implemented as a special kind of\n    \"pair\" that doesn't satisfy the\n    pair?\n    predicate.\n    For simulation purposes,\n    pointer-to-pair?\n    can be implemented as\n    pair?.","5.4":"5.4  The Explicit-Control Evaluator","5.4#p1":"\n    In section 5.1 we saw how to\n    transform simple\n    Scheme\n    programs into descriptions of register\n    machines.  We will now perform this transformation on a more complex\n    program, the metacircular evaluator of\n    sections 4.1.1–4.1.4,\n    which shows how the behavior of a\n    Scheme\n    interpreter can be described in terms of the\n    \n\tprocedures\n\teval\n    and apply.\n    The explicit-control\n    evaluator that we develop in this section shows how the underlying\n    procedure-calling\n    and argument-passing mechanisms used in the\n    evaluation process can be described in terms of operations on\n    registers and stacks.  In addition, the explicit-control evaluator can\n    serve as an implementation of a\n    Scheme\n    interpreter, written in a language that is very similar to the native machine\n    language of conventional computers.  The evaluator can be executed by the\n    register-machine simulator of section 5.2.\n    Alternatively, it can be used as a starting point for building a\n    machine-language implementation of a\n    Scheme\n    evaluator, or even a\n    \n    special-purpose machine for evaluating\n    Scheme expressions.\n    Figure 5.16 shows such a hardware\n    implementation: a silicon chip that acts as an evaluator for\n    Scheme.\n    The chip designers started with the data-path and controller specifications\n    for a register machine similar to the evaluator described in this section\n    and used design automation programs to construct the\n    integrated-circuit layout.","5.4#footnote-link-1":"1","5.4#fig-5.16":"","5.4#h1":"Registers and operations","5.4#p2":"\n    In designing the explicit-control evaluator, we must specify the\n    operations to be used in our register machine.  We described the\n    metacircular evaluator in terms of abstract syntax, using\n    procedures\n    such as\n    quoted?\n    and\n    make-procedure.\n    In implementing the\n    register machine, we could expand these\n    procedures\n    into sequences of\n    elementary list-structure memory operations, and implement these\n    operations on our register machine.  However, this would make our\n    evaluator very long, obscuring the basic structure with\n    details.  To clarify the presentation, we will include as primitive\n    operations of the register machine the syntax\n    procedures\n    given in\n    section 4.1.2 and the\n    procedures\n    for representing environments and other runtime data given in\n    sections 4.1.3\n    and 4.1.4.\n    In order to completely specify an evaluator that could be programmed\n    in a low-level machine language or implemented in hardware, we would\n    replace these operations by more elementary operations, using the\n    list-structure implementation we described in\n    section 5.3.\n  ","5.4#p3":"\n    Our\n    Scheme\n    evaluator register machine includes a stack and seven\n    registers:\n    exp, env,\n    val,\n    continue,\n    proc,\n      argl, and\n    unev.\n    Exp\n    is used to hold the\n    \n\texpression\n      \n    to be evaluated, and env contains the environment in\n    which the evaluation is to be performed. At the end of an evaluation,\n    val contains the value obtained by evaluating the\n    expression\n    in the designated environment. The continue register is\n    used to implement recursion, as explained in section 5.1.4. (The evaluator needs to call itself recursively, since\n    evaluating an expression requires  evaluating its\n    subexpressions.) The registers\n    proc,\n      argl, and unev are\n    used in evaluating function applications. \n  ","5.4#p4":"\n    We will not provide a data-path diagram to show how the registers and\n    operations of the evaluator are connected, nor will we give the\n    complete list of machine operations.  These are implicit in the\n    evaluator's controller, which will be presented in detail.\n  ","5.4#footnote-1":"See \n    Batali et al. 1982 for more\n    information on the chip and the method by which it was designed.","5.4.1":"5.4.1","5.4.1#p1":"\n    The central element in the evaluator is the sequence of instructions beginning at\n    eval-dispatch.\n    This corresponds to the\n    evalprocedure\n    of the metacircular evaluator described in section 4.1.1. When the controller starts at\n    eval-dispatch,\n    it evaluates the \n    expression\n    specified by\n    exp\n    in the environment specified by env. When evaluation is\n    complete, the controller will go to the entry point stored in\n    continue, and the val\n    register will hold the value of the\n    expression.\n    As with the metacircular\n    eval,\n    the structure of\n    eval-dispatch\n    is a case analysis on the syntactic type of the\n    expression\n    to be evaluated.eval-dispatch\n(test (op self-evaluating?) (reg exp))\n(branch (label ev-self-eval))\n(test (op variable?) (reg exp))\n(branch (label ev-variable))\n(test (op quoted?) (reg exp))\n(branch (label ev-quoted))\n(test (op assignment?) (reg exp))\n(branch (label ev-assignment))\n(test (op definition?) (reg exp))\n(branch (label ev-definition))\n(test (op if?) (reg exp))\n(branch (label ev-if))\n(test (op lambda?) (reg exp))\n(branch (label ev-lambda))\n(test (op begin?) (reg exp))\n(branch (label ev-begin))\n(test (op application?) (reg exp))\n(branch (label ev-application))\n(goto (label unknown-expression-type))","5.4.1#footnote-link-1":"1","5.4.1#h1":"Evaluating simple expressions","5.4.1#p2":"\n    Numbers and strings (which are self-evaluating),\n    variables, quotations,\n    and\n    lambda\n    expressions have no subexpressions to be evaluated. For these, the evaluator simply\n    places the correct value in the val register and\n    continues execution at the entry point specified by\n    continue. Evaluation of simple expressions is performed\n    by the following controller code:\n    ev-self-eval\n(assign val (reg exp))\n(goto (reg continue))\nev-variable\n(assign val (op lookup-variable-value) (reg exp) (reg env))\n(goto (reg continue))\nev-quoted\n(assign val (op text-of-quotation) (reg exp))\n(goto (reg continue))\nev-lambda\n(assign unev (op lambda-parameters) (reg exp))\n(assign exp (op lambda-body) (reg exp))\n(assign val (op make-procedure)\n(reg unev) (reg exp) (reg env))\n(goto (reg continue))\n    Observe how\n    ev-lambda\n    uses the\n    unev\n    and\n    exp\n    registers to hold the parameters and body of the lambda expression so\n    that they can be passed to the\n    make-procedure\n    operation, along with the environment in env.\n  ","5.4.1#h2":"\n\t  Evaluating procedure applications\n\t","5.4.1#p3":"\n\tA procedure application is specified by a combination containing an\n\toperator and operands. The operator is a subexpression whose value is a\n\tprocedure, and the operands are subexpressions whose values are the\n\targuments to which the procedure should be applied. The metacircular\n\teval\n\thandles applications by calling itself recursively to\n\tevaluate each element of the combination, and then passing the results\n\tto apply, which performs the actual\n\tprocedure\n\tapplication.  The\n\texplicit-control evaluator does the same thing; these recursive calls\n\tare implemented by\n\tgoto\n\tinstructions, together with \n\t\n\tuse of the stack to save registers that will be restored after the recursive\n\tcall returns.  Before each call we will be careful to identify which\n\tregisters must be saved (because their values will be needed\n\tlater).","5.4.1#footnote-link-2":"2","5.4.1#p4":"\n\tWe begin the evaluation of an application by evaluating the\n\toperator to produce a procedure, which will later be applied to the evaluated\n\toperands. To evaluate the operator, we move it to the \n\texp\n\tregister and go to eval-dispatch.\n\tThe environment in the env register is already\n\tthe correct one in which to evaluate the operator.\n\tHowever, we save env because we will need it\n\tlater to evaluate the operands. We also extract the operands\n\tinto unev and save this on the stack.  We set\n\tup continue so that\n\teval-dispatch will resume at\n\tev-appl-did-operator\n\tafter the operator has been evaluated.  First, however, we save the old value of\n\tcontinue, which tells the controller where to\n\tcontinue after the application.\n\tev-application\n  (save continue)\n  (save env)\n  (assign unev (op operands) (reg exp))\n  (save unev)\n  (assign exp (op operator) (reg exp))\n  (assign continue (label ev-appl-did-operator))\n  (goto (label eval-dispatch))","5.4.1#p5":"\n\tUpon returning from evaluating the operator subexpression,\n\twe proceed to evaluate the operands of the combination\n\tand to accumulate the resulting arguments in a list, held in\n\targl.\n\tFirst we restore the unevaluated operands and the environment.  We\n\tinitialize argl to an empty list.  Then we\n\tassign to the\n\tproc\n\tregister the procedure that was produced by evaluating the operator.\n\tIf there are no operands, we go directly to\n\tapply-dispatch. Otherwise we save\n\tproc\n\ton the stack and start the argument-evaluation\n\tloop:ev-appl-did-operator\n  (restore unev)                  ; the operands\n  (restore env)\n  (assign argl (op empty-arglist))\n  (assign proc (reg val))         ; the operator\n  (test (op no-operands?) (reg unev))\n  (branch (label apply-dispatch))\n  (save proc)","5.4.1#footnote-link-3":"3","5.4.1#p6":"\n\tEach cycle of the argument-evaluation loop evaluates an \n\toperand\n\tfrom the list in unev and accumulates the\n\tresult into argl. To evaluate an\n\toperand, we place it in the \n\texp\n\tregister and go to\n\teval-dispatch,\n\tafter setting continue so that execution will\n\tresume with the argument-accumulation phase.  But first we save the\n\targuments accumulated so far (held in argl), the\n\tenvironment (held in env), and the remaining\n\toperands\n\tto be evaluated (held in unev).  A special case\n\tis made for the evaluation of the last\n\toperand which is handled at\n\tev-appl-last-arg.\n\tev-appl-operand-loop\n  (save argl)\n  (assign exp (op first-operand) (reg unev))\n  (test (op last-operand?) (reg unev))\n  (branch (label ev-appl-last-arg))\n  (save env)\n  (save unev)\n  (assign continue (label ev-appl-accumulate-arg))\n  (goto (label eval-dispatch))","5.4.1#p7":"\n\tWhen an operand has been evaluated, the value is accumulated into the list\n\theld in argl.  The operand\n\tis then removed from the list of unevaluated operands\n\tin unev, and\n\tthe argument-evaluation loop continues.\n\tev-appl-accumulate-arg\n  (restore unev)\n  (restore env)\n  (restore argl)\n  (assign argl (op adjoin-arg) (reg val) (reg argl))\n  (assign unev (op rest-operands) (reg unev))\n  (goto (label ev-appl-operand-loop))","5.4.1#p8":"\n\tEvaluation of the last argument is handled differently.  There is no\n\tneed to save the environment or the list of unevaluated \n\toperands before going to\n\teval-dispatch,\n\tsince they will not be required after the last\n\toperand\tis evaluated.\n\tThus, we return from the evaluation to a special entry point\n\tev-appl-accum-last-arg,\n\twhich restores the argument list, accumulates the new argument, restores the\n\tsaved procedure, and goes off to perform the application.ev-appl-last-arg\n  (assign continue (label ev-appl-accum-last-arg))\n  (goto (label eval-dispatch))\nev-appl-accum-last-arg\n  (restore argl)\n  (assign argl (op adjoin-arg) (reg val) (reg argl))\n  (restore proc)\n  (goto (label apply-dispatch))","5.4.1#footnote-link-4":"4","5.4.1#p9":"\n\tThe details of the argument-evaluation loop determine the\n\t\n\torder in which the interpreter evaluates the\n\toperands of a combination (e.g.,\n\tleft to right or right to left—see\n\texercise 3.8).  This order is not\n\tdetermined by the metacircular evaluator, which inherits its control\n\tstructure from the underlying Scheme in which it is implemented.\n        Because the first-operand selector (used in\n        ev-appl-operand-loop to extract successive\n        operands from unev) is implemented as\n        car and the\n        rest-operands selector is implemented as\n        cdr, the explicit-control evaluator will\n        evaluate the operands of a combination in left-to-right order.\n      ","5.4.1#footnote-link-5":"5","5.4.1#h3":"\n\t  Procedure application\n\t","5.4.1#p10":"\n\tThe entry point\n\tapply-dispatch\n\tcorresponds to the apply\n\tprocedure of the metacircular evaluator.  By the time we get to\n\tapply-dispatch,\n\tthe proc register contains the\n\tprocedure\n\tto apply and argl contains the list of\n\tevaluated arguments to which it must be applied.  The saved value of\n\tcontinue (originally passed to\n\teval-dispatch\n\tand saved at\n\tev-application),\n\twhich tells where to return with the result of the\n\tprocedure application, is on the stack.  When the application is complete, the\n\tcontroller transfers to the entry point specified by the saved\n\tcontinue, with the result of the application in\n\tval.  As with the metacircular\n\tapply, there are two cases to consider.  Either\n\tthe procedure to be applied is a primitive or it is a compound\n\tprocedure.\n\tapply-dispatch\n  (test (op primitive-procedure?) (reg proc))\n  (branch (label primitive-apply))\n  (test (op compound-procedure?) (reg proc))  \n  (branch (label compound-apply))\n  (goto (label unknown-procedure-type))","5.4.1#p11":"\n\tWe assume that each\n\t\n\tprimitive is implemented so as to obtain its\n\targuments from argl and place its result in\n\tval.  To specify how the machine handles\n\tprimitives, we would have to provide a sequence of controller instructions\n\tto implement each primitive and arrange for\n\tprimitive-apply\n\tto dispatch to the\n\tinstructions for the primitive identified by the\n\tcontents of\n\tproc.\n\tSince we are interested in the structure of the evaluation process rather\n\tthan the details of the primitives, we will instead just use an\n\tapply-primitive-procedure operation\n\tthat applies the\n\tprocedure in proc\n\tto the arguments in argl.  For the purpose of\n\tsimulating the evaluator with the simulator\n\tof section 5.2 we use the\n\tprocedure apply-primitive-procedure,\n\twhich calls on the underlying Scheme system to perform the application,\n\tjust as we did for the metacircular\n\tevaluator in section 4.1.4.  After computing\n\tthe value of the primitive application, we restore\n\tcontinue and go\n\tto the designated entry point.\n\tprimitive-apply\n  (assign val (op apply-primitive-procedure)\n              (reg proc)\n              (reg argl))\n  (restore continue)\n  (goto (reg continue))","5.4.1#p12":"\n\tTo apply a compound procedure, we proceed just as with\n\tthe metacircular evaluator.  We construct a frame\n\tthat binds the\n\tprocedure's\n\tparameters to the arguments, use this frame to extend the environment\n\tcarried by the procedure,\n\tand evaluate in this extended environment \n\tthe sequence of expressions that forms the body of the\n\tprocedure.\n        Ev-sequence, described below\n        in section ,\n        handles the evaluation of the sequence.\n\tcompound-apply\n  (assign unev (op procedure-parameters) (reg proc))\n  (assign env (op procedure-environment) (reg proc))\n  (assign env (op extend-environment)\n              (reg unev) (reg argl) (reg env))\n  (assign unev (op procedure-body) (reg proc))\n  (goto (label ev-sequence))","5.4.1#p13":"Compound-apply is the only\n\tplace in the interpreter where the\n\tenv register is ever assigned a\n\tnew value. Just as in the metacircular evaluator, the new environment\n\tis constructed from the environment carried by the\n\tprocedure, together with the argument list and the corresponding list of\n\tvariables to be bound.\n      ","5.4.1#footnote-1":"In our controller, the dispatch is written as a sequence of\n    test and branch\n    instructions. Alternatively, it could have been written in a data-directed \n    style (and in a real system it probably would have been) to avoid\n    the need to perform sequential tests and\n    to facilitate\n    the definition of new\n    expression\n    types.\n    \n        A machine designed to run Lisp would probably include a\n        dispatch-on-type instruction\n        that would efficiently execute such data-directed\n        dispatches.\n      ","5.4.1#footnote-2":"This is an important but subtle point in translating\n\talgorithms from a procedural language, such as Lisp,\n\tto a register-machine language.  As an alternative to saving only what is\n\tneeded, we could save all the registers (except\n\tval) before each recursive call.\n\tThis is called a \n\tframed-stack discipline.  This\n\twould work but might save more registers than necessary; this could be\n\tan important consideration in a system where stack operations are\n\texpensive.  Saving registers whose contents will not be needed later\n\tmay also hold on to useless data that could otherwise be\n\tgarbage-collected, freeing space to be reused.","5.4.1#footnote-3":"We add to the evaluator data-structure\n\tprocedures\n\tin section 4.1.3 the following two\n\tprocedures for manipulating argument lists:\n\t(define (empty-arglist) '())\n\n(define (adjoin-arg arg arglist)\n  (append arglist (list arg)))\n\tWe also use an additional syntax procedure\n\tto test for the last\n\toperand in a combination:\n\t(define (last-operand? ops)\n(null? (cdr ops)))","5.4.1#footnote-4":"The\n\toptimization of\ttreating the last operand specially is known as\n\tevlis tail recursion (see \n\tWand 1980). We could be somewhat more efficient\n\tin the argument evaluation loop if we made evaluation of the first\n\toperand a special case too.  This would permit us to postpone\n\tinitializing argl until after evaluating the\n\tfirst operand, so\n\tas to avoid saving argl in this case.  The\n\tcompiler in section 5.5 performs this\n\toptimization. (Compare the\n\tconstruct-arglist procedure\n\tof section 5.5.3.)","5.4.1#footnote-5":"\n        The order of operand evaluation in the metacircular evaluator is\n        determined by the order of evaluation of the arguments to\n        cons in the procedure\n        list-of-values of section 4.1.1 (see exercise ).","5.4.2":"5.4.2\n        The metacircular implementation of the evaluator in chapter 4\n        does not specify whether the evaluator is tail-recursive, because that\n        evaluator inherits its mechanism for saving state from the underlying\n        Scheme. With the explicit-control evaluator, however, we can trace\n        through the evaluation process to see when procedure calls cause a net\n        accumulation of information on the stack.\n      ","5.4.2#p1":"\n        The portion of the explicit-control evaluator at\n        ev-sequence is analogous to the metacircular\n        evaluator's eval-sequence procedure. It\n        handles sequences of expressions in procedure bodies or in explicit\n        begin expressions.