Applying Simple Procedures

When we introduced the substitution model in section  we showed how the combination (f 5) evaluates to 136, given the following procedure definitions:

(define (square x)
  (* x x))

(define (sum-of-squares x y)
  (+ (square x) (square y)))

(define (f a)
  (sum-of-squares (+ a 1) (* a 2)))

We can analyze the same example using the environment model. Figure  shows the three procedure objects created by evaluating the definitions of f, square, and sum-of-squares in the global environment. Each procedure object consists of some code, together with a pointer to the global environment.

 

In figure  we see the environment structure created by evaluating the expression (f 5). The call to f creates a new environment E1 beginning with a frame in which a, the formal parameter of f, is bound to the argument 5. In E1, we evaluate the body of f:

(sum-of-squares (+ a 1) (* a 2))

 

To evaluate this combination, we first evaluate the subexpressions. The first subexpression, sum-of-squares, has a value that is a procedure object. (Notice how this value is found: We first look in the first frame of E1, which contains no binding for sum-of-squares. Then we proceed to the enclosing environment, i.e. the global environment, and find the binding shown in figure .) The other two subexpressions are evaluated by applying the primitive operations + and * to evaluate the two combinations (+ a 1) and (* a 2) to obtain 6 and 10, respectively.

Now we apply the procedure object sum-of-squares to the arguments 6 and 10. This results in a new environment E2 in which the formal parameters x and y are bound to the arguments. Within E2 we evaluate the combination (+ (square x) (square y)). This leads us to evaluate (square x), where square is found in the global frame and x is 6. Once again, we set up a new environment, E3, in which x is bound to 6, and within this we evaluate the body of square, which is (* x x). Also as part of applying sum-of-squares, we must evaluate the subexpression (square y), where y is 10. This second call to square creates another environment, E4, in which x, the formal parameter of square, is bound to 10. And within E4 we must evaluate (* x x).

The important point to observe is that each call to square creates a new environment containing a binding for x. We can see here how the different frames serve to keep separate the different local variables all named x. Notice that each frame created by square points to the global environment, since this is the environment indicated by the square procedure object.

After the subexpressions are evaluated, the results are returned. The values generated by the two calls to square are added by sum-ofsquares, and this result is returned by f. Since our focus here is on the environment structures, we will not dwell on how these returned values are passed from call to call; however, this is also an important aspect of the evaluation process, and we will return to it in detail in chapter 5.

Exercise. In section  we used the substitution model to analyze two procedures for computing factorials, a recursive version

(define (factorial n)
  (if (= n 1)
      1
      (* n (factorial (- n 1)))))

and an iterative version

(define (factorial n)
  (fact-iter 1 1 n))

(define (fact-iter product counter max-count)
  (if (> counter max-count)
      product
      (fact-iter (* counter product)
                 (+ counter 1)
                 max-count)))

Show the environment structures created by evaluating (factorial 6) using each version of the factorial procedure.