The function sqrt is our first example of a process defined by a set of mutually defined functions. Notice that the declaration of sqrt_iter is recursive; that is, the function is defined in terms of itself. The idea of being able to define a 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 function. The entire sqrt program can be viewed as a cluster of 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 function accomplishes an identifiable task that can be used as a module in defining other functions.

For example, when we define the is_good_enough function in terms of square, we are able to regard the square function as a black box. We are not at that moment concerned with how the 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 is_good_enough function is concerned, square is not quite a function but rather an abstraction of a function, a so-called functional abstraction. At this level of abstraction, any function that computes the square is equally good.

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

function square(x) {
    return x * x;
}

function square(x) {
    return math_exp(double(math_log(x)));
}
function double(x) {
    return x + x;
}

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

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

function square(x) {
    return x * x;
}

function square(y) {
    return y * y;
}

This principle—that the meaning of a 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 function must be local to the body of the function. For example, we used square in the declaration of is_good_enough in our square-root function:

function is_good_enough(guess, x) {
    return abs(square(guess) - x) < 0.001;
}

The intention of the author of 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 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 is_good_enough must be a different x than the one in square. Running the function square must not affect the value of x that is used by is_good_enough, because that value of x may be needed by is_good_enough after square is done computing.

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

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

In the declaration of is_good_enough above, guess and x are bound names but abs and square are free. The meaning of 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 name abs. It would have changed from free to bound.) The meaning of is_good_enough is not independent of the choice of its free names, however. It surely depends upon the fact (external to this declaration) that the name abs refers to a function for computing the absolute value of a number. The function is_good_enough will compute a different function if we substitute math_cos (the primitive cosine function) for abs in its declaration.

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

function sqrt(x) {
    return sqrt_iter(1, x);
}
function sqrt_iter(guess, x) {
    return is_good_enough(guess, x)
           ? guess
           : sqrt_iter(improve(guess, x), x);
}
function is_good_enough(guess, x) {
    return abs(square(guess) - x) < 0.001;
}
function improve(guess, x) {
    return average(guess, x / guess);
}

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

function sqrt(x) {
    function is_good_enough(guess, x) {
        return abs(square(guess) - x) < 0.001;
    }
    function improve(guess, x) {
        return average(guess, x / guess);
    }
    function sqrt_iter(guess, x) {
        return is_good_enough(guess, x) 
               ? guess
               : sqrt_iter(improve(guess, x), x);
    }
    return sqrt_iter(1, x);
}

Any matching pair of braces designates a block, and declarations inside the block are local to the block. Such nesting of declarations, 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 declarations of the auxiliary functions, we can simplify them. Since x is bound in the declaration of sqrt, the functions is_good_enough, improve, and sqrt_iter, which are declared internally to sqrt, are in the scope of x. Thus, it is not necessary to pass x explicitly to each of these functions. Instead, we allow x to be a free name in the internal declarations, as shown below. Then x gets its value from the argument with which the enclosing function sqrt is called. This discipline is called lexical scoping.

function sqrt(x) {
    function is_good_enough(guess) {
        return abs(square(guess) - x) < 0.001;
    }
    function improve(guess) {
        return average(guess, x / guess);
    }
    function sqrt_iter(guess) {
        return is_good_enough(guess)
               ? guess
               : sqrt_iter(improve(guess));
    }
    return sqrt_iter(1);
}

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

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.