\n      ","5.4.2#p2":"\n        Explicitbegin expressions are evaluated by\n        placing the sequence of expressions to be evaluated in\n        unev, saving\n        continue on the stack, and jumping to\n        ev-sequence.\n        ev-begin\n  (assign unev (op begin-actions) (reg exp))\n  (save continue)\n  (goto (label ev-sequence))\n    \n        The implicit sequences in procedure bodies are handled by jumping to\n        ev-sequence from\n        compound-apply, at which point\n        continue is already on the stack, having\n        been saved at ev-application.\n      ","5.4.2#p3":"\n        The entries at ev-sequence and\n        ev-sequence-continue form a loop that successively\n        evaluates each expression in a sequence. The list of unevaluated expressions is\n        kept in unev. Before evaluating each expression, we\n        check to see if there are additional expressions to be evaluated in the\n        sequence. If so, we save the rest of the unevaluated expressions (held in\n        unev) and the environment in which these must be\n        evaluated (held in env) and call\n        eval-dispatch to evaluate the expression. The two\n        saved registers are restored upon the return from this evaluation, at\n        ev-sequence-continue.\n      ","5.4.2#p4":"\n        The final expression in the sequence is handled differently, at the entry point\n        ev-sequence-last-exp. Since there are no more\n        expressions to be evaluated after this one, we need not save\n        unev or env before\n        going to eval-dispatch. expression, so after the\n        evaluation of the last expression there is nothing left to do except continue at\n        the entry point currently held on the stack (which was saved by\n        ev-application or\n        ev-begin.) Rather than setting up\n        continue to arrange for\n        eval-dispatch to return here and then restoring\n        continue from the stack and continuing at that\n        entry point, we restore continue from the stack\n        before going to eval-dispatch, so that\n        eval-dispatch will continue at that entry point\n        after evaluating the expression.\n        ev-sequence\n  (assign exp (op first-exp) (reg unev))\n  (test (op last-exp?) (reg unev))\n  (branch (label ev-sequence-last-exp))\n  (save unev)\n  (save env)\n  (assign continue (label ev-sequence-continue))\n  (goto (label eval-dispatch))\nev-sequence-continue\n  (restore env)\n  (restore unev)\n  (assign unev (op rest-exps) (reg unev))\n  (goto (label ev-sequence))\nev-sequence-last-exp\n  (restore continue)\n  (goto (label eval-dispatch))","5.4.2#h1":"\n      Tail Recursion\n    ","5.4.2#p5":"\n    In chapter 1 we said that the process described by a\n    procedure\n    such as\n        \n    (define (sqrt-iter guess x)\n  (if (good-enough? guess x)\n      guess\n      (sqrt-iter (improve guess x)\n                 x)))\n\n    is an iterative process.  Even though the\n    procedure\n    is syntactically recursive (defined in terms of itself), it is not logically\n    necessary for an evaluator to save information in passing from one call to\n    sqrt-iter\n    to the next. An\n    evaluator that can execute a\n    procedure\n    such as\n    sqrt-iter\n    without requiring increasing storage as the\n    procedure\n    continues to call itself is called a \n    tail-recursive evaluator.  \n  ","5.4.2#footnote-link-1":"1","5.4.2#p6":"\n        Our evaluator is tail-recursive, because in order to evaluate the final\n        expression of a sequence we transfer directly to\n        eval-dispatch without saving any information on the\n        stack. Hence, evaluating the final expression in a sequence—even if it\n        is a procedure call (as in sqrt-iter, where the\n        if expression, which is the last expression in the\n        procedure body, reduces to a call to\n        sqrt-iter)—will not cause any information\n        to be accumulated on the stack.\n\n        If we did not think to take advantage of the fact that it was unnecessary to\n        save information in this case, we might have implemented\n        eval-sequence by treating all the expressions in a\n        sequence in the same way—saving the registers, evaluating the\n        expression, returning to restore the registers, and repeating this until all the\n        expressions have been evaluated:ev-sequence\n  (test (op no-more-exps?) (reg unev))\n  (branch (label ev-sequence-end))\n  (assign exp (op first-exp) (reg unev))\n  (save unev)\n  (save env)\n  (assign continue (label ev-sequence-continue))\n  (goto (label eval-dispatch))\nev-sequence-continue\n  (restore env)\n  (restore unev)\n  (assign unev (op rest-exps) (reg unev))\n  (goto (label ev-sequence))\nev-sequence-end\n  (restore continue)\n  (goto (reg continue))","5.4.2#footnote-link-2":"2","5.4.2#p7":"\n    This may seem like a minor change to our previous code for evaluation of \n    a sequence:\n    \n    The only difference is that we go through the save-restore\n    cycle for the last expression in a sequence as well as for the\n    others.\n\n    The interpreter will still give the same value for any expression. But this change\n    is fatal to the tail-recursive implementation, because we must now come back after\n    evaluating the final expression in a\n    sequence in order\n    to undo the (useless) register saves.\n\n    These extra saves will accumulate during a nest of\n    procedure\n    calls.\n\n    Consequently, processes such as\n    sqrt-iter\n    will require space proportional to the number of iterations rather than requiring\n    constant space.\n\n    This difference can be significant. For example,\n    \n    with tail recursion, an infinite loop can be expressed using only the\n    procedure-call mechanism:(define (count n)\n  (newline)\n  (display n)\n  (count (+ n 1)))\n\n    Without tail recursion, such a\n    procedure\n    would eventually run out of stack space, and expressing a true iteration\n    would require some control mechanism other than\n    procedure\n    call.\n  ","5.4.2#footnote-1":"We saw in\n    section 5.1 how to\n    implement such a process with a register machine that had no stack; the\n    state of the process was stored in a fixed set of registers.","5.4.2#footnote-2":"We can define\n        no-more-exps? as follows:\n\n        (define (no-more-exps? seq) (null? seq))","5.4.3":"5.4.3","5.4.3#p1":"\n    As with the metacircular evaluator, special syntactic forms are handled by selectively\n    evaluating fragments of the expression. For an if\n    expression, we must evaluate the predicate and decide, based on the value of\n    predicate, whether to evaluate the consequent or the alternative.\n  ","5.4.3#p2":"\n    Before evaluating the predicate, we save the if\n    expression itself so that we can later extract the consequent or alternative. We also\n    save the environment, which we will need later in order to evaluate the consequent or\n    the alternative, and we save continue, which we will need\n    later in order to return to the evaluation of the expression that is waiting for the\n    value of the if.\n    ev-if\n(save exp)                    \n(save env)\n(save continue)\n(assign continue (label ev-if-decide))\n(assign exp (op if-predicate) (reg exp))\n(goto (label eval-dispatch))","5.4.3#p3":"\n    When we return from evaluating the predicate, we test whether it was true or false\n    and, depending on the result, place either the consequent or the alternative in\n    exp before going to\n    eval-dispatch. Notice that restoring\n    env and continue here sets\n    up eval-dispatch to have the correct environment and to\n    continue at the right place to receive the value of the\n    if expression.\n    ev-if-decide\n(restore continue)\n(restore env)\n(restore exp)\n(test (op true?) (reg val))\n(branch (label ev-if-consequent))\nev-if-alternative\n(assign exp (op if-alternative) (reg exp))\n(goto (label eval-dispatch))\nev-if-consequent\n(assign exp (op if-consequent) (reg exp))\n(goto (label eval-dispatch))","5.4.3#h1":"\n      Assignments and\n      definitions","5.4.3#p4":"\n    Assignments\n    \n    are handled by\n    ev-assignment, which is\n    reached from\n    eval-dispatch\n    with the assignment expression in exp. The\n    code at\n    ev-assignment\n    first evaluates the value part of the expression and then installs the new\n    value in the environment.\n    Set-variable-value!\n    is assumed to be available as a machine operation.\n    ev-assignment\n(assign unev (op assignment-variable) (reg exp))\n(save unev)                   \n(assign exp (op assignment-value) (reg exp))\n(save env)\n(save continue)\n(assign continue (label ev-assignment-1))\n(goto (label eval-dispatch))  \nev-assignment-1\n(restore continue)\n(restore env)\n(restore unev)\n(perform\n(op set-variable-value!) (reg unev) (reg val) (reg env))\n(assign val (const ok))\n(goto (reg continue))","5.4.3#p5":"\n        Definitions are handled in a similar way:\n      ev-definition\n(assign unev (op definition-variable) (reg exp))\n(save unev)                   \n(assign exp (op definition-value) (reg exp))\n(save env)\n(save continue)\n(assign continue (label ev-definition-1))\n(goto (label eval-dispatch))  \nev-definition-1\n(restore continue)\n(restore env)\n(restore unev)\n(perform\n(op define-variable!) (reg unev) (reg val) (reg env))\n(assign val (const ok))\n(goto (reg continue))","5.4.3#ex-5.23":"\n\tExtend the evaluator to handle derived expressions such as\n\tcond,\n\tlet, and so on\n\t(section 4.1.2).\n\tYou may \"cheat\" and assume that the syntax\n\ttransformers such as cond->if are available\n\tas machine operations.","5.4.3#footnote-link-1":"1","5.4.3#ex-5.24":"\n\tImplement cond as a new basic\n        \n\tspecial form\n\twithout reducing it to if. You will have to\n\tconstruct a loop that tests the predicates of successive\n\tcond clauses until you find one that is\n\ttrue, and then use ev-sequence to evaluate\n\tthe actions of the clause.\n        ","5.4.3#ex-5.25":"\n    Modify the evaluator so that it uses\n    \n    normal-order evaluation,\n    based on the lazy evaluator of\n    section 4.2.\n    ","5.4.3#footnote-1":"This isn't really cheating.  In an\n\tactual implementation built from scratch, we would use our explicit-control\n\tevaluator to interpret a Scheme program that performs source-level\n\ttransformations\n\tlike cond->if in a syntax phase that runs before execution.","5.4.4":"5.4.4  \n    Running the Evaluator","5.4.4#p1":"\n    With the implementation of the explicit-control evaluator we come to\n    the end of a development, begun in chapter 1, in which we have\n    explored successively more precise\n    \n    models of the evaluation process.\n    We started with the relatively informal substitution model, then\n    extended this in chapter 3 to the environment model, which enabled us\n    to deal with state and change.  In the metacircular evaluator of\n    chapter 4, we used\n    Scheme\n    itself as a language for making more\n    explicit the environment structure constructed during evaluation of an\n    expression.\n    Now, with register machines, we have taken a close look\n    at the evaluator's mechanisms for storage management,\n    argument passing, and control.  At\n    each new level of description, we have had to raise issues and resolve\n    ambiguities that were not apparent at the previous, less precise\n    treatment of evaluation.  To understand the behavior of the\n    explicit-control evaluator, we can simulate it and monitor its\n    performance.\n  ","5.4.4#p2":"\n    We will install a\n    \n    driver loop in our evaluator machine.  This plays\n    the role of the\n    driver-loopprocedure\n    of section 4.1.4. The evaluator\n    will repeatedly print a prompt, read \n    an expression,\n    evaluate\n    the expression\n    by going to\n    eval-dispatch,\n      \n    and print the result.\n    \n    The following instructions form the beginning of the\n    explicit-control evaluator's controller sequence:\n      read-eval-print-loop\n      (perform (op initialize-stack))\n      (perform\n      (op prompt-for-input) (const \";;; EC-Eval input:\"))\n      (assign exp (op read))\n      (assign env (op get-global-environment))\n      (assign continue (label print-result))\n      (goto (label eval-dispatch))\n      print-result\n      (perform\n      (op announce-output) (const \";;; EC-Eval value:\"))\n      (perform (op user-print) (reg val))\n      (goto (label read-eval-print-loop))\n      ","5.4.4#footnote-link-1":"1","5.4.4#p3":"\n    When we encounter an\n    \n    error in a\n    procedure\n    (such as the \n    \"unknown procedure type error\"\n    indicated at\n    apply-dispatch),\n    we print an error message and return to the driver loop.\n      unknown-expression-type\n      (assign val (const unknown-expression-type-error))\n      (goto (label signal-error))\n\n      unknown-procedure-type\n      (restore continue)    \n      (assign val (const unknown-procedure-type-error))\n      (goto (label signal-error))\n\n      signal-error\n      (perform (op user-print) (reg val))\n      (goto (label read-eval-print-loop))\n      ","5.4.4#footnote-link-2":"2","5.4.4#p4":"\n    For the purposes of the simulation, we initialize the stack each time\n    through the driver loop, since it might not be empty after an error\n    \n\t(such as an undefined variable)\n      \n    interrupts an evaluation.","5.4.4#footnote-link-3":"3","5.4.4#p5":"\n    If we combine all the code fragments presented in sections\n    5.4.1–5.4.4,\n    we can create an\n    \n    evaluator machine model that we can run using the\n    register-machine simulator of section 5.2.\n\n    \n      (define eceval\n      (make-machine\n      '(exp env val proc argl continue unev)\n      eceval-operations\n      '(\n      read-eval-print-loop\n      $\\langle$entire machine controller as given above$\\rangle$\n      )))\n      \n\n    We must define \n    Scheme procedures\n    to simulate the operations used as primitives by the evaluator.  These are\n    the same\n    procedures\n    we used for the metacircular evaluator in\n    section 4.1, together with the few additional\n    ones defined in footnotes throughout section 5.4.\n    \n(define eceval-operations\n  (list (list 'self-evaluating? self-evaluating)\n        ))\n      ","5.4.4#p6":"\n    Finally, we can initialize the global environment and run the evaluator:\n    (define the-global-environment (setup-environment))\n\n      (start eceval) ","5.4.4#p7":"\n    Of course, evaluating\n    expressions\n    in this way will take much longer\n    than if we had directly typed them into\n    Scheme,\n    because of the\n    multiple levels of simulation involved.  Our\n    expressions\n    are evaluated\n    by the explicit-control-evaluator machine, which is being simulated by\n    a\n    Scheme\n    program, which is itself being evaluated by the\n    Scheme\n    interpreter.\n  ","5.4.4#h1":"Monitoring the performance of the evaluator","5.4.4#p8":"\n    Simulation can be a powerful tool to guide the implementation of\n    evaluators.\n    \n    Simulations make it easy not only to explore variations\n    of the register-machine design but also to monitor the performance of\n    the simulated evaluator.  For example, one important factor in\n    performance is how efficiently the evaluator uses the stack.  We can\n    observe the number of stack operations required to evaluate various\n    expressions\n    by defining the evaluator register machine with the\n    version of the simulator that collects statistics on stack use\n    (section 5.2.4), and adding an instruction at the\n    evaluator's\n    print-result\n    entry point to print the statistics:\n    \n      print-result\n      (perform (op print-stack-statistics))\n      (perform\n      (op announce-output) (const \"EC-Eval value:\"))\n      $\\ldots$ ; same as before\n      \n    Interactions with the evaluator now look like this:\n    \n\n    Note that the driver loop of the evaluator reinitializes the stack\n    at the start of\n    each interaction, so that the statistics printed will refer only to\n    stack operations used to evaluate the previous\n    expression.","5.4.4#ex-5.26":"\n    Use the monitored stack to explore the\n    \n    tail-recursive property of the\n    evaluator (section 5.4.2).  Start the\n    evaluator and define the\n    \n    iterative factorialprocedure\n    from section 1.2.1:\n    (define (factorial n)\n      (define (iter product counter)\n      (if (> counter n)\n            product\n            (iter (* counter product)\n            (+ counter 1))))\n      (iter 1 1))\n    Run the\n    procedure\n    with some small values of $n$.  Record the\n    maximum stack depth and the number of pushes required to compute\n    $n!$ for each of these values.\n    \n\tYou will find that the maximum depth required to evaluate\n\t$n!$ is independent of\n\t$n$.  What is that depth?\n      \n\tDetermine from your data a formula in terms of\n\t$n$ for the total number of push operations\n\tused in evaluating $n!$ for any\n\t$n \\geq 1$. Note that the number of\n\toperations used is a linear function of $n$\n\tand is thus determined by two constants.\n      ","5.4.4#ex-5.27":"\n    For comparison with exercise 5.26, explore\n    the behavior of the following\n    procedure\n    for computing\n    \n    factorials recursively:\n    (define (factorial n)\n      (if (= n 1)\n      1\n      (* (factorial (- n 1)) n)))\n    By running this\n    procedure\n    with the monitored stack, determine, as a function of\n    $n$, the maximum depth of the stack and the total\n    number of pushes used in evaluating $n!$ for\n    $n \\geq 1$.  (Again, these functions will be\n    linear.)  Summarize your experiments by filling in the following table with\n    the appropriate expressions in terms of $n$:\n    \n    The maximum depth is a measure of the amount of space used by the\n    evaluator in carrying out the computation, and the number of pushes\n    correlates well with the time required.\n  ","5.4.4#fig-":"","5.4.4#ex-5.28":"\n    Modify the definition of the evaluator by changing\n    eval-sequence\n      as described in section \n    so that the evaluator is no longer\n    \n    tail-recursive.  Rerun your experiments from\n    exercises 5.26\n    and 5.27 to demonstrate that both versions of\n    the factorialprocedure\n    now require space that grows linearly with their input.\n    ","5.4.4#ex-5.29":"\n    Monitor the stack operations in the tree-recursive\n    \n    Fibonacci computation:\n    (define (fib n)\n      (if (< n 2)\n      n\n      (+ (fib (- n 1)) (fib (- n 2)))))\n\tGive a formula in terms of $n$ for the\n\tmaximum depth of the stack required to compute\n\t${\\textrm{Fib}}(n)$ for\n\t$n \\geq 2$.  Hint: In\n\tsection 1.2.2 we argued that the space\n\tused by this process grows linearly with $n$.\n      \n\tGive a formula for the total number of pushes used to compute\n\t${\\textrm{Fib}}(n)$ for\n\t$n \\geq 2$.  You should find that the number\n\tof pushes (which correlates well with the time used) grows exponentially\n\twith $n$.  