` Sqrt` is our first example of a process defined by a set of
mutually defined procedures. Notice that the definition of `
sqrt-iter` is * recursive*; that is, the procedure is
defined in terms of itself. The idea of being able to define a
procedure 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. The entire ` sqrt`
program can be viewed as a cluster of procedures (shown in figure 1.2) 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 accomplishes an identifiable task that can be used
as a module in defining other procedures. For example, when we define
the ` good-enough?` procedure in terms of ` square`, we
are able to regard the ` square` procedure as a "black box. We
are not at that moment concerned with * how* the procedure
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 concerned, ` square` is not
quite a procedure but rather an abstraction of a procedure, a
so-called * procedural abstraction*. At this level of
abstraction, any procedure that computes the square is equally good.

**Figure 1.2** Procedural decomposition of the `sqrt` program.

Thus, considering only the values they return, the following two
procedures for squaring a number should be indistinguishable. Each
takes a numerical argument and produces the square of that number as
the value.^{25}

(define (square x) ( x x)) (define (square x) (exp (double (log x)))) (define (double x) (+ x x))

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

**Local names**

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

This principle that the meaning of a procedure 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 must be local to the body of the procedure. For example, we used(define (square x) ( x x)) (define (square y) ( y y))

The intention of the author of(define (good-enough? guess x) (< (abs (-(square guess) x)) 0.001))

If the parameters were not local to the bodies of their respective
procedures, then the parameter ` x` in ` square` could
be confused with the parameter ` x` in ` good-enough?`,
and the behavior of ` 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 has a very special role in the
procedure 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 * binds* its formal
parameters. The meaning of a procedure definition is unchanged if a
bound variable is consistently renamed throughout the definition. ^{26}

If a variable is not bound, we say that it is * free*. The set
of expressions for which a binding defines a name is called the *
scope* of that name. In a procedure definition, the bound
variables declared as the formal parameters of the procedure have the
body of the procedure as their scope.

In the definition of ` good-enough?` above, ` guess` and
` x` are bound variables but ` <;`, ` -`,
` abs`, and ` square` are free. The meaning of `
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 not independent of the names of its free
variables, however. It surely depends upon the fact (external to this
definition) that the symbol ` abs` names a procedure for
computing the absolute value of a number. ` Good-enough?` will
compute a different function if we substitute ` cos` for `
abs` in its definition.

**Internal definitions and block structure**

We have one kind of name isolation available to us so far: The formal parameters of a procedure are local to the body of the procedure. 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:

(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 (- (squre guess) x)) 0.001)) (define (improve guess x) (average guess (/ x guess)))

The problem with this program is that the only procedure that is
important to users of ` sqrt` is ` sqrt`. The other
procedures (` sqrt-iter`, ` good-enough?`, and `
improve`) only clutter up their minds. They may not define any
other procedure called ` 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, many numerical functions are computed as successive
approximations and thus might have procedures named `
good-enough?` and ` improve` as auxiliary procedures. We
would like to localize the subprocedures, hiding them inside `
sqrt` so that ` sqrt` could coexist with other successive
approximations, each having its own private ` good-enough?`
procedure. To make this possible, we allow a procedure to have
internal definitions that are local to that procedure. For example,
in the square-root problem we can write

(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))

Such nesting of 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, we can simplify them.
Since ` x` is bound in the definition of ` sqrt`, the
procedures ` good-enough?`, ` improve`, and `
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. Instead, we allow `
x` to be a free variable in the internal definitions, as shown
below. Then ` x` gets its value from the argument with which
the enclosing procedure ` sqrt` is called. This discipline is
called lexical scoping * lexical scoping*. ^{27}

We will use block structure extensively to help us break up large
programs into tractable pieces. ^{28}
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.