Sqrt The function sqrt is our first example of a process defined by a set of mutually defined procedures. defined functions. Notice that the definition of sqrt-iter definition of sqrt_iter is recursive; that is, the procedure function is defined in terms of itself. The idea of being able to define a procedure function in terms of itself may be disturbing; it may seem unclear how such a circular definition could make sense at all, much less specify a well-defined process to be carried out by a computer. This will be addressed more carefully in section 1.2. But first let's consider some other important points illustrated by the sqrt example.

Observe that the problem of computing square roots breaks up naturally into a number of subproblems: how to tell whether a guess is good enough, how to improve a guess, and so on. Each of these tasks is accomplished by a separate procedure. function. The entire sqrt program can be viewed as a cluster of procedures (shown in figure 1.3) functions (shown in figure 1.4) that mirrors the decomposition of the problem into subproblems.

The importance of this decomposition strategy is not simply that one is dividing the program into parts. After all, we could take any large program and divide it into parts—the first ten lines, the next ten lines, the next ten lines, and so on. Rather, it is crucial that each procedure function accomplishes an identifiable task that can be used as a module in defining other procedures. functions. For example, when we define the good-enough? procedure is_good_enough function in terms of square, we are able to regard the square procedure function as a black box. We are not at that moment concerned with how the procedure function computes its result, only with the fact that it computes the square. The details of how the square is computed can be suppressed, to be considered at a later time. Indeed, as far as the good-enough? procedure is_good_enough function is concerned, square is not quite a procedure function but rather an abstraction of a procedure, function, a so-called procedural abstraction. functional abstraction. At this level of abstraction, any procedure function that computes the square is equally good.

Thus, considering only the values they return, the following two procedures functions squaring a number should be indistinguishable. Each takes a numerical argument and produces the square of that number as the value.[1]

Original	Python
(define (square x) (* x x))	def square(x): return x * x

Original	Python
(define (square x) (exp (double (log x)))) (define (double x) (+ x x))	def square(x): return math_exp(double(math_log(x))) def double(x): return x + x

So a procedure function should be able to suppress detail. The users of the procedure function may not have written the procedure function themselves, but may have obtained it from another programmer as a black box. A user should not need to know how the procedure function is implemented in order to use it.

Local names

One detail of a procedure's function's implementation that should not matter to the user of the procedure function is the implementer's choice of names for the procedure's formal parameters. function's parameters. Thus, the following procedures functions should not be distinguishable:

Original	Python
(define (square x) (* x x))	def square(x): return x * x

Original	Python
(define (square y) (* y y))	def square(y): return y * y

This principle—that the meaning of a procedure function should be independent of the parameter names used by its author—seems on the surface to be self-evident, but its consequences are profound. The simplest consequence is that the parameter names of a procedure function must be local to the body of the procedure. function. For example, we used square in the definition of good-enough? in the definition of is_good_enough in our square-root procedure: function:

Original	Python
(define (good-enough? guess x) (< (abs (- (square guess) x)) 0.001))	def is_good_enough(guess, x): return abs(square(guess) - x) < 0.001

The intention of the author of good-enough? is_good_enough is to determine if the square of the first argument is within a given tolerance of the second argument. We see that the author of good-enough? is_good_enough used the name guess to refer to the first argument and x to refer to the second argument. The argument of square is guess. If the author of square used x (as above) to refer to that argument, we see that the x in good-enough? is_good_enough must be a different x than the one in square. Running the procedure function square must not affect the value of x that is used by good-enough?, is_good_enough, because that value of x may be needed by good-enough? is_good_enough after square is done computing.

If the parameters were not local to the bodies of their respective procedures, functions, then the parameter x in square could be confused with the parameter x in good-enough?, is_good_enough, and the behavior of good-enough? is_good_enough would depend upon which version of square we used. Thus, square would not be the black box we desired.

A formal parameter of a procedure parameter of a function has a very special role in the procedure definition, function definition, in that it doesn't matter what name the formal parameter has. Such a name is called a bound variable, and we say that the procedure definition bound, and we say that the function definition binds its formal parameters. parameters. The meaning of a procedure definition is unchanged if a bound variable function definition is unchanged if a bound name is consistently renamed throughout the definitiondefinition.[2] If a variable name is not bound, we say that it is free. The set of expressions statements for which a binding defines defines a name is called the scope of that name. In a procedure definition, the bound variables function definition, the bound names declared as the formal parameters of the procedure parameters of the function have the body of the procedure function as their scope.