Hint: Let\n\t$S(n)$ be the number of pushes used in\n\tcomputing ${\\textrm{Fib}}(n)$.  You should be\n\table to argue that there is a formula that expresses\n\t$S(n)$ in terms of\n\t$S(n-1)$, $S(n-2)$,\n\tand some fixed \"overhead\" constant\n\t$k$ that is independent of\n\t$n$.  Give the formula, and say what\n\t$k$ is.  Then show that\n\t$S(n)$ can be expressed as\n\t$a {\\textrm{Fib}}(n+1) + b$ and give the\n\tvalues of $a$ and\n\t$b$.\n      ","5.4.4#ex-5.30":"\n    Our evaluator currently catches and signals only two kinds of\n    \n    errors—unknown\n    \n        expression\n      \n    types and unknown\n    procedure\n    types.  Other errors will take us out of the evaluator\n    \n\tread-eval-print\n      \n    loop.\n    When we run the evaluator using the register-machine simulator, these\n    errors are caught by the underlying\n    Scheme\n    system.  This is analogous\n    to the computer crashing when a user program makes an error.\n    It is a large project to\n    make a real error system work, but it is well worth the effort to understand\n    what is involved here.\n    \n\tErrors that occur in the evaluation process, such as an attempt to\n\taccess an unbound\n\tvariable,\n\tcould be caught by changing the lookup\n\toperation to make it return a distinguished condition code, which cannot\n\tbe a possible value of any user\n\tvariable.\n\tThe evaluator can test\n\tfor this condition code and then do what is necessary to go to\n\tsignal-error.\n\tFind all of the places in the evaluator where such a\n\tchange is necessary and fix them.  This is lots of work.\n      \n\tMuch worse is the problem of handling errors that are signaled by\n\tapplying primitive\n\tprocedures\n\tsuch as an attempt to divide by zero or an attempt to extract the\n\tcar\n\t    of a symbol.\t    \n\t  \n\tIn a professionally written high-quality system, each\n\tprimitive application is checked for safety as part of the primitive.\n\tFor example, every call to\n\tcar\n\tcould first check that the argument is a pair.  If the argument is not\n\ta pair, the application would return a distinguished condition code to\n\tthe evaluator, which would then report the failure.  We could arrange\n\tfor this in our register-machine simulator by making each primitive\n\tprocedure\n\tcheck for applicability and returning an appropriate distinguished\n\tcondition code on failure. Then the\n\tprimitive-apply\n\tcode in the evaluator can check for the condition code and go to\n\tsignal-error\n\tif necessary.  Build this structure and make it work.\n\tThis is a major project.\n      ","5.4.4#footnote-link-4":"4","5.4.4#footnote-1":"We assume\n    here that\n    read\n    and the various printing\n    operations are available as primitive machine operations, which is useful\n    for our simulation, but completely unrealistic in practice.  These are\n    actually extremely complex operations.  In practice,\n    they\n    would be\n    implemented using low-level input-output operations such as transferring\n    single characters to and from a device.\n    \n\tTo support the\n\tget-global-environment operation we define\n\t(define the-global-environment (setup-environment))\n\n\t    (define (get-global-environment)\n\t    the-global-environment)","5.4.4#footnote-2":"There are\n    other errors that we would like the interpreter to handle, but these are not\n    so simple.  See exercise 5.30.","5.4.4#footnote-3":"We\n    could perform the stack initialization only after errors, but doing it in\n    the driver loop will be convenient for monitoring the evaluator's\n    performance, as described below.","5.4.4#footnote-4":"\n\tRegrettably, this is the normal state of affairs in\n\t\n\tconventional compiler-based language systems such as C.  \n\t\n\tIn UNIX$^{\\textrm{TM}}$   the system \"dumps\n\tcore,\" and in\n\t\n\tDOS/Windows$^{\\textrm{TM}}$  \n\tit becomes catatonic. The\n\t\n\tMacintosh$^{\\textrm{TM}}$   displays a picture of\n\tan exploding bomb and offers you the opportunity to reboot the\n\tcomputer—if you're lucky.\n      ","5.5":"5.5  Compilation","5.5#p1":"\n    The explicit-control evaluator of\n    section 5.4 is a\n    register machine whose controller interprets\n    Scheme\n    programs.  In this\n    section we will see how to run\n    Scheme\n    programs on a register machine whose controller is not a\n    Scheme\n    interpreter.\n  ","5.5#p2":"\n    The explicit-control evaluator machine is\n    \n    universal—it\n    can carry out any computational process that can be described in\n    Scheme.\n    The\n    evaluator's controller orchestrates the use of its data\n    paths to perform the desired computation.  Thus, the\n    evaluator's data paths are universal: They are sufficient\n    to perform any computation we desire, given an appropriate\n    controller.","5.5#footnote-link-1":"1","5.5#p3":"\n    Commercial\n    \n    general-purpose computers are\n    register machines organized\n    around a collection of registers and operations that constitute\n    an efficient and convenient universal set of data paths.\n    The controller for a general-purpose machine is an interpreter for\n    a register-machine language like the one we have been using.  This\n    language is called the\n    native language of the machine, or simply\n    machine language.  Programs written in machine language are\n    sequences of instructions that use the machine's data paths.\n    For example, the\n    \n    explicit-control evaluator's instruction sequence\n    can be thought of as a machine-language program for a general-purpose\n    computer rather than as the controller for a specialized interpreter\n    machine.\n  ","5.5#p4":"\n    There are two common strategies for bridging the gap between\n    higher-level languages and register-machine languages.\n    The explicit-control evaluator illustrates the\n    strategy of interpretation.  An interpreter written in the native\n    language of a machine configures the machine to execute programs\n    written in a language (called the\n    source language) that may\n    differ from the native language of the machine performing the\n    evaluation.  The primitive\n    procedures\n    of the source language are implemented as a library of subroutines written\n    in the native language of the given machine.  A program to be interpreted\n    (called the\n    source program) is represented as a data structure.  The interpreter\n      traverses this data structure, analyzing the source program.  As it\n      does so, it simulates the intended behavior of the source program by\n      calling appropriate primitive subroutines from the library.\n  ","5.5#p5":"\n    In this section, we explore the alternative strategy of \n    compilation.  A compiler for a given source language and machine\n    translates a source program into an equivalent program (called the\n    object program) written in the machine's native language.\n    The compiler that we implement in this section translates programs written in\n    Scheme\n    into sequences of instructions to be executed using the explicit-control\n    evaluator machine's data paths.","5.5#footnote-link-2":"2","5.5#p6":"\n    Compared with interpretation, compilation can provide a great increase\n    in the efficiency of program execution, as we will explain below in\n    the overview of the compiler.\n    On the other hand, an interpreter\n    provides a more powerful environment for interactive program\n    development and debugging, because the source program being executed\n    is available at run time to be examined and modified.  In addition,\n    because the entire library of primitives is present, new programs can\n    be constructed and added to the system during debugging.\n  ","5.5#p7":"\n    In view of the complementary advantages of compilation and\n    interpretation, modern\n    program-development environments\n    pursue a mixed\n    strategy.\n    Lisp interpreters\n    are generally organized so that interpreted\n    procedures\n    and compiled\n    procedures\n    can call each other.\n    This enables a programmer to compile those parts of a\n    program that are assumed to be debugged, thus gaining the efficiency\n    advantage of compilation, while retaining the interpretive mode of execution\n    for those parts of the program that are in the flux of interactive\n    development and\n    debugging.\n    In section 5.5.7, after\n    we have implemented the compiler, we will show how to interface it\n    with our interpreter to produce an integrated\n    interpreter-compiler\n    development\n    system.\n  ","5.5#h1":"An overview of the compiler","5.5#p8":"\n    Our compiler is much like our interpreter, both in its structure and in\n    the function it performs.  Accordingly, the mechanisms used by the\n    compiler for analyzing\n    \n\texpressions\n      \n    will be similar to those used by\n    the interpreter.  Moreover, to make it easy to interface compiled and\n    interpreted code, we will design the compiler to generate code that\n    obeys the same conventions of\n    \n    register usage as the interpreter: The\n    environment will be kept in the env register,\n    argument lists will be accumulated in argl, a\n    procedure\n    to be applied will be in\n    proc,\n      procedures\n    will return their answers in val,\n    and the location to which a\n    procedure\n    should return will be kept in\n    continue.\n    In general, the compiler translates a source program into an object\n    program that performs essentially the same register operations as\n    would the interpreter in evaluating the same source program.\n  ","5.5#p9":"\n    This description suggests a strategy for implementing a rudimentary\n    compiler: We traverse the\n    \n\texpression\n      \n    in the same way the\n    interpreter does.  When we encounter a register instruction that the\n    interpreter would perform in evaluating the\n    \n\texpression,\n      \n    we do not\n    execute the instruction but instead accumulate it into a sequence.  The\n    resulting sequence of instructions will be the object code.  Observe\n    the\n    \n    efficiency advantage of compilation over interpretation.  Each\n    time the interpreter evaluates\n    \n\tan expression—for example,\n      (f 84 96)—it\n      \n    performs the work of classifying the\n    \n\texpression\n      \n    (discovering that this is a\n    procedure\n    application) and\n\ttesting for the end of the\n    \n\toperand list (discovering that there are two operands).\n      \n    With a\n    compiler, the\n    \n\texpression\n      \n    is analyzed only once, when the\n    instruction sequence is generated at compile time.  The object code\n    produced by the compiler contains only the instructions that evaluate\n    the\n    \n\toperator and the two operands,\n      \n    assemble the argument list, and apply the\n    procedure (in proc)\n    to the arguments (in argl).\n  ","5.5#p10":"\n    This is the same kind of optimization we implemented in the\n    \n    analyzing evaluator of section 4.1.7.\n    But there are further opportunities to gain efficiency in compiled code.\n    As the interpreter runs, it follows a process that must be applicable\n    to any\n    \n        expression\n      \n    in the language.  In contrast, a given segment of\n    compiled code is meant to execute some particular\n    \n        expression.\n      \n    This can make a big difference, for example in the use of the\n    stack to save registers.  When the interpreter evaluates\n    \n        an expression,\n      \n    it must be prepared for any contingency. Before evaluating a\n    subexpression,\n    the interpreter saves all registers that will be needed later, because the\n    subexpression\n    might require an arbitrary evaluation.\n    A compiler, on the other hand, can exploit the structure of the particular\n    \n        expression\n      \n    it is processing to generate code that avoids\n    unnecessary stack operations.\n  ","5.5#p11":"\n    As a case in point, consider the\n    combination (f 84 96).\n    Before the interpreter evaluates the\n    \n\toperator of the combination,\n      \n    it prepares\n    for this evaluation by saving the registers containing the\n    \n\toperands\n      \n    and the environment, whose values will be needed later.  The interpreter then\n    evaluates the\n    \n\toperator\n      \n    to obtain the result in\n    val, restores the saved registers, and finally\n    moves the result from val to\n    proc.\n    However, in the particular expression we\n    are dealing with, the\n    \n\toperator\n      \n    is the\n    \n\tsymbol\n      f, whose evaluation is\n    accomplished by the machine operation\n    lookup-variable-value,\n    which does not alter any registers.  The compiler that we implement in\n    this section will take advantage of this fact and generate code that\n    evaluates the\n    \n\t      operator\n      \n      using the instruction\n    (assign proc (op lookup-variable-value) (const f) (reg env))\n    This code not only avoids the unnecessary saves and\n    restores but also assigns the value of the lookup directly to\n    proc,\n    whereas the interpreter would obtain the\n    result in val and then move this to\n    proc.","5.5#p12":"\n    A compiler can also optimize access to the environment.  Having\n    analyzed the code, the compiler can\n    \n\tin many cases\n      \n    know in which frame\n    a particular variable\n    will be located and access that frame directly,\n    rather than performing the\n    lookup-variable-value\n    search.  We will discuss how to implement such\n    variable access\n    in\n    section 5.5.6.  Until then, however,\n    we will focus on the kind of register and stack optimizations described\n    above. There are many other optimizations that can be performed by a\n    compiler, such as coding primitive operations \"in line\" instead\n    of using a general apply mechanism (see\n    exercise 5.38); but we will not emphasize these\n    here. Our main goal in this section is to illustrate the compilation process\n    in a simplified (but still interesting) context.\n  ","5.5#footnote-1":"This is a theoretical statement.  We are\n    not claiming\n    that the evaluator's data paths are a particularly convenient or\n    efficient set of data paths for a general-purpose computer.  For example,\n    they are not very good for implementing high-performance floating-point\n    calculations or calculations that intensively manipulate bit\n    vectors.","5.5#footnote-2":"Actually, the machine that\n    runs compiled code can be simpler than the interpreter machine, because we\n    \n    won't use the\n    exp\n    and\n    unev registers.  The interpreter\n    used these to hold pieces of unevaluated\n    \n\texpressions.\n      \n    With the\n    compiler, however, these\n    \n\texpressions\n      \n    get built into the\n    compiled code that the register machine will run.  For the same\n    reason,\n    \n    we don't need the machine operations that deal with\n    \n\texpression\n      \n    syntax.  But compiled code will use a few additional machine\n    operations (to represent compiled\n    procedure\n    objects) that didn't\n    appear in the explicit-control evaluator machine.","5.5.1":"5.5.1  \n    Structure of the Compiler\n      \\[\n      \\begin{array}{l|l|l|l}\n        \\textit{seq}_1 & \\texttt{(save}\\ \\textit{reg}_1\\texttt{)}    & \\texttt{(save}\\ \\textit{reg}_2\\texttt{)}     & \\texttt{(save}\\ \\textit{reg}_2\\texttt{)}    \\\\\n        \\textit{seq}_2 & \\textit{seq}_1                             & \\textit{seq}_1                              & \\texttt{(save}\\ \\textit{reg}_1\\texttt{)}    \\\\\n                       & \\texttt{(restore}\\ \\textit{reg}_1\\texttt{)} & \\texttt{(restore}\\ \\textit{reg}_2\\texttt{)}  & \\textit{seq}_1                             \\\\\n                       & \\textit{seq}_2                             & \\textit{seq}_2                              & \\texttt{(restore}\\ \\textit{reg}_1\\texttt{)} \\\\\n                       &                                            &                                             & \\texttt{(restore}\\ \\textit{reg}_2\\texttt{)} \\\\\n                       &                                            &                                             & \\textit{seq}_2\n      \\end{array}\n      \\]\n    ","5.5.1#p1":"\n    In section 4.1.7 we modified our\n    original metacircular interpreter to separate\n    \n    analysis from execution.  We\n    analyzed each\n    \n        expression\n      \n    to produce an execution\n    procedure\n    that took an environment as argument and performed the required operations.\n    In our compiler, we will do essentially the same analysis.  Instead of\n    producing execution\n    procedures,\n    however, we will generate sequences of instructions to be run by our\n    register machine.\n  ","5.5.1#p2":"\n    The\n    procedurecompile is the top-level dispatch in the\n    compiler. It corresponds to the evalprocedure\n    of section 4.1.1, the\n    analyzeprocedure\n    of section 4.1.7, and the\n    eval-dispatch\n    entry point of the explicit-control-evaluator in\n    section 5.4.1. The compiler, like the\n    interpreters, uses the\n    expression-syntax procedures\n    defined in\n    section 4.1.2.Compile performs a case analysis on the\n    syntactic type of the\n    \n\texpression\n      \n    to be compiled.  For\n    each type of\n    \n\texpression,\n      \n    it dispatches to a\n    specialized\n    code generator:\n    (define (compile exp target linkage)\n  (cond ((self-evaluating? exp)\n         (compile-self-evaluating exp target linkage))\n        ((quoted? exp) (compile-quoted exp target linkage))\n        ((variable? exp)\n         (compile-variable exp target linkage))\n        ((assignment? exp)\n         (compile-assignment exp target linkage))\n        ((definition? exp)\n         (compile-definition exp target linkage))\n        ((if? exp) (compile-if exp target linkage))\n        ((lambda? exp) (compile-lambda exp target linkage))\n        ((begin? exp)\n         (compile-sequence (begin-actions exp)\n                           target\n                           linkage))\n        ((cond? exp) (compile (cond->if exp) target linkage))\n                               ((application? exp)\n                                (compile-application exp target linkage))\n                               (else\n                                (error \"Unknown expression type - - COMPILE\" exp))))","5.5.1#footnote-link-1":"1","5.5.1#h1":"Targets and linkages","5.5.1#p3":"Compile\n    and the code generators that it calls\n    take two\n    \n    arguments in addition to the\n    \n        expression\n      \n    to compile.  There is a\n    target, which specifies the register in which the compiled code is\n    to return the value of the\n    \n        expression.\n      \n    There is also a\n    linkage descriptor, which describes how the code resulting from the\n    compilation of the\n    \n        expression\n      \n    should proceed when it has finished its\n    execution.  