10

Exercise 1.10: The following procedure computes a mathematical function called Ackermann’s function.

(define (A x y)
  (cond ((= y 0) 0)
        ((= x 0) (* 2 y))
        ((= y 1) 2)
        (else (A (- x 1)
                 (A x (- y 1))))))

What are the values of the following expressions?

(A 1 10)

The steps under the hood:

(A 1 10)

(A 2 4)

The steps under the hood:

(A 2 4)

(A 3 3)

The steps under the hood:

(A 3 3)

Consider the following procedures, where A is the procedure defined above:

(define (f n) (A 0 n))

(define (g n) (A 1 n))

(define (h n) (A 2 n))

(define (k n) (* 5 n n))

Give concise mathematical definitions for the functions computed by the procedures f, g, and h for positive integer values of n. For example, (k n) computes $| 5n^2 |$.

$$(f\;n) = 2n$$

$$(g\;n) = 2^n$$

$$(h\;1) = h(1) = 2$$

$$(h\;2) = h(2) = 2^{2}$$

$$(h\;3) = h(3) = 2^{2^{2}}$$

$$...$$

$$(h\;n) = h(n) = h(n-1)$$