In the definition of good-enough? definition of is_good_enough above, guess and x are bound variables but <, -, abs and square are free. The meaning of good-enough? is_good_enough should be independent of the names we choose for guess and x so long as they are distinct and different from <, -, abs and square. (If we renamed guess to abs we would have introduced a bug by capturing the variable abs. It would have changed from free to bound.) The meaning of good-enough? is_good_enough is not independent of the names of its free variables, however. It surely depends upon the fact (external to this definition) (external to this definition) that the symbol abs names a procedure that the name abs refers to a function for computing the absolute value of a number. Good-enough? The function is_good_enough will compute a different function if we substitute cos math_cos (the primitive cosine function) for abs in its definition. definition.

Internal definitions definitions and block structure

We have one kind of name isolation available to us so far: The formal parameters of a procedure The parameters of a function are local to the body of the procedure. function. The square-root program illustrates another way in which we would like to control the use of names. The existing program consists of separate procedures: functions:

Original	Python
(define (sqrt x) (sqrt-iter 1.0 x)) (define (sqrt-iter guess x) (if (good-enough? guess x) guess (sqrt-iter (improve guess x) x))) (define (good-enough? guess x) (< (abs (- (square guess) x)) 0.001)) (define (improve guess x) (average guess (/ x guess)))	def sqrt(x): return sqrt_iter(1, x) def sqrt_iter(guess, x): return (guess if is_good_enough(guess, x) else sqrt_iter(improve(guess, x), x)) def is_good_enough(guess, x): return abs(square(guess) - x) < 0.001 def improve(guess, x): return average(guess, x / guess)

The problem with this program is that the only procedure function that is important to users of sqrt is sqrt. The other procedures functions (sqrt-iter, good-enough?, (sqrt_iter, is_good_enough, and improve) only clutter up their minds. They may not define any other procedure define any other function called good-enough? is_good_enough as part of another program to work together with the square-root program, because sqrt needs it. The problem is especially severe in the construction of large systems by many separate programmers. For example, in the construction of a large library of numerical procedures, functions, many numerical functions are computed as successive approximations and thus might have procedures functions named good-enough? is_good_enough and improve as auxiliary procedures. functions. We would like to localize the subprocedures, subfunctions, hiding them inside sqrt so that sqrt could coexist with other successive approximations, each having its own private good-enough? procedure. is_good_enough function. To make this possible, we allow a procedure function to have internal definitions that are local to that procedure. function. For example, in the square-root problem we can write

Original	Python
(define (sqrt x) (define (good-enough? guess x) (< (abs (- (square guess) x)) 0.001)) (define (improve guess x) (average guess (/ x guess))) (define (sqrt-iter guess x) (if (good-enough? guess x) guess (sqrt-iter (improve guess x) x))) (sqrt-iter 1.0 x))	def sqrt(x): def is_good_enough(guess, x): return abs(square(guess) - x) < 0.001 def improve(guess, x): return average(guess, x / guess) def sqrt_iter(guess, x): return (guess if is_good_enough(guess, x) else sqrt_iter(improve(guess, x), x)) return sqrt_iter(1, x)

The body of a function definition forms a block; definitions inside it are local to the function. Such nesting of definitions, definitions, called block structure, is basically the right solution to the simplest name-packaging problem. But there is a better idea lurking here. In addition to internalizing the definitions of the auxiliary procedures, definitions of the auxiliary functions, we can simplify them. Since x is bound in the definition definition of sqrt, the procedures functions good-enough?, is_good_enough, improve, and sqrt-iter, which are defined internally to sqrt_iter, which are defined internally to sqrt, are in the scope of x. Thus, it is not necessary to pass x explicitly to each of these procedures. functions. Instead, we allow x to be a free variable name in the internal definitions, definitions, as shown below. Then x gets its value from the argument with which the enclosing procedure function sqrt is called. This discipline is called lexical scoping.[3]

Original	Python
(define (sqrt x) (define (good-enough? guess) (< (abs (- (square guess) x)) 0.001)) (define (improve guess) (average guess (/ x guess))) (define (sqrt-iter guess) (if (good-enough? guess) guess (sqrt-iter (improve guess)))) (sqrt-iter 1.0))	def sqrt(x): def is_good_enough(guess): return abs(square(guess) - x) < 0.001 def improve(guess): return average(guess, x / guess) def sqrt_iter(guess): return (guess if is_good_enough(guess) else sqrt_iter(improve(guess))) return sqrt_iter(1)

We will use block structure extensively to help us break up large programs into tractable pieces.[4] The idea of block structure originated with the programming language Algol 60. It appears in most advanced programming languages and is an important tool for helping to organize the construction of large programs.