The linkage descriptor can require the code to do one of\n    the following three things:\n    \n\tproceed to the next instruction in sequence (this is\n\tspecified by the linkage descriptor\n\tnext),\n\t  \n\t    return from the procedure being compiled\n\t  \n\t(this is specified\n\tby the linkage descriptor\n\treturn),\n\t  \n\tor\n      \n\tjump to a named entry point (this is specified by using the\n\tdesignated label as the linkage descriptor).\n      ","5.5.1#p4":"\n    For example, compiling the\n    \n        expression\n      5\n        (which is self-evaluating)\n      \n    with a target of the val\n    register and a linkage of\n    next\n    should produce\n    the instruction\n    (assign val (const 5))\n    Compiling the same expression with a linkage of\n    return\n    should produce the instructions\n    (assign val (const 5))\n(goto (reg continue))\n    In the first case, execution will continue with the next instruction\n    in the sequence. In the second case,\n    we will return from a procedure call.\n    In both cases, the value of the expression will be placed into\n    the target val register.\n    ","5.5.1#h2":"Instruction sequences and stack usage","5.5.1#p5":"\n    Each code generator returns an\n    instruction sequence containing\n    the object code it has generated for the\n    \n        expression.\n      \n    Code generation for a\n    compound expression\n    is accomplished by combining the output from simpler code\n    generators for\n    component expressions,\n    just as evaluation of a\n    compound expression\n    is accomplished by evaluating the\n    component expressions.","5.5.1#p6":"\n    The simplest method for combining instruction sequences is a\n    procedure\n    called\n    append-instruction-sequences.\n\tIt takes as arguments\n\tany number of instruction sequences\n      \n    that are to be\n    executed\n    \n\tsequentially; it\n      \n    appends them and returns the combined sequence.\n    That is, if $seq_1$ and\n    $seq_2$ are sequences of instructions, then\n    evaluating\n    \n      (append-instruction-sequences $seq_1$ $seq_2$)\n      \n    produces the sequence\n    \n      $seq_1$\n      $seq_2$\n      ","5.5.1#p7":"\n    Whenever registers might need to be saved, the compiler's code\n    generators use\n    preserving, which is a more subtle method for\n    combining instruction sequences.\n    Preserving\n    takes three arguments: a set of registers and two instruction sequences that\n    are to be executed sequentially.  It appends the sequences in such a way\n    that the contents of each register in the set is preserved over the\n    execution of the first sequence, if this is needed for the execution of the\n    second sequence.  That is, if the first sequence modifies the register\n    and the second sequence actually needs the register's original\n    contents, then preserving wraps a\n    save and a restore\n    of the register around the first sequence before appending the sequences.\n    Otherwise, preserving simply returns the\n    appended instruction sequences.  Thus, for example,\n    \n      (preserving (list $reg_1$ $reg_2$) $seq_1$ $seq_2$)\n      \n    produces one of the following four sequences of instructions, depending on\n    how\n    seq$_1$ and\n    seq$_2$ use\n    reg$_1$ and\n    reg$_2$:\n  ","5.5.1#p8":"\n    By using preserving to combine instruction\n    sequences the compiler avoids unnecessary\n    \n    stack operations.  This also\n    isolates the details of whether or not to generate\n    save and restore\n    instructions within the preservingprocedure,\n    separating them from the concerns that arise in writing each of the\n    individual code  generators.\n    In fact no save or\n    restore instructions are explicitly\n    produced by the code\n    generators.","5.5.1#p9":"\n    In principle, we could represent an instruction sequence simply as a\n    list of instructions.\n    Append-instruction-sequences\n    could then combine instruction sequences by performing an ordinary list\n    append.  However,\n    preserving would then be a complex operation,\n    because it would have to analyze each instruction sequence to\n    determine how the sequence uses its registers.\n    Preserving\n    would be inefficient as well as complex, because it would have to\n    analyze each of its instruction sequence arguments, even though these\n    sequences might themselves have been constructed by calls to\n    preserving, in which case their parts would\n    have already been analyzed.  To avoid such repetitious analysis we will\n    associate with each instruction sequence some information about its register\n    use. When we construct a basic instruction sequence we\n    will provide this information explicitly,\n    and the\n    procedures\n    that combine instruction sequences will derive\n    register-use information for the combined sequence from the\n    information associated with the sequences being combined.\n  ","5.5.1#p10":"\n    An instruction sequence will contain three pieces of information:\n    the set of registers that must be initialized before the\n      instructions in the sequence are executed (these registers are said to\n      be needed by the sequence),\n      the set of registers whose values are modified by the\n      instructions in the sequence, and\n      the actual instructions\n      \n\t  (also called statements)\n\t\n      in the sequence.\n      \n    We will represent an instruction sequence as a list of its three\n    parts.  The constructor for instruction sequences is thus\n    (define (make-instruction-sequence needs modifies statements)\n  (list needs modifies statements))","5.5.1#p11":"\n    For example, the two-instruction\n    sequence that looks up the value of the\n    \n\tvariable\n\tx\n    in the current environment,\n    assigns the result to val,\n    and then returns,\n    requires registers env and\n    continue to have been initialized, and\n    modifies register val.\n    This sequence would therefore be constructed as\n    (make-instruction-sequence '(env continue) '(val)\n      '((assign val\n            (op lookup-variable-value) (const x) (reg env))\n      (goto (reg continue))))","5.5.1#p12":"\n    The\n    procedures\n    for combining instruction sequences are shown in\n    section 5.5.4.\n  ","5.5.1#ex-5.31":"\n    In evaluating a\n    procedure\n    application, the explicit-control evaluator always saves and restores\n    the env register around the evaluation of the\n    operator,\n    saves and restores env around the\n    evaluation of each\n    operand\n    (except the final one), saves and restores\n    argl around the evaluation of each\n    operand,\n    and saves and restores\n    proc\n    around the\n    evaluation of the\n    operand\n    sequence.  For each of the following\n    combinations,\n    say which of these save and\n    restore operations are superfluous and\n    thus could be eliminated by the compiler's\n    preserving mechanism:\n    (f 'x 'y)\n\n      ((f) 'x 'y)\n\n      (f (g 'x) y)\n\n      (f (g 'x) 'y)","5.5.1#ex-5.32":"\n    Using the\n    preserving mechanism, the compiler\n    will avoid saving and restoring env around the\n    evaluation of the\n    \n\toperator of a combination\n      \n    in the case where the\n    \n\toperator is a symbol.\n      \n    We could also build such optimizations into the evaluator.\n    Indeed, the explicit-control evaluator of\n    section 5.4 already performs a similar\n    optimization, by treating\n    \n\tcombinations with no operands\n      \n    as a special case.\n    \n\tExtend the explicit-control evaluator to recognize as a separate class\n\tof\n\t\n\texpressions combinations whose operator is a symbol,\n\t  \n\tand to take\n\tadvantage of this fact in evaluating such\n\t\n\t    expressions.\n\t  \n\tAlyssa P. Hacker suggests that by extending the evaluator to recognize\n\tmore and more special cases we could incorporate all the compiler's\n\toptimizations, and that this would eliminate the advantage of compilation\n\taltogether.  What do you think of this idea?\n      ","5.5.1#footnote-1":"Notice,\n    however, that our compiler is a\n    Scheme\n    program, and the syntax\n    procedures\n    that it uses to manipulate expressions are the actual\n    Scheme procedures\n    used with the metacircular evaluator.  For the explicit-control evaluator, in\n    contrast, we assumed that equivalent syntax operations were available\n    as operations for the register machine.  (Of course, when we simulated\n    the register machine in\n    Scheme,\n    we used the actual\n    Scheme procedures\n    in our register machine simulation.)","5.5.2":"5.5.2  \n    Compiling","5.5.2#p1":"\n    In this section and the next we implement the code generators to which the\n    compileprocedure\n    dispatches.\n  ","5.5.2#h1":"Compiling linkage code","5.5.2#p2":"\n    In general, the output of each code generator will end with\n    instructions—generated by the\n    procedurecompile-linkage—that\n    implement the required linkage.  If the linkage is\n    return\n    then\n    we must generate the instruction\n    (goto (reg continue)).\n    This needs the continue register and does not\n    modify any registers.\n    If the linkage is\n    next,\n      \n    then we needn't include any additional instructions.  Otherwise, the\n    linkage is a label, and we generate a\n    goto\n    to that label, an instruction that does not need or modify\n    any registers.(define (compile-linkage linkage)\n  (cond ((eq? linkage 'return)\n         (make-instruction-sequence '(continue) '()\n          '((goto (reg continue)))))\n        ((eq? linkage 'next)\n         (empty-instruction-sequence))\n        (else\n         (make-instruction-sequence '() '()\n          `((goto (label ,linkage)))))))\n    The linkage code is appended to an instruction sequence by\n    preserving the\n    continue register, since a\n    return\n    linkage will\n    require the continue register:\n    If the given instruction sequence modifies\n    continue and the linkage code needs it,\n    continue will be saved and restored.\n    (define (end-with-linkage linkage instruction-sequence)\n  (preserving '(continue)\n   instruction-sequence\n   (compile-linkage linkage)))","5.5.2#footnote-link-1":"1","5.5.2#h2":"Compiling simple\n    \n\texpressions\n      ","5.5.2#p3":"\n    The code generators for\n    self-evaluating expressions, quotations, and variables\n    construct instruction\n    sequences that assign the required value to the target register\n    and then proceed as specified by the linkage descriptor.\n  ","5.5.2#p4":"(define (compile-self-evaluating exp target linkage)\n  (end-with-linkage linkage\n   (make-instruction-sequence '() (list target)\n    `((assign ,target (const ,exp))))))\n\n(define (compile-quoted exp target linkage)\n  (end-with-linkage linkage\n   (make-instruction-sequence '() (list target)\n    `((assign ,target (const ,(text-of-quotation exp)))))))\n\n(define (compile-variable exp target linkage)\n  (end-with-linkage linkage\n   (make-instruction-sequence '(env) (list target)\n    `((assign ,target\n              (op lookup-variable-value)\n              (const ,exp)\n              (reg env))))))All these\n    assignment instructions modify the target register,\n    and the one that looks up a\n    \n\tvariable\n      \n    needs the env register.\n  ","5.5.2#p5":"\n    Assignments and\n    definitions\n    are handled much as they are in the\n    interpreter.\n    \n        We recursively generate code that computes the value to be\n        assigned to the variable, and append to it a\n        two-instruction sequence that actually sets or defines the\n        variable and assigns the value of the whole expression\n        (the symbol ok) to the target\n        register. The recursive compilation has target\n        val and linkage\n        next so that the code will\n        put its result into val and\n        continue with the code that is appended after it. The\n        appending is done preserving\n        env, since the environment is\n        needed for setting or defining the variable and the code\n        for the variable value could be the compilation of a\n        complex expression that might modify the registers in\n        arbitrary ways.\n      (define (compile-assignment exp target linkage)\n  (let ((var (assignment-variable exp))\n        (get-value-code\n         (compile (assignment-value exp) 'val 'next)))\n    (end-with-linkage linkage\n     (preserving '(env)\n      get-value-code\n      (make-instruction-sequence '(env val) (list target)\n       `((perform (op set-variable-value!)\n                  (const ,var)\n                  (reg val)\n                  (reg env))\n         (assign ,target (const ok))))))))\n\n(define (compile-definition exp target linkage)\n  (let ((var (definition-variable exp))\n        (get-value-code\n         (compile (definition-value exp) 'val 'next)))\n    (end-with-linkage linkage\n     (preserving '(env)\n      get-value-code\n      (make-instruction-sequence '(env val) (list target)\n       `((perform (op define-variable!)\n                  (const ,var)\n                  (reg val)\n                  (reg env))\n         (assign ,target (const ok))))))))\n    The appended two-instruction sequence requires\n    env and val\n    and modifies the target.  Note that although we preserve\n    env for this sequence, we do not preserve\n    val, because the\n    get-value-code\n    is designed to explicitly place its result in\n    val for use by this sequence.\n    (In fact, if we did preserve val, we would\n    have a bug, because this would cause the previous contents of\n    val to be restored right after the\n    get-value-code\n    is run.)\n  ","5.5.2#h3":"Compiling\n    \n        conditional expressions\n      ","5.5.2#p6":"\n    The code for\n    \n\tan if expression\n    compiled with a given target and linkage has the form\n    \n $\\langle compilation\\ of\\ predicate,\\ target$ val$,\\ linkage$ next$\\rangle$\n (test (op false?) (reg val))\n (branch (label false-branch))\ntrue-branch\n $\\langle compilation\\ of\\ consequent\\ with\\ given\\ target\\ and\\ given\\ linkage or$ after-if$\\rangle$\nfalse-branch\n $\\langle compilation\\ of\\ alternative\\ with\\ given\\ target\\ and\\ linkage\\rangle$\nafter-if\n      ","5.5.2#p7":"\n    To generate this code, we compile the predicate, consequent,\n    and alternative, and combine the resulting code with instructions\n    to test the predicate result and with newly generated labels to mark the\n    true and false branches and the end of the\n    conditional.\n    In this arrangement of code, we must branch around the true branch\n    if the test is false.  The only slight complication is in how the\n    linkage for the true branch should be handled.  If the linkage for the\n    conditional is\n    return\n    or a label, then the\n    true and false branches will both use this same linkage.  If the linkage is\n    next,\n      \n    the true branch ends with a jump around\n    the code for the false branch to the label at the end of the conditional.\n    (define (compile-if exp target linkage)\n  (let ((t-branch (make-label 'true-branch))\n        (f-branch (make-label 'false-branch))\n        (after-if (make-label 'after-if)))\n    (let ((consequent-linkage\n           (if (eq? linkage 'next) after-if linkage)))\n      (let ((p-code (compile (if-predicate exp) 'val 'next))\n            (c-code\n             (compile\n              (if-consequent exp) target consequent-linkage))\n            (a-code\n             (compile (if-alternative exp) target linkage)))\n        (preserving '(env continue)\n         p-code\n         (append-instruction-sequences\n          (make-instruction-sequence '(val) '()\n           `((test (op false?) (reg val))\n             (branch (label ,f-branch))))\n          (parallel-instruction-sequences\n           (append-instruction-sequences t-branch c-code)\n           (append-instruction-sequences f-branch a-code))\n          after-if))))))Env\n    is preserved around the predicate code\n    because it could be needed by the true and false \n      branches, and continue is preserved because it could\n    be needed by the linkage code in those branches.\n  The code for the true and\n    false branches (which are not executed sequentially) is appended using a\n    special combiner\n    parallel-instruction-sequences\n    described in\n    section 5.5.4.\n  ","5.5.2#footnote-link-2":"2","5.5.2#p8":"\n\tNote that cond is a derived expression,\n\tso all that the compiler needs to do to handle it is to apply the\n\tcond->if transformer\n\t(from section 4.1.2) and\n\tcompile the resulting if expression.\n      ","5.5.2#h4":"Compiling sequences","5.5.2#p9":"\n    The compilation of\n    \n        sequences (from procedure bodies or explicit begin\n\texpressions)\n\tparallels their evaluation.\n      \n    Each\n    \n        expression\n      \n    of the sequence is compiled—the last\n    \n        expression\n      \n    with\n    the linkage specified for the sequence, and the other\n    \n        expressions\n      \n    with\n    linkage\n    next\n    (to execute the rest of the\n    sequence). The instruction sequences for the individual\n    \n        expressions\n      \n    are\n    appended to form a single instruction sequence, such that\n    env (needed for the rest of the\n    sequence)\n      and continue (possibly needed\n      for the linkage at the end of the sequence) are \n      preserved.\n    (define (compile-sequence seq target linkage)\n  (if (last-exp? seq)\n      (compile (first-exp seq) target linkage)\n      (preserving '(env continue)\n       (compile (first-exp seq) target 'next)\n       (compile-sequence (rest-exps seq) target linkage))))","5.5.2#h5":"Compiling\n\tlambda\n\texpressions","5.5.2#p10":"\n\tLambda expressions\n\t\n\tconstruct\n\tprocedures.\n\tThe object code for a lambda expression must have the form\n\t\n$\\langle construct\\ procedure\\ object\\ and\\ assign\\ it\\ to\\ target\\ register\\rangle$\n$\\langle linkage\\rangle$\n\t  \n\tWhen we compile the lambda expression, we also generate the code for the\n\tprocedure\n\tbody. Although the body won't be executed at the time of\n\tprocedure\n\tconstruction, it is convenient to insert it into the object code right after\n\tthe code for the\n\tlambda.\n\tIf the linkage for the lambda expression is a label or\n\treturn,\n\t  \n\tthis is fine.  But if the linkage is\n\tnext,\n\t  \n\twe will need to skip around the code for\n\tthe\n\tprocedure\n\tbody by using a linkage that jumps to a label that is inserted after the\n\tbody.  The object code thus has the form\n\t\n $\\langle construct\\ procedure\\ object\\ and\\ assign\\ it\\ to\\ target\\ register\\rangle$\n $\\langle code\\ for\\ given\\ linkage\\rangle\\ or$ (goto (label after-lambda))\n $\\langle compilation\\ of\\ procedure\\ body\\rangle$\nafter-lambda\n\t  ","5.5.2#p11":"Compile-lambda\n\tgenerates the code for constructing the\n\tprocedure\n\tobject followed by the code for the\n\tprocedure\n\tbody. The\n\tprocedure\n\tobject will be constructed at run time by combining the current environment\n\t(the environment at the point of definition) with the entry point to the\n\tcompiled\n\tprocedure\n\tbody (a newly generated label).(define (compile-lambda exp target linkage)\n  (let ((proc-entry (make-label 'entry))\n        (after-lambda (make-label 'after-lambda)))\n    (let ((lambda-linkage\n           (if (eq? linkage 'next) after-lambda linkage)))\n      (append-instruction-sequences\n       (tack-on-instruction-sequence\n        (end-with-linkage lambda-linkage\n         (make-instruction-sequence '(env) (list target)\n          `((assign ,target\n                    (op make-compiled-procedure)\n                    (label ,proc-entry)\n                    (reg env)))))\n        (compile-lambda-body exp proc-entry))\n       after-lambda))))Compile-lambda\n\tuses the special combiner\n\ttack-on-instruction-sequence\n\t(section 5.5.4) rather\n\tthan\n\tappend-instruction-sequences\n\tto append the\n\tprocedure\n\tbody to the lambda expression code, because the\n\tbody is not part of the sequence of instructions that will be executed when\n\tthe combined sequence is entered; rather, it is in the sequence only because\n\tthat was a convenient place to put it.\n      ","5.5.2#footnote-link-3":"3","5.5.2#p12":"Compile-lambda-body\n\tconstructs the code for the body of the\n\tprocedure.\n\tThis code begins with a label for the entry point.  Next come instructions\n\tthat will cause the runtime evaluation environment to switch to the correct\n\tenvironment for evaluating the\n\tprocedure\n\tbody—namely, the\n\tdefinition\n\tenvironment of the\n\tprocedure,\n\textended to include the bindings of the\n\tformal\n\tparameters to the arguments\n\twith which the\n\tprocedure\n\tis called.  After this comes the code for the\n\t\n\t\n\t    sequence of expressions that makes up the\n\t  procedure body.\n\t    The sequence is compiled with linkage return and target\n\t    val so that it will end by returning from\n\t    the procedure with the procedure result in\n\t    val.\n\t  (define (compile-lambda-body exp proc-entry)\n  (let ((formals (lambda-parameters exp)))\n    (append-instruction-sequences\n     (make-instruction-sequence '(env proc argl) '(env)\n      `(,proc-entry\n        (assign env (op compiled-procedure-env) (reg proc))\n        (assign env\n                (op extend-environment)\n                (const ,formals)\n                (reg argl)\n                (reg env))))\n     (compile-sequence (lambda-body exp) 'val 'return))))","5.5.2#footnote-1":"This procedure\n    uses a feature of Lisp called\n    backquote (or quasiquote) that is handy for constructing\n    lists. Preceding a list with a backquote symbol is much like quoting it,\n    except that anything in the list that is flagged with a comma is\n    evaluated.\n    \n    For example, if the value of linkage is the\n    symbol branch25,\n    then the expression\n    `((goto (label ,linkage)))\n    evaluates to the list\n    ((goto (label branch25))).\n    Similarly, if the value of x is the list\n    (a b c),\n    then\n    `(1 2 ,x)\n    evaluates to the list\n    (1 2 a).","5.5.2#footnote-2":"We can't just use the labels\n    true-branch,false-branch,\n    and\n    after-if\n    as shown above, because there might be more than one\n    if\n    in the program.\n    \n    The compiler uses the\n    proceduremake-label\n    to generate labels.\n    Make-label\n    takes a symbol as argument and returns a new\n    symbol that begins with the\n    given symbol.  For example, successive calls to\n    (make-label 'a)\n    would return\n    a1,\n    a2,\n    and so on.\n    Make-label\n    can be implemented similarly to the generation of unique variable names in\n    the query language, as follows:(define label-counter 0)\n\n(define (new-label-number)\n  (set! label-counter (+ 1 label-counter))\n  label-counter)\n\n(define (make-label name)\n  (string->symbol\n    (string-append (symbol->string name)\n                   (number->string (new-label-number)))))","5.5.2#footnote-3":"We need machine operations to\n\timplement a data structure for representing compiled\n\tprocedures,\n\tanalogous to the structure for compound\n\tprocedures\n\tdescribed in section 4.1.3:\n\t(define (make-compiled-procedure entry env)\n  (list 'compiled-procedure entry env))\n\n(define (compiled-procedure? proc)\n  (tagged-list? proc 'compiled-procedure))\n\n(define (compiled-procedure-entry c-proc) (cadr c-proc))\n\n(define (compiled-procedure-env c-proc) (caddr c-proc))","5.5.3":"5.5.3  \n    Compiling","5.5.3#p1":"\n    The essence of the compilation process is the compilation of\n    procedure\n    applications. The code for\n    a combination\n    compiled with a given target and\n    linkage has the form\n    \n$\\langle compilation\\ of\\ operator,\\ target$ proc$,\\ linkage$ next$\\rangle$\n$\\langle evaluate\\ operands\\ and\\ construct\\ argument\\ list\\ in$ argl$\\rangle$\n$\\langle compilation\\ of\\ procedure\\ call\\ with\\ given\\ target\\ and\\ linkage\\rangle$\n      \n\n    The registers env,\n    proc,\n      \n    and argl may\n    have to be saved and restored during evaluation of the\n    \n\toperator and operands.\n      \n    Note that this is the only place in the compiler where a target other\n    than val is specified.\n  ","5.5.3#p2":"\n    The required code is generated by\n    compile-application.\n    This recursively compiles the\n    operator,\n    to produce code that puts the\n    procedure\n    to be applied into\n    proc,\n    and compiles the\n    operands,\n    to produce code that evaluates the individual\n    operands\n    of the\n    application.  The instruction sequences for the\n    operands\n    are combined\n    (by\n    construct-arglist)\n    with code that constructs the list of\n    arguments in argl, and the resulting\n    argument-list code is combined with the\n    procedure\n    code and the code that performs the\n    procedure\n    call (produced by\n    compile-procedure-call).\n    In appending the code sequences, the env\n    register must be preserved around the evaluation of the\n    operator\n    (since\n    evaluating the\n    operator\n    might modify env, which\n    will be needed to evaluate the\n    operands),\n    and the\n    proc\n    register must be preserved around the\n    construction of the argument list (since evaluating the\n    operands\n    might\n    modify\n    proc,\n      \n    which will be needed for the actual\n    procedure\n    application). Continue\n    must also be preserved throughout, since\n    it is needed for the linkage in the\n    procedure\n    call.\n    (define (compile-application exp target linkage)\n  (let ((proc-code (compile (operator exp) 'proc 'next))\n        (operand-codes\n         (map (lambda (operand) (compile operand 'val 'next))\n              (operands exp))))\n    (preserving '(env continue)\n     proc-code\n     (preserving '(proc continue)\n      (construct-arglist operand-codes)\n      (compile-procedure-call target linkage)))))","5.5.3#p3":"\n    The code to construct the argument list will evaluate each\n    \n    operand\n      \n    into\n    val and then\n    cons\n\tthat value onto the argument list being accumulated in\n\targl.\n      \n    Since we\n    cons\n\tthe arguments onto argl\n\tin sequence,\n      \n    we must start with the last argument and end with the first, so that the\n    arguments will appear in order from first to last in the resulting list.\n    Rather than waste an instruction by initializing\n    argl to the empty list\n    to set up for this sequence of evaluations,\n    we make the first code sequence construct the initial\n    argl.\n    The general form of the argument-list construction is thus as follows:\n    \n$\\langle compilation\\ of\\ last\\ operand,\\ targeted\\ to$ val$\\rangle$\n(assign argl (op list) (reg val))\n$\\langle compilation\\ of\\ next\\ operand,\\ targeted\\ to$ val$\\rangle$\n(assign argl (op cons) (reg val) (reg argl))\n$\\ldots$\n$\\langle compilation\\ of\\ first\\ operand,\\ targeted\\ to$ val$\\rangle$\n(assign argl (op cons) (reg val) (reg argl))\n      Argl\n\tmust be preserved around each operand\n      \n    evaluation except the first (so that arguments accumulated so far\n    won't be lost), and env must be\n    preserved around each\n    \n\toperand evaluation except the last (for use by subsequent operand evaluations).\n      ","5.5.3#p4":"\n    Compiling this argument code is a bit tricky, because of the special\n    treatment of the first\n    operand\n    to be evaluated and the need to preserve\n    argl and env in\n    different places. The\n    construct-arglistprocedure\n    takes as arguments the code that evaluates the individual\n    operands.\n    If there are no\n    operands\n    at all, it simply emits the instruction\n    (assign argl (const ()))\n    Otherwise,\n    construct-arglist\n    creates code that initializes argl with the\n    last argument, and appends code that evaluates the rest of the arguments and\n    adjoins them to argl in succession.  In order\n    to process the arguments from last to first, we must reverse the list of\n    operand\n    code sequences from the order supplied by\n    compile-application.(define (construct-arglist operand-codes)\n  (let ((operand-codes (reverse operand-codes)))\n    (if (null? operand-codes)\n        (make-instruction-sequence '() '(argl)\n         '((assign argl (const ()))))\n        (let ((code-to-get-last-arg\n               (append-instruction-sequences\n                (car operand-codes)\n                (make-instruction-sequence '(val) '(argl)\n                 '((assign argl (op list) (reg val)))))))\n          (if (null? (cdr operand-codes))\n              code-to-get-last-arg\n              (preserving '(env)\n               code-to-get-last-arg\n               (code-to-get-rest-args\n                (cdr operand-codes))))))))\n\n(define (code-to-get-rest-args operand-codes)\n  (let ((code-for-next-arg\n         (preserving '(argl)\n          (car operand-codes)\n          (make-instruction-sequence '(val argl) '(argl)\n           '((assign argl\n              (op cons) (reg val) (reg argl)))))))\n    (if (null? (cdr operand-codes))\n        code-for-next-arg\n        (preserving '(env)\n         code-for-next-arg\n         (code-to-get-rest-args (cdr operand-codes))))))","5.5.3#h1":"\n      Applying\n      procedures","5.5.3#p5":"\n    After evaluating the elements of a\n    combination,\n    the compiled code must apply the\n    procedure\n    in\n    proc\n    to the arguments in\n    argl.  The code performs essentially the same\n    dispatch as the applyprocedure\n    in the metacircular evaluator of\n    section 4.1.1 or the\n    apply-dispatch\n    entry point in the explicit-control evaluator of\n    section 5.4.2.  It checks whether\n    the\n    procedure\n    to be applied is a primitive\n    procedure\n    or a compiled\n    procedure.\n    For a primitive\n    procedure,\n    it uses\n    apply-primitive-procedure;\n    we will see shortly how it handles compiled\n    procedures.\n    The\n    procedure-application\n    code has the following form:\n    \n (test (op primitive-procedure?) (reg proc))\n (branch (label primitive-branch))\ncompiled-branch\n $\\langle code\\ to\\ apply\\ compiled\\ procedure\\ with\\ given\\ target\\ and\\ appropriate\\ linkage\\rangle$\nprimitive-branch\n (assign $\\langle target\\rangle$\n         (op apply-primitive-procedure)\n         (reg proc)\n         (reg argl))\n $\\langle linkage \\rangle$\nafter-call\n      \n    Observe that the compiled branch must skip around the primitive\n    branch.  Therefore, if the linkage for the original\n    procedure\n    call was\n    next,\n      \n    the compound branch must use a\n    linkage that jumps to a label that is inserted after the primitive branch.\n    (This is similar to the linkage used for the true branch in\n    compile-if.)(define (compile-procedure-call target linkage)\n  (let ((primitive-branch (make-label 'primitive-branch))\n        (compiled-branch (make-label 'compiled-branch))\n        (after-call (make-label 'after-call)))\n    (let ((compiled-linkage\n           (if (eq? linkage 'next) after-call linkage)))\n      (append-instruction-sequences\n       (make-instruction-sequence '(proc) '()\n        `((test (op primitive-procedure?) (reg proc))\n          (branch (label ,primitive-branch))))\n       (parallel-instruction-sequences\n        (append-instruction-sequences\n         compiled-branch\n         (compile-proc-appl target compiled-linkage))\n        (append-instruction-sequences\n         primitive-branch\n         (end-with-linkage linkage\n          (make-instruction-sequence '(proc argl)\n                                     (list target)\n           `((assign ,target\n                     (op apply-primitive-procedure)\n                     (reg proc)\n                     (reg argl)))))))\n       after-call))))\n    The primitive and compound branches, like the true\n    and false branches in\n    compile-if,\n    are appended using\n    parallel-instruction-sequences\n    rather than the ordinary\n    append-instruction-sequences,\n      \n    because they will not be executed sequentially.\n  ","5.5.3#h2":"\n      Applying compiled\n    procedures","5.5.3#p6":"The code that handles procedure\n    application\n    \n      is the most subtle part of the\n      \n          compiler, even though the instruction sequences it generates are very short.\n        \n      A compiled\n    procedure\n    (as constructed by\n    compile-lambda)\n    has an entry point, which is a label that designates where the code for the\n    procedure\n    starts. The code at this entry point computes a result in\n    valand returns by executing the instruction (goto (reg continue)).","5.5.3#p7":"\n        Thus, we might expect the code for a\n        compiled-procedure application (to be\n        generated by\n        compile-proc-appl) with a\n        given target and linkage to look like this if the linkage\n        is a label:\n      \n (assign continue (label proc-return))\n (assign val (op compiled-procedure-entry) (reg proc))\n (goto (reg val))\nproc-return\n (assign $target$ (reg val)) ; included if target is not $\\texttt{val}$\n (goto (label $\\langle linkage\\rangle$))  ; linkage code\n      \n    or like\n    \n        this if the linkage is return.\n      \n (save continue)\n (assign continue (label proc-return))\n (assign val (op compiled-procedure-entry) (reg proc))\n (goto (reg val))\nproc-return\n (assign $\\langle target\\rangle$ (reg val)) ; included if target is not $\\texttt{val}$\n (restore continue)\n (goto (reg continue))      ; linkage code\n      \n    This code sets up continue so that the\n    procedure\n    will return to a label\n    proc-return\n    and jumps to the\n    procedure's\n    entry point. The code at\n    proc-return\n    transfers the\n    procedure's\n    result from val to the target register (if\n    necessary) and then jumps to the location specified by the linkage.\n    (The linkage is always\n    return\n    or a label,\n    because\n    compile-procedure-call\n    replaces a\n    next\n    linkage for the\n    \n\tcompound-procedure branch by an\n      after-call\n    label.)\n\n    ","5.5.3#p8":"\n    In fact, if\n    the target is not val,\n    that is\n    exactly the code our compiler will generate.\n    Usually, however, the target is val (the only\n    time the compiler specifies a different register is when targeting the\n    evaluation of\n    an operator\n    to\n    proc),\n      \n    so the\n    procedure\n    result is put directly into\n    the target register and there is no need to\n    return\n    to a special\n    location that copies it. Instead we simplify the code by\n    setting up continue so that the\n    procedure\n    will\n    \"return\"\n    directly to the place specified by the caller's linkage:\n    \n $\\langle set\\ up$ continue $for\\ linkage\\rangle$\n (assign val (op compiled-procedure-entry) (reg proc))\n (goto (reg val))\n      \n    If the linkage is a label, we set up continue\n    so that the\n    procedure\n    will return to\n    that label.  (That is, the\n    (goto (reg continue))\n    the\n    procedure\n    ends with becomes equivalent to the\n    (goto (label linkage))\n    at\n    proc-return\n    above.)\n    \n      (assign continue (label linkage))\n      (assign val (op compiled-procedure-entry) (reg proc))\n      (goto (reg val))\n      \n    If the linkage is\n    return,\n      \n    we don't need to \n\n    set up continue at all:\n\n    It already holds the desired location.\n    (That is, the\n    (goto (reg continue))\n    the\n    procedure\n    ends with goes directly to the\n    place where the\n    (goto (reg continue))\n    at\n    proc-return\n    would have gone.)\n    (assign val (op compiled-procedure-entry) (reg proc))\n      (goto (reg val))\n    With this implementation of the\n    return\n    linkage,\n    the compiler generates\n    \n    tail-recursive code.\n    Calling a procedure as the final step in a procedure body\n    does a direct transfer, without saving\n    any information on the stack.","5.5.3#footnote-link-1":"1","5.5.3#p9":"\n    Suppose instead that we had handled the case of a\n    procedure\n    call with a\n    linkage of return and a target of\n    val in the same way as for a\n    non-val target. This would destroy tail\n    recursion. Our system would still\n    \n        give\n      \n      the same value for any\n    \n\texpression.\n      \n    But each time we called a\n    procedure,\n    we would save\n    continue \n      and return after the call to undo the (useless) save. These extra\n    saves would accumulate during a nest of\n    procedure\n    calls.","5.5.3#footnote-link-2":"2","5.5.3#p10":"Compile-proc-appl\n    generates the above\n    procedure-application\n    code by considering four cases,\n    depending on whether the target for the call\n    is val and whether the linkage is\n    return.\n      \n    Observe that the instruction sequences\n    are declared to modify all the registers, since executing the\n    procedure\n    body can change the registers in arbitrary ways.\n\tAlso note that the code sequence for the case with target\n\tval and linkage\n\treturn\n\tis declared to need\n\tcontinue:  Even though\n\tcontinue is not explicitly used in the\n\ttwo-instruction sequence, we must be sure that\n\tcontinue will have the correct\n\tvalue when we enter the compiled procedure.\n      \n      (define (compile-proc-appl target linkage)\n      (cond ((and (eq? target 'val) (not (eq? linkage 'return)))\n            (make-instruction-sequence '(proc) all-regs\n            `((assign continue (label ,linkage))\n            (assign val (op compiled-procedure-entry)\n            (reg proc))\n            (goto (reg val)))))\n            ((and (not (eq? target 'val))\n            (not (eq? linkage 'return)))\n            (let ((proc-return (make-label 'proc-return)))\n            (make-instruction-sequence '(proc) all-regs\n            `((assign continue (label ,proc-return))\n            (assign val (op compiled-procedure-entry)\n            (reg proc))\n            (goto (reg val))\n            ,proc-return\n            (assign ,target (reg val))\n            (goto (label ,linkage))))))\n            ((and (eq? target 'val) (eq? linkage 'return))\n            (make-instruction-sequence '(proc continue) all-regs\n            '((assign val (op compiled-procedure-entry)\n            (reg proc))\n            (goto (reg val)))))\n            ((and (not (eq? target 'val)) (eq? linkage 'return))\n            (error \"return linkage, target not val - - COMPILE\"\n            target))))\n      ","5.5.3#footnote-link-3":"3","5.5.3#footnote-1":"Actually, we signal\n    an error when the target is not val and the\n    linkage is\n    return,\n      \n    since the only place we request\n    returnlinkages \n      \n    is in compiling\n    procedures,\n    and our convention is that\n    procedures\n    return their values in val.","5.5.3#footnote-2":"Making a\n    \n    compiler generate tail-recursive\n    \n        code might seem like a straightforward idea.  But\n      \n    However, compilers for common languages,\n    including\n    C and Pascal,\n    do not always do this, and therefore these languages\n    cannot represent iterative processes in terms of\n    procedure\n    call alone. The difficulty with\n    \n    tail recursion in these languages is that their\n    implementations use the stack to store\n    procedure\n    arguments and local\n    variables\n    as well as return addresses.  The\n    \n\tScheme\n      \n    implementations described in this book store arguments and\n    variables\n    in memory to be garbage-collected.  The reason for using the stack for\n    variables\n    and arguments is that it avoids the need for garbage collection\n    in languages that would not otherwise require it, and is generally\n    believed to be more efficient.  Sophisticated\n    Lisp\n    compilers can, in fact, use the stack for arguments without destroying tail\n    recursion. (See\n    Hanson 1990 for a description.)  There is also some\n    debate about whether stack allocation is actually more efficient than garbage\n    collection in the first place, but the details seem to hinge on fine\n    points of computer architecture.  (See\n    Appel 1987 and\n    Miller and Rozas 1994\n    for opposing views on this issue.)","5.5.3#footnote-3":"The\n    variableall-regs\n    is bound to the list of names of all the registers:\n    (define all-regs '(env proc val argl continue))","5.5.4":"5.5.4  \n    Combining Instruction Sequences","5.5.4#p1":"\n    This section describes the details on how instruction sequences are\n    represented and combined.  Recall from\n    section 5.5.1 that an instruction\n    sequence is represented as a list of the registers needed, the registers\n    modified, and the actual instructions.  We will also consider a label\n    (symbol)\n    to be a degenerate case of an instruction sequence, which\n    doesn't need or modify any registers.\n    So to determine the registers needed\n    and modified by instruction sequences we use the selectors\n    (define (registers-needed s)\n  (if (symbol? s) '() (car s)))\n\n (define (registers-modified s)\n   (if (symbol? s) '() (cadr s)))\n\n(define (statements s)\n  (if (symbol? s) (list s) (caddr s)))\n    and to determine whether a given\n    sequence needs or modifies a given register we use the predicates\n    (define (needs-register? seq reg)\n  (memq reg (registers-needed seq)))\n\n(define (modifies-register? seq reg)\n  (memq reg (registers-modified seq)))\n    In terms of these predicates and selectors, we can implement the\n    various instruction sequence combiners used throughout the compiler.\n  ","5.5.4#p2":"\n    The basic combiner is\n    append-instruction-sequences.\n      \n    This takes as\n    arguments\n    an arbitrary number of\n    instruction sequences that are to be\n    executed sequentially and returns an instruction sequence whose statements\n    are the statements of\n    all the\n    sequences appended together.\n    The subtle point is to determine the registers that are needed and modified by the resulting sequence.\n    It modifies those registers that\n    are modified by any of the sequences;\n    it needs those registers that must be initialized before the\n    first sequence can be run (the registers needed by the first sequence), together with those registers needed by\n    any of the other sequences that are not initialized (modified) by sequences preceding it.","5.5.4#p3":"\n\tThe sequences are appended two at a time by\n      append-2-sequences. This\n      \n    is given two instruction sequences seq1 and\n    seq2 and returns the instruction sequence whose\n    \n        statements\n      \n    are the\n    \n        statements\n      \n    of seq1 followed\n    by the\n    \n        statements\n      \n    of seq2, whose modified\n    registers are those registers that are modified by either\n    seq1 or seq2, and\n    whose needed registers are the registers needed by\n    seq1 together with those registers needed by\n    seq2 that are not modified by\n    seq1.  (In terms of set operations, the new set\n    of needed registers is the union of the set of registers needed by\n    seq1 with the set difference of the registers\n    needed by seq2 and the registers modified by\n    seq1.)  Thus,\n    append-instruction-sequences\n    is implemented as follows:\n    (define (append-instruction-sequences . seqs)\n  (define (append-2-sequences seq1 seq2)\n    (make-instruction-sequence\n     (list-union (registers-needed seq1)\n                 (list-difference (registers-needed seq2)\n                                  (registers-modified seq1)))\n     (list-union (registers-modified seq1)\n                 (registers-modified seq2))\n     (append (statements seq1) (statements seq2))))\n  (define (append-seq-list seqs)\n    (if (null? seqs)\n        (empty-instruction-sequence)\n        (append-2-sequences (car seqs)\n                            (append-seq-list (cdr seqs)))))\n  (append-seq-list seqs))","5.5.4#p4":"\n    This\n    procedure\n    uses some simple operations for manipulating sets\n    represented as lists, similar to the (unordered) set representation\n    described in section 2.3.3:\n    (define (list-union s1 s2)\n  (cond ((null? s1) s2)\n        ((memq (car s1) s2) (list-union (cdr s1) s2))\n        (else (cons (car s1) (list-union (cdr s1) s2)))))\n\n(define (list-difference s1 s2)\n  (cond ((null? s1) '())\n        ((memq (car s1) s2) (list-difference (cdr s1) s2))\n        (else (cons (car s1)\n                    (list-difference (cdr s1) s2)))))","5.5.4#p5":"Preserving,\n      \n    the second major instruction\n    sequence combiner, takes a list of registers\n    regs and two instruction sequences\n    seq1 and seq2 that\n    are to be executed sequentially.  It returns an instruction sequence whose\n    \n        statements\n      \n    are the\n    \n        statements\n      \n    of seq1 followed\n    by the\n    \n        statements\n      \n    of seq2, with appropriate\n    save and restore\n    instructions around seq1 to protect the\n    registers in regs that are modified by\n    seq1 but needed by\n    seq2.  To accomplish this,\n    preserving first creates a sequence that has\n    the required saves followed by the\n    \n        statements\n      \n    of seq1 followed by the required\n    restores.  This sequence needs the registers\n    being saved and restored in addition to the registers needed by\n    seq1, and modifies the registers modified by\n    seq1 except for the ones being saved and\n    restored.  This augmented sequence and seq2\n    are then appended in the usual way.  The following\n    procedure\n    implements this strategy recursively, walking down the list of registers to\n    be preserved:(define (preserving regs seq1 seq2)\n  (if (null? regs)\n      (append-instruction-sequences seq1 seq2)\n      (let ((first-reg (car regs)))\n        (if (and (needs-register? seq2 first-reg)\n                 (modifies-register? seq1 first-reg))\n            (preserving (cdr regs)\n             (make-instruction-sequence\n              (list-union (list first-reg)\n                          (registers-needed seq1))\n              (list-difference (registers-modified seq1)\n                               (list first-reg))\n              (append `((save ,first-reg))\n                      (statements seq1)\n                      `((restore ,first-reg))))\n             seq2)\n            (preserving (cdr regs) seq1 seq2)))))","5.5.4#footnote-link-1":"1","5.5.4#p6":"\n    Another sequence combiner,\n    tack-on-instruction-sequence,\n      \n    is used by\n    compile-lambda\n    to append a\n    procedure\n    body to another sequence.  Because the\n    procedure\n    body is not \"in line\" to be executed as part of the combined\n    sequence, its register use has no impact on the register use of the sequence\n    in which it is embedded. We thus ignore the\n    procedure\n    body's sets of needed and modified\n    registers when we tack it onto the other sequence.\n    (define (tack-on-instruction-sequence seq body-seq)\n  (make-instruction-sequence\n   (registers-needed seq)\n   (registers-modified seq)\n   (append (statements seq) (statements body-seq))))","5.5.4#p7":"Compile-if\n    and\n    compile-procedure-call\n    use a special combiner called\n    parallel-instruction-sequences\n    to append the two alternative branches that follow a test.  The two branches\n    will never be executed sequentially; for any particular evaluation of the\n    test, one branch or the other will be entered.  Because of this, the\n    registers needed by the second branch are still needed by the combined\n    sequence, even if these are modified by the first branch.\n    (define (parallel-instruction-sequences seq1 seq2)\n  (make-instruction-sequence\n   (list-union (registers-needed seq1)\n               (registers-needed seq2))\n   (list-union (registers-modified seq1)\n               (registers-modified seq2))\n   (append (statements seq1) (statements seq2))))","5.5.4#footnote-1":"Note that preserving\n    calls append with three\n    \n    arguments.  Though the definition of append\n    shown in this book accepts only two arguments, Scheme standardly provides an\n    append procedure\n    that takes an arbitrary number of arguments.","5.5.5":"5.5.5  \n    An Example of Compiled Code","5.5.5#p1":"\n    Now that we have seen all the elements of the compiler, let us examine\n    an example of compiled code to see how things fit together.  We will\n    compile the\n    definition\n    of a recursive\n    factorialprocedure\n\tby calling\n\tcompile:\n      (compile\n '(define (factorial n)\n    (if (= n 1)\n        1\n        (* (factorial (- n 1)) n)))\n 'val\n 'next)\n    We have specified that the value of the\n    define expression\n    should be placed in the val register.\n    We don't care what the compiled\n    code does after executing the\n    define,\n    so our choice of\n    next\n    as the linkage\n    descriptor is arbitrary.\n  ","5.5.5#p2":"Compile\n\tdetermines that the\n\texpression is a definition, so it\n      \ncalls    \n    compile-definition to compile \n    code to compute the value to be assigned (targeted to\n    val), followed by code to install the\n    definition,\n    followed by code to put the value of the\n    define (which is the symbol\n\tok)\n      \n    into the target register, followed finally by the linkage code.\n    Env\n    is preserved around the computation of the\n    value, because it is needed in order to install the\n    \n\tdefinition.\n      \n    Because\n    the linkage is\n    next,\n      \n    there is no linkage code\n    in this case.  The skeleton of the compiled code is thus\n    \n$\\langle save$ env $if\\ modified\\ by\\ code\\ to\\ compute\\ value\\rangle$\n$\\langle compilation\\ of\\ definition\\ value, target$ val$, linkage$ next$\\rangle$\n$\\langle restore$ env $if\\ saved\\ above\\rangle$\n(perform (op define-variable!)\n         (const factorial)\n         (reg val)\n         (reg env))\n(assign val (const ok))\n      ","5.5.5#p3":"\n    The expression that is\n    to be\n    compiled to produce the value for the\n    variablefactorial\n    is a\n    lambda\n    expression whose value is the\n    procedure\n    that computes factorials.\n    Compile\n    handles this\n    by calling\n    compile-lambda,\n    which compiles the\n    procedure\n    body, labels it as a new entry point, and generates the instruction that\n    will combine the\n    procedure\n    body at the new entry point with the runtime environment and assign the\n    result to val.  The sequence then skips around\n    the compiled\n    procedure\n    code, which is inserted at this point.  The\n    procedure\n    code itself begins by extending the\n    procedure's definition\n    environment by a frame that binds the\n    formal\n    parameter n to the\n    procedure\n    argument.  Then comes the actual\n    procedure\n    body.  Since this code for the value of the\n    variable\n    doesn't modify the env register, the\n    optional save\n    and restore shown above aren't\n    generated.  (The\n    procedure\n    code at\n    entry2\n     isn't executed at this point,\n    so its use of env is irrelevant.)\n    Therefore, the skeleton for the compiled code becomes\n    \n  (assign val (op make-compiled-procedure)\n              (label entry2)\n              (reg env))\n  (goto (label after-lambda1))\nentry2\n  (assign env (op compiled-procedure-env) (reg proc))\n  (assign env (op extend-environment)\n              (const (n))\n              (reg argl)\n              (reg env))\n  $\\langle compilation\\ of\\ procedure\\ body\\rangle$\nafter-lambda1\n  (perform (op define-variable!)\n           (const factorial)\n           (reg val)\n           (reg env))\n  (assign val (const ok))\n      ","5.5.5#p4":"\n    A\n    procedure\n    body is always compiled (by\n    compile-lambda-body)as a sequence\n    with target val and linkage\n    return.\n      \n    The\n    \n\tsequence\n      \n    in this case consists of\n    a single\n    if expression:\n      (if (= n 1)\n    1\n    (* (factorial (- n 1)) n))Compile-if\n    generates code that first computes the predicate (targeted to\n    val), then checks the result and branches\n    around the true branch if the predicate is false.\n    Env\n    and continue\n    are preserved around the predicate code, since they may be needed for the\n    rest of the\n    if\n    expression.\n    \n\tSince the if expression\n\tis the final expression (and only expression) in the sequence making up\n\tthe procedure\n\tbody, its target is val and its linkage is\n\treturn,\n\tso the\n      \n    true and false branches are both\n    compiled with target val and linkage\n    return.\n      \n    (That is, the value of the conditional,\n    which is the value computed by either of its branches, is the value of the\n    procedure.)\n  $\\langle save$ continue, env $if\\ modified\\ by\\ predicate\\ and\\ needed\\ by\\ branches\\rangle$\n  $\\langle compilation\\ of\\ predicate, target$ val$,\\ linkage$ next$\\rangle$\n  $\\langle restore$ continue, env $if\\ saved\\ above\\rangle$\n  (test (op false?) (reg val))\n  (branch (label false-branch4))\ntrue-branch5\n  $\\langle compilation\\ of\\ true\\ branch, target$ val$,\\ linkage$ return$\\rangle$\nfalse-branch4\n  $\\langle compilation\\ of\\ false\\ branch, target$ val$,\\ linkage$ return$\\rangle$\nafter-if3\n      ","5.5.5#p5":"\n    The predicate\n    (= n 1)\n    is a\n    procedure call.\n    This looks up the\n    \n    operator\n    (the symbol =)\n      \n    and places this value in\n    proc.\n    It then assembles the arguments 1 and the value\n    of n into argl.\n    Then it tests whether\n    proc\n    contains a primitive or a compound\n    procedure,\n    and dispatches to a primitive branch or a compound branch accordingly.\n    Both branches resume at the\n    after-call\n    label.\n    \n    The requirements to preserve registers around the evaluation of the\n    \n\toperator and operands\n      \n    don't result in\n    any saving of registers, because in this case those evaluations don't\n    modify the registers in question.\n    \n  (assign proc\n          (op lookup-variable-value) (const =) (reg env))\n  (assign val (const 1))\n  (assign argl (op list) (reg val))\n  (assign val (op lookup-variable-value) (const n) (reg env))\n  (assign argl (op cons) (reg val) (reg argl))\n  (test (op primitive-procedure?) (reg proc))\n  (branch (label primitive-branch17))\ncompiled-branch16\n  (assign continue (label after-call15))\n  (assign val (op compiled-procedure-entry) (reg proc))\n  (goto (reg val))\nprimitive-branch17\n  (assign val (op apply-primitive-procedure)\n              (reg proc)\n              (reg argl))\nafter-call15\n      ","5.5.5#p6":"\n    The true branch, which is the constant 1, compiles (with target\n    val and linkage\n    return)\n      \n    to\n    \n(assign val (const 1))\n(goto (reg continue))\n      \n    The code for the false branch is another\n    procedure\n    call, where the\n    procedure\n    is the value of the symbol\n    *,\n    and the arguments\n    are n and the result of another\n    procedure\n    call (a call to factorial).\n    Each of these calls sets up\n    proc\n    and argl and its own primitive\n    and compound branches.  Figure 5.17\n    shows the complete compilation of the\n    \n\tdefinition\n      \n    of the factorialprocedure.\n    Notice that the possible save and\n    restore of\n    continue and\n    env around the predicate, shown above,\n    are in fact generated, because these registers are modified by the\n    procedure\n    call in the predicate and needed for the\n    procedure\n    call and the\n    return\n    linkage in the branches.\n  ","5.5.5#ex-5.33":"\n    Consider the following declaration of a factorial\n    procedure,\n    which is slightly different from the one given above:\n    (define (factorial-alt n)\n  (if (= n 1)\n      1\n      (* n (factorial-alt (- n 1)))))\n    Compile this\n    procedure\n    and compare the resulting code with that produced for\n    factorial.  Explain any differences you find.\n    Does either program execute more efficiently than the other?\n    ","5.5.5#ex-5.34":"\n    Compile the\n    \n    iterative factorial\n    procedure(define (factorial n)\n  (define (iter product counter)\n    (if (> counter n)\n        product\n        (iter (* counter product)\n        (+ counter 1))))\n  (iter 1 1))\n    Annotate the resulting code, showing the essential difference between\n    the code for iterative and recursive versions of\n    factorial that makes one process build up\n    stack space and the other run in constant stack space.\n    ","5.5.5#ex-5.35":"\n  What\n  \n      expression\n    \n  was compiled to produce the code shown in\n  figure 5.19?\n  ","5.5.5#ex-5.36":"\n    What\n    \n    order of evaluation does our compiler produce for\n    operands of a combination?\n    Is it left-to-right (as mandated by the ECMAScript specification), right-to-left, or some other order?\n    Where in the compiler is this order determined?  Modify the compiler\n    so that it produces some other order of evaluation.  (See the\n    discussion of order of evaluation for the explicit-control evaluator\n    in section 5.4.1.)  How does changing the\n    order of\n    operand\n    evaluation affect the efficiency of the code that\n    constructs the argument list?\n    ","5.5.5#ex-5.37":"\n    One way to understand the compiler's\n    preserving mechanism for\n    optimizing stack usage is to see what extra operations would\n    be generated if we did not use this idea.  Modify\n    preserving so\n    that it always generates the save and\n    restore operations.\n    Compile some simple expressions and identify the unnecessary stack\n    operations that are generated.\n    Compare the code to that generated with the\n    preserving mechanism intact.\n    ","5.5.5#ex-5.38":"\n    Our compiler is clever about avoiding unnecessary stack operations,\n    but it is not clever at all when it comes to compiling calls to the primitive\n    procedures\n    of the language in terms of the primitive operations\n    supplied by the machine.  For example, consider how much code is\n    compiled to compute\n    (+ a 1):\n    The code sets up an argument list in argl, puts\n    the primitive addition\n    procedure\n    (which it finds by looking up the symbol\n    +\n    in the environment) into\n    proc,\n    and tests whether the\n    procedure\n    is primitive or compound.  The\n    compiler always generates code to perform the test, as well as code\n    for primitive and compound branches (only one of which will be executed).\n    We have not shown the part of the controller that implements\n    primitives, but we presume that these instructions make use of\n    primitive arithmetic operations in the machine's data paths.  Consider\n    how much less code would be generated if the compiler could\n    open-code primitives—that is, if it could generate code to\n    directly use these primitive machine operations.  The expression\n    (+ a 1)\n    might be compiled into something as simple as(assign val (op lookup-variable-value) (const a) (reg env))\n(assign val (op +) (reg val) (const 1))\n    In this exercise we will extend our compiler to support open coding of\n    selected primitives.  Special-purpose code will be generated for calls to these primitive\n    procedures\n    instead of the general\n    procedure-application\n    code.  In order to support this, we will augment\n    our machine with special argument registers\n    arg1 and arg2.\n    The primitive arithmetic operations of the machine will take their\n    inputs from arg1 and\n    arg2. The results may be put into\n    val, arg1, or\n    arg2.\n    \n    The compiler must be able to recognize the application of an\n    open-coded primitive in the source program.  We will augment the\n    dispatch in the compileprocedure\n    to recognize the names of these primitives in addition to the\n    \n\treserved words (the special forms)\n\tit currently recognizes.\n    For each\n    special\n    form our compiler has a code\n    generator.  In this exercise we will construct a family of code generators\n    for the open-coded primitives.\n    \n\tThe open-coded primitives, unlike the\n\tspecial\n\tforms, all need their\n\toperands\n\tevaluated.  Write a code generator\n\tspread-arguments\n\tfor use by all the open-coding code generators.\n\tSpread-arguments\n\tshould take\n\tan operand list\n\tand compile the given\n\toperands\n\ttargeted to\n\tsuccessive argument registers.  Note that an\n\toperand\n\tmay contain a call\n\tto an open-coded primitive, so argument registers will have to be\n\tpreserved during\n\toperand\n\tevaluation.\n      \n\tFor each of the primitive\n\tprocedures=,*,\n\t-, and +,\n\twrite a code generator that takes\n\ta combination with that operator,\n\t  \n\ttogether with a target and a linkage descriptor, and\n\tproduces code to spread the arguments into the registers and then\n\tperform the operation targeted to the given target with the given\n\tlinkage.\n\tYou need only handle expressions with two operands.\n\tMake compile dispatch to these code\n\tgenerators.\n      \n\tTry your new compiler on the factorial\n\texample.  Compare the resulting code with the result produced without\n\topen coding.\n      \n\t    Extend your code generators for + and\n\t    * so that they\n\t    can handle expressions with arbitrary numbers of operands.  An\n\t    expression with more than two operands will have to be compiled into a\n\t    sequence of operations, each with only two inputs.\n\t  ","5.5.5#footnote-link-1":"1","5.5.5#footnote-link-2":"2","5.5.5#footnote-1":"We have used\n    the same symbol + here to denote both the\n    source-language\n    procedure\n    and the machine operation.  In general there will not be a\n    one-to-one correspondence between primitives of the source language\n    and primitives of the machine.","5.5.5#footnote-2":"Making\n\tthe primitives into reserved words is in general a bad idea, since a user\n\tcannot then rebind these names to different procedures.\n\tMoreover, if we add reserved words to\n\ta compiler that is in use, existing programs that define procedures\n\twith these names will stop working.  See\n\texercise 5.44 for ideas on how to avoid\n\tthis problem.","5.5.6":"5.5.6  \n    Lexical Addressing","5.5.6#p1":"\n    One of the most common optimizations performed by compilers is the\n    optimization of\n    variable\n    lookup. Our compiler, as we have implemented it so far, generates code that\n    uses the\n    lookup-variable-value\n    operation of the evaluator machine.\n    \n        This searches for a\n      variable\n    by comparing\n    it\n    with each\n    \n\tvariable\n      \n    that is currently bound, working frame\n    by frame outward through the runtime environment.  This search can be\n    expensive if the frames are deeply nested or if there are many\n    variables.\n    For example, consider the problem of looking up the value\n    of x while evaluating the expression\n    (* x y z)\n    in an application of the\n    procedure\n    that is returned by\n    (let ((x 3) (y 4))\n  (lambda (a b c d e)\n    (let ((y (* a b x))\n          (z (+ c d x)))\n      (* x y z)))) \n\tSince a let expression is just syntactic\n\tsugar for a lambda\n\tcombination, this expression is equivalent to\n\t((lambda (x y)\n   (lambda (a b c d e)\n     ((lambda (y z) (* x y z))\n      (* a b x)\n      (+ c d x))))\n  3\n  4) \n    Each time\n    lookup-variable-value\n    searches for x, it must determine that\n    the symbol\n    x is not\n\teq? to\n\ty or\n\tz (in the first frame), nor to\n\ta, b,\n\tc, d, or\n\te (in the second frame).\n      \n\tWe will assume, for the moment, that our programs do not use\n\tdefine—that variables are bound only\n\twith lambda.\n      \n    Because our language is\n    \n    lexically scoped, the runtime environment for any\n    \n        expression\n      \n    will have a\n    structure that parallels the lexical structure of the program in which\n    the\n    \n        expression\n      \n    appears.\n    Thus, the compiler can know, when it analyzes the\n    above expression,\n    that each time the\n    procedure\n    is applied the\n    \n\tvariable\n\tx\n    in\n    (* x y z)\n    will be found two frames out from the\n    current frame and will be the first\n    variable\n    in that frame.\n  ","5.5.6#footnote-link-1":"1","5.5.6#p2":"\n    We can exploit this fact by inventing a new kind of\n    variable-lookup\n    operation,\n    lexical-address-lookup,\n    that takes as arguments an environment and a\n    lexical address that\n    consists of two numbers: a frame number, which specifies how many\n    frames to pass over, and a displacement number, which specifies\n    how many\n    variables\n    to pass over in that frame.\n    Lexical-address-lookup\n    will produce the value of the\n    variable\n    stored at that lexical address\n    relative to the current environment.  If we add the\n    lexical-address-lookup\n    operation to our machine, we can make the compiler generate code that\n    references\n    variables\n    using this operation, rather than\n    lookup-variable-value.\n    Similarly, our compiled code can use a new\n    lexical-address-set!\n    operation instead of\n    set-variable-value!.","5.5.6#p3":"\n    In order to generate such code, the compiler must be able to determine\n    the lexical address of a\n    variable\n    it is about to compile a reference\n    to.  The lexical address of a\n    variable\n    in a program depends on where\n    one is in the code.  For example, in the following program, the\n    address of x in expression\n    $e_1$ is (2,0)—two frames back\n    and the first\n    variable\n    in the frame.  At that point\n    y is at\n    address (0,0) and c is at address (1,2).\n    In expression\n    $e_2$,\n    x is at (1,0),\n    y is at (1,1), and\n    c is at (0,2).\n    \n((lambda (x y)\n   (lambda (a b c d e)\n     ((lambda (y z) e1)\n        e2\n        (+ c d x))))\n 3\n 4)\n      ","5.5.6#p4":"\n    One way for the compiler to produce code that uses lexical addressing\n    is to maintain a data structure called a\n    compile-time environment.  This keeps track of which\n    variables\n    will be at which\n    positions in which frames in the runtime environment when a\n    particular\n    variable-access\n    operation is executed.  The compile-time\n    environment is a list of frames, each containing a list of\n    variables.\n\t(There will of course be no values bound to the\n\tvariables,\n\tsince values are not computed at compile time.)\n      \n    The compile-time\n    environment becomes an additional argument to\n    compile and is\n    passed along to each code generator.  The top-level call to\n    compile uses\n    \n\tan empty compile-time environment.\n      \n    When\n    \n\ta lambda body\n      \n    is compiled,\n    compile-lambda-body\n    extends the compile-time environment by a frame containing the\n    \n\tprocedure's\n      \n    parameters, so that the\n    sequence making up the\n    body is compiled with that extended environment.\n    \n    At each point in the compilation,\n    compile-variable\n    and\n    compile-assignment\n    use the compile-time\n    environment in order to generate the appropriate lexical addresses.\n  ","5.5.6#p5":"\n\tExercises 5.39\n\tthrough 5.43 describe how to\n\tcomplete this sketch of the lexical-addressing strategy in order to\n\tincorporate lexical lookup into the compiler.\n\tExercise 5.44 describes another use for the\n\tcompile-time environment.\n      ","5.5.6#ex-5.39":"\n    Write a\n    procedurelexical-address-lookup\n    that implements the new lookup operation.  It should take two\n    arguments—a lexical address and a runtime environment—and\n    return the value of the\n    variable\n    stored at the specified lexical address.\n    Lexical-address-lookup\n    should signal an error if the value\n    of the variable\n    is the\n    symbol *unassigned*.\n    Also write a\n    procedurelexical-address-set!\n    that implements the operation that changes the value\n    of the variable\n    at a specified lexical address.\n    ","5.5.6#footnote-link-2":"2","5.5.6#ex-5.40":"\n    Modify the compiler to maintain the\n    \n    compile-time environment as\n    described above.  That is, add a compile-time-environment argument to\n    compile and the various code generators, and\n    extend it in\n    compile-lambda-body.","5.5.6#ex-5.41":"\n    Write a\n    procedurefind-variable\n    that takes as arguments a\n    variable\n    and a\n    \n    compile-time environment and\n    returns the lexical address of the\n    variable\n    with respect to that\n    environment.  For example, in the program fragment that is shown above, the\n    compile-time environment during the compilation of expression\n    $e_1$ is\n    ((y z) (a b c d e) (x y)).\n      Find-variable\n    should produce\n    (find-variable 'c '((y z) (a b c d e) (x y))) (find-variable 'x '((y z) (a b c d e) (x y))) (find-variable 'w '((y z) (a b c d e) (x y))) ","5.5.6#ex-5.42":"\n    Using\n    find-variable\n    from exercise 5.41,\n    rewrite\n    compile-variable\n\tand\n\tcompile-assignment\n    to output lexical-address instructions.\n    \n\tIn cases where find-variable\n\treturns not-found\n\t(that is, where the variable is not in the\n\tcompile-time environment), you should have the code generators use the\n\tevaluator operations, as before, to search for the binding.\n\t(The only place a variable that is not found at compile time can be is in\n\tthe global environment, which is part of the runtime environment but\n\tis not part of the compile-time\n\tenvironment.\n\tThus, if you wish, you may have the evaluator operations looks directly\n\tin the global environment, which can be obtained with the operation\n\t(op get-global-environment),\n\tinstead of having them search the whole runtime\n\tenvironment found in env.)\n\t\n    Test the modified compiler on a few simple cases, such as the nested\n    lambda\n    combination at the beginning of this section.\n    ","5.5.6#footnote-link-3":"3","5.5.6#ex-5.43":"\n\tWe argued in section 4.1.6 that\n\tinternal definitions for block structure should not be considered\n\t\"real\"defines.  Rather, a\n\tprocedure\n\tbody should be interpreted as if the internal variables being defined\n\twere installed as ordinary lambda variables\n\tinitialized to their correct values using\n\tset!.\n\tSection 4.1.6 and\n\texercise 4.17 showed how to modify the\n\tmetacircular interpreter to accomplish this by\n\t\n\tscanning out internal\n\tdefinitions.  Modify the compiler to perform the same transformation\n\tbefore it compiles a\n\tprocedure\n\tbody.\n\t","5.5.6#ex-5.44":"\n\tIn this section we have focused on the use of the compile-time\n\tenvironment to produce lexical addresses.  But there are other uses\n\tfor compile-time environments.  For instance, in\n\texercise 5.38 we increased the efficiency of\n\tcompiled code by\n\t\n\topen-coding primitive\n\tprocedures. Our implementation treated the names of open-coded\n\tprocedures as reserved words.  If a program were to rebind such a name, the\n\tmechanism described in exercise 5.38 would still\n\topen-code it as a primitive, ignoring the new binding.  For example,\n\tconsider the\n\tprocedure\n\t(lambda (+ * a b x y)\n  (+ (* a x) (* b y))) \n\twhich computes a linear combination of x and\n\ty.  We might call it with arguments\n\t+matrix, *matrix,\n\tand four matrices, but the open-coding compiler would still open-code the\n\t+ and the * in\n\t(+ (* a x) (* b y))\n\tas primitive + and\n\t*.  Modify the open-coding compiler to consult\n\tthe compile-time environment in order to compile the correct code for\n\texpressions involving the names of primitive\n\tprocedures.\n\t(The code will work correctly as long as the program does not\n\tdefine\n\tor\n\tset!\n\tthese names.)\n\t","5.5.6#footnote-1":"This is not true if\n    we allow internal definitions, unless we scan them out.\n    See exercise 5.43.\n    ","5.5.6#footnote-2":"This\n      is the modification to variable lookup\n      required if we implement the\n      \n      scanning method to eliminate internal\n      definitions (exercise 5.43).\n      We will need to eliminate these definitions in order for lexical addressing\n      to work.","5.5.6#footnote-3":"Lexical addresses cannot be used to access\n\tvariables in the global environment, because these names can be defined\n\tand redefined interactively at any time.  With internal definitions\n\tscanned out, as\n\tin exercise 5.43,\n\tthe only definitions the\n\tcompiler sees are those at top level, which act on the global\n\tenvironment.  Compilation of a definition does not cause the defined\n\tname to be entered in the compile-time environment.","5.5.7":"5.5.7  \n    Interfacing Compiled Code to the Evaluator","5.5.7#p1":"\n    We have not yet explained how to load compiled code into the evaluator\n    machine or how to run it.  We will assume that the explicit-control-evaluator\n    machine has been defined as in\n    section 5.4.4, with the additional\n    operations specified in footnote 4 (section 5.5.2).\n    We will implement a\n    procedurecompile-and-go\n    that compiles a\n    Scheme expression,\n    loads the resulting object code into the evaluator machine,\n    and causes the machine to run the code in the\n    evaluator global environment, print the result, and\n    enter the evaluator's driver loop.  We will also modify the evaluator\n    so that interpreted\n    \n\texpressions\n      \n    can call compiled\n    procedures\n    as well as interpreted ones.  We can then put a compiled\n    procedure\n    into the machine and use the\n    evaluator to call it:\n    (compile-and-go\n '(define (factorial n)\n    (if (= n 1)\n        1\n        (* (factorial (- n 1)) n)))) (factorial 5)","5.5.7#p2":"\n    To allow the evaluator to handle compiled\n    procedures\n    (for example,\n    to evaluate the call to factorial above),\n    we need to change the code at\n    apply-dispatch\n    (section 5.4.2) so that it\n    recognizes compiled\n    procedures\n    (as distinct from compound or primitive\n    procedures)\n    and transfers control directly to the entry point of the\n    compiled code:apply-dispatch\n  (test (op primitive-procedure?) (reg proc))\n  (branch (label primitive-apply))\n  (test (op compound-procedure?) (reg proc))  \n  (branch (label compound-apply))\n  (test (op compiled-procedure?) (reg proc))  \n  (branch (label compiled-apply))\n  (goto (label unknown-procedure-type))\n\ncompiled-apply\n  (restore continue)\n  (assign val (op compiled-procedure-entry) (reg proc))\n  (goto (reg val))\n        Note the restore of continue at\n        compiled-apply.\n        Recall that the evaluator was arranged so that at\n        apply-dispatch,\n        the continuation would be at the top of the stack.  The compiled code entry\n        point, on the other hand, expects the continuation to be in\n        continue, so\n        continue must be\n        restored before the compiled code is executed.\n      ","5.5.7#footnote-link-1":"1","5.5.7#p3":"\n    To enable us to run some compiled code when we start the evaluator\n    machine, we add a branch instruction at\n    the beginning of the evaluator machine, which causes the machine to\n    go to a new entry point if the flag register \n    is set.\n  (branch (label external-entry))      ; branches if $\\texttt{flag}$ is set\nread-eval-print-loop\n  (perform (op initialize-stack))\n  $\\ldots$\n      External-entry\n    assumes that the machine is started with val\n    containing the location of an instruction sequence that puts a result into\n    val and ends with\n    (goto (reg continue)).\n    Starting at this entry point jumps to the location designated\n    by val, but first assigns\n    continue so that execution will return to\n    print-result,\n    which prints the value in val and then goes to\n    the beginning of the evaluator's\n    \n\tread-eval-print\n      \n    loop.external-entry\n  (perform (op initialize-stack))\n  (assign env (op get-global-environment))\n  (assign continue (label print-result))\n  (goto (reg val))","5.5.7#footnote-link-2":"2","5.5.7#footnote-link-3":"3","5.5.7#p4":"\n    Now we can use the following\n    procedure\n    to compile a\n    procedure definition,\n    execute the compiled code, and run the\n    \n\tread-eval-print\n      \n    loop so\n    we can try the\n    procedure.\n    Because we want the compiled code to\n    return\n    to the location in\n    continue with its result in\n    val, we compile the\n    expression\n    with a target of val and a\n    linkage of\n    return.\n      \n    In order to transform the\n    object code produced by the compiler into executable instructions\n    for the evaluator register machine, we use the\n    procedureassemble from the\n    register-machine simulator\n    (section 5.2.2).  \n    \nWe then initialize\n    the val register to point to the list\n    of instructions, set the\n    flag so that the evaluator will go to\n    external-entry,\n    and start the evaluator.\n    (define (compile-and-go expression)\n  (let ((instructions\n        (assemble (statements\n                   (compile expression 'val 'return))\n                  eceval)))\n    (set! the-global-environment (setup-environment))\n    (set-register-contents! eceval 'val instructions)\n    (set-register-contents! eceval 'flag true)\n    (start eceval))) ","5.5.7#p5":"\n    If we have set up\n    \n    stack monitoring, as at the end of\n    section 5.4.4, we can examine the\n    stack usage of compiled code:\n    (compile-and-go\n '(define (factorial n)\n    (if (= n 1)\n        1\n        (* (factorial (- n 1)) n))))(factorial 5)\n    Compare\n    \n    this example with the evaluation of\n    (factorial 5)\n    using the interpreted version of the same\n    procedure,\n    shown at the end of section 5.4.4.\n    \n        The interpreted version required 144 pushes and a maximum stack depth of 28.\n      \n    This illustrates the optimization that results from our compilation strategy.\n  ","5.5.7#h1":"Interpretation and compilation","5.5.7#p6":"\n    With the programs in this section, we can now experiment with the\n    alternative execution strategies of interpretation and\n    compilation.\n    An interpreter raises the machine to the level of the user program; a\n    compiler lowers the user program to the level of the machine language.\n    We can regard the \n    \n        Scheme\n      \n    language (or any programming language) as a\n    coherent family of abstractions erected on the machine language.\n    Interpreters are good for interactive program development and\n    debugging because the steps of program execution are organized in\n    terms of these abstractions, and are therefore more intelligible\n    to the programmer.\n    Compiled code can execute faster, because the steps of program execution\n    are organized in terms of the machine language, and the compiler is free\n    to make optimizations that cut across the higher-level\n    abstractions.","5.5.7#footnote-link-4":"4","5.5.7#footnote-link-5":"5","5.5.7#p7":"\n    The alternatives of interpretation and compilation also lead to\n    different strategies for\n    \n    porting languages to new computers. Suppose\n    that we wish to implement\n    Lisp\n    for a new machine.  One strategy is\n    to begin with the explicit-control evaluator of\n    section 5.4\n    and translate its instructions to instructions for the\n    new machine.  A different strategy is to begin with the compiler and\n    change the code generators so that they generate code for the new\n    machine.  The second strategy allows us to run any\n    Lisp\n    program on the new machine by first compiling it with the compiler running\n    on our\n    original Lisp system, and linking it with a compiled version of the runtime\n    library.  Better yet, we can compile the compiler itself, and run\n    this on the new machine to compile other\n    Lisp programs.  Or we can compile one of the interpreters of\n    section 4.1 to produce an interpreter that\n    runs on the new machine.\n  ","5.5.7#footnote-link-6":"6","5.5.7#footnote-link-7":"7","5.5.7#ex-5.45":"\n    By\n    \n    comparing the stack operations used by compiled code to the stack\n    operations used by the evaluator for the same computation, we can\n    determine the extent to which the compiler optimizes use of the stack,\n    both in speed (reducing the total number of stack operations) and in\n    space (reducing the maximum stack depth).  Comparing this optimized\n    stack use to the performance of a special-purpose machine for the same\n    computation gives some indication of the quality of the compiler.\n    \n\tExercise 5.27 asked you to determine, as a\n\tfunction of $n$, the number of pushes and\n\tthe maximum stack depth needed by the evaluator to compute\n\t$n!$ using the recursive factorial\n\tprocedure\n\tgiven above.  Exercise 5.14 asked you\n\tto do the same measurements for the special-purpose factorial machine\n\tshown in figure 5.11. Now perform the\n\tsame analysis using the compiled factorialprocedure.\n\tCan you suggest improvements to the compiler that would help it\n\tgenerate code that would come closer in performance to the\n\thand-tailored version?\n      ","5.5.7#p8":"\n\t  Take the ratio of the number of pushes in the compiled version to the\n\t  number of pushes in the interpreted version, and do the same for the\n\t  maximum stack depth.  Since the number of operations and the stack\n\t  depth used to compute $n!$ are linear in\n\t  $n$, these ratios should\n\t  approach constants as $n$ becomes large.\n\t  What are these constants? Similarly, find the ratios of the stack usage\n\t  in the special-purpose machine to the usage in the interpreted version.\n\t","5.5.7#p9":"\n\tCompare the ratios for special-purpose versus interpreted code to the\n\tratios for compiled versus interpreted code.  You should find that the\n\tspecial-purpose machine is much more efficient than the compiled code, since\n\tthe hand-tailored controller code should be much better than what is\n\tproduced by our rudimentary general-purpose compiler.\n\t","5.5.7#ex-5.46":"\n    Carry out an analysis like the one in\n    exercise 5.45 to determine the\n    effectiveness of compiling the tree-recursive\n    \n    Fibonacci\n    procedure(define (fib n)\n  (if (< n 2)\n      n\n      (+ (fib (- n 1)) (fib (- n 2))))) \n    compared to the effectiveness of using the special-purpose Fibonacci machine\n    of figure 5.12.  (For measurement of the\n    interpreted performance, see exercise 5.29.)\n    For Fibonacci, the time resource used is not linear in\n    $n$; hence the ratios of stack operations will not\n    approach a limiting value that is independent of\n    $n$.\n    ","5.5.7#ex-5.47":"\n    This section described how to modify the explicit-control evaluator so\n    that interpreted code can call compiled\n    procedures.\n    Show how to modify the compiler so that compiled\n    procedures\n    can call not only primitive\n    procedures\n    and compiled\n    procedures,\n    but interpreted\n    procedures\n    as well.  This requires modifying\n    compile-procedure-call\n    to handle the case of compound (interpreted)\n    procedures.\n    Be sure to handle all the same target and\n    linkage combinations\n    as in\n    compile-proc-appl.\n      \n    To do the actual\n    procedure\n    application,\n    the code needs to jump to the evaluator's\n    compound-apply\n    entry point. This label cannot be directly referenced in object code\n    (since the assembler requires that all labels referenced by the\n    code it is assembling be defined there), so we will add a register\n    called compapp to the evaluator machine to\n    hold this entry point, and add an instruction to initialize it:\n    \n  (assign compapp (label compound-apply))\n  (branch (label external-entry))      ; branches if $\\texttt{flag}$ is set\nread-eval-print-loop\n  $\\ldots$\n      \n    To test your code, start by\n    \n\tdefining\n      \n    a\n    \n\tprocedure\n      f that calls a\n    \n\tprocedure\n      g.  Use\n    compile-and-go\n    to compile the\n    \n\tdefinition\n      \n    of f\n    and start the evaluator.  Now, typing at the evaluator,\n    \n\tdefine\n\tg and try to call\n\tf.\n      ","5.5.7#ex-5.48":"\n    The\n    compile-and-go\n    interface implemented in this section is\n    awkward, since the compiler can be called only once (when the\n    evaluator machine is started).  Augment the compiler–interpreter \n    interface by providing a\n    compile-and-run\n    primitive that can be called from within the explicit-control evaluator\n    as follows:\n    (compile-and-run\n '(define (factorial n)\n    (if (= n 1)\n        1\n        (* (factorial (- n 1)) n))))(factorial 5) ","5.5.7#ex-5.49":"\n    As an alternative to using the explicit-control evaluator's\n    \n\tread-eval-print\n      \n    loop, design a register machine that performs a\n    read-compile-execute-print loop.  That is, the machine should run a\n    loop that reads\n    \n\tan expression,\n      \n    compiles it, assembles and\n    executes the resulting code, and prints the result.  This is easy to\n    run in our simulated setup, since we can arrange to call the\n    procedurescompile and\n    assemble as \"register-machine\n    operations.\"","5.5.7#ex-5.50":"\n    Use the compiler to compile the\n    \n    metacircular evaluator of\n    section 4.1 and run this program using the\n    register-machine simulator.\n    \n\t      (To compile more than one definition at a time,\n\t      you can package the definitions in a\n\t      begin.)\n      \n    The resulting interpreter will run very slowly because of the multiple\n    levels of interpretation, but getting all the details to work is an\n    instructive exercise.\n    ","5.5.7#ex-5.51":"\n    Develop a rudimentary implementation of\n    Scheme\n    in\n    \n    C (or some other low-level language of your choice) by translating the\n    explicit-control evaluator of section 5.4\n    into C.  In order to run this code you will need to also\n    provide appropriate storage-allocation routines and other runtime\n    support.\n    ","5.5.7#ex-5.52":"\n    As a counterpoint to exercise 5.51, modify\n    the compiler so that it compiles \n    Scheme procedures\n    into sequences of\n    \n    C instructions.  Compile the metacircular evaluator of\n    section 4.1 to produce a\n    Scheme  \n    interpreter written in C.\n    ","5.5.7#footnote-1":"Of course, compiled\n    procedures\n    as well as interpreted\n    procedures\n    are compound (nonprimitive).  For compatibility with the terminology used\n    in the explicit-control evaluator, in this section we will use\n    \"compound\" to mean interpreted (as opposed to\n    compiled).","5.5.7#footnote-2":"Now that the evaluator machine starts\n    with a branch, we must always initialize the\n    flag register before starting the evaluator\n    machine.  To start the machine at its ordinary\n    \n\tread-eval-print\n      \n    loop, we\n    could use\n    (define (start-eceval)\n  (set! the-global-environment (setup-environment))\n  (set-register-contents! eceval 'flag false)\n  (start eceval)) ","5.5.7#footnote-3":"Since\n    a compiled\n    procedure\n    is an object that the system may try to print, we also modify the system\n    print operation\n    user-print\n    (from section 4.1.4) so that it will not\n    attempt to print the components of a compiled\n    procedure:(define (user-print object)\n        (cond ((compound-procedure? object)\n              (display (list 'compound-procedure\n              (procedure-parameters object)\n              (procedure-body object)\n              '<procedure-env>)))\n              ((compiled-procedure? object)\n              (display '<compiled-procedure>))\n              (else (display object))))","5.5.7#footnote-4":"We can do even better by extending the compiler\n    to allow compiled code to call interpreted\n    procedures.\n    See exercise 5.47.","5.5.7#footnote-5":"Independent of the strategy of execution, we\n    incur significant overhead if we insist that\n    \n    errors encountered in\n    execution of a user program be detected and signaled, rather than being\n    allowed to kill the system or produce wrong answers.  For example, an\n    out-of-bounds array reference can be detected by checking the validity\n    of the reference before performing it.  The overhead of checking,\n    however, can be many times the cost of the array reference itself, and\n    a programmer should weigh speed against safety in determining whether\n    such a check is desirable.  A good compiler should be able to produce\n    code with such checks, should avoid redundant checks, and should allow\n    programmers to control the extent and type of error checking in the\n    compiled code.\n    \n      Compilers for popular languages, such as\n      \n      C and C++,\n      put hardly any error-checking operations into\n      running code, so as to make things run as fast as possible.  As a\n      result, it falls to programmers to explicitly provide error checking.\n      Unfortunately, people often neglect to do this, even in\n      critical applications where speed is not a constraint.  Their programs\n      lead fast and dangerous lives.  For example, the notorious \n      \"Worm\"\n      that paralyzed the Internet in 1988 exploited the \n      \n      UNIX$^{\\textrm{TM}}$\n      operating system's failure to check whether the input buffer has\n      overflowed in the finger daemon. (See\n      Spafford 1989.)\n  ","5.5.7#footnote-6":"Of course, with either the interpretation or the\n    compilation strategy we must also implement for the new machine storage\n    allocation, input and output, and all the various operations that we took\n    as \"primitive\" in our discussion of\n    the evaluator and compiler.  One strategy for minimizing work here is\n    to write as many of these operations as possible in\n    Lisp\n    and then compile them for the new machine.  Ultimately, everything reduces\n    to a small kernel (such as garbage collection and the mechanism for\n    applying actual machine primitives) that is hand-coded for the new\n    machine.","5.5.7#footnote-7":"\n    This strategy leads to amusing tests of correctness of\n    the compiler, such as checking\n    whether the compilation of a program on the new machine, using the\n    compiled compiler, is identical with the\n    compilation of the program on the original\n    Lisp\n    system.  Tracking down the source of differences is fun but often\n    frustrating, because the results are extremely sensitive to minuscule\n    details.","references":"References\n    Abelson, Harold, Andrew Berlin, Jacob Katzenelson,\n    William McAllister,\n    Guillermo Rozas, Gerald Jay Sussman, and Jack Wisdom. 1992.  The\n    Supercomputer Toolkit: A general framework for special-purpose\n    computing.  International Journal of High-Speed Electronics\n    3(3):337–361.\n    Allen, John.  1978.  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Artificial Intelligence.  3rd edition.\n    Reading, MA: Addison-Wesley.Zabih, Ramin, David McAllester, and David Chapman.  1987.\n    Non-deterministic Lisp with dependency-directed backtracking.\n    AAAI-87, pp. 59–64.Zippel, Richard.  1979.  Probabilistic algorithms for sparse\n    polynomials.  Ph.D. dissertation, Department of Electrical Engineering\n    and Computer Science, MIT.Zippel, Richard.  1993.  Effective Polynomial Computation.\n    Boston, MA: Kluwer Academic Publishers.","making-of":"Background","making-of#h1":"Background","making-of#p1":"\n      The JavaScript adaptation of SICP is an open-source community effort.\n      The software and data required for making these web pages and the PDF edition are contained\n      in the github repository\n      source-academy / sicp,\n      and improvements, extensions and discussions are handled in this repository using git.\n    ","making-of#p2":"\n      Martin Henz started translating SICP to JavaScript in 2008. He obtained the original\n      $\\LaTeX$ sources of the second edition from Gerald Jay Sussman, and converted them to\n      a custom-built XML format. The original sources are retained in the XML format, which\n      allows for generating the\n      comparison edition.\n      A processing system written in XSLT resulted in the\n      first version of the JavaScript adaptation around 2009, covering the first few sections of SICP.\n      The content of SICP JS contained in the\n      XML files are undergoing continuous improvement by the adapters Martin Henz and Tobias\n      Wrigstad, and by the community of SICP JS readers, using the github repository.\n    ","making-of#p3":"\n      In the book, program fragments often require other program fragments.\n      In order to collect and execute the necessary programs, the corresponding\n      SNIPPET tags in the xml files include REQUIRES tags. The \n      XML processors use these tags in order to assemble the executable programs.\n      The project thus can be seen as a literate programming system, custom-made\n      for authoring SICP JS.\n    ","making-of#h2":"Interactive SICP JS","making-of#p4":"Interactive SICP JS\n      was designed and implemented by Samuel Fang\n      in 2021. The XML textbook sources are translated to a JSON format, which\n      are then read and rendered by a dedicated component of the Source Academy.\n    ","making-of#h3":"Comparison Edition","making-of#p5":"\n      The precursor of the comparison edition was the \n      mobile-friendly web edition\n      of SICP JS, designed and implemented by Liu Hang\n      in 2017. Feng Piaopiao improved the system in 2018.\n      He Xinyue and Wang Qian developed the\n      comparison edition\n      in 2020.\n      Formulas are retained in the resulting HTML files\n      and are typeset by the reader's browser on the fly,\n      using the MathJax system.\n      The comparison edition is maintained by Martin Henz.\n    ","making-of#h4":"PDF Edition","making-of#p6":"\n      The early PDF editions (2010-2018) used XSLT for generating $\\LaTeX$ from\n      the XML sources.\n      The first Node.js version of the PDF edition\n      was designed and implemented by Chan Ger Hean in 2019.\n      Tobias Wrigstad and Martin Henz are the main developers of this system.\n    ","making-of#h5":"Figures","making-of#p7":"\n      Most figures are adapted from the\n      HTML5/EPUB3 version of SICP\n      by Andres Raba. The figures are licensed under Creative Commons\n      Attribution-ShareAlike 4.0 International\n      License (cc by-sa).\n      JavaScript adaptations\n      of figures were done by Tobias Wrigstad using Inkscape and gratuitous use of\n      sed. \n    "}}