SICP Exercises 1.9–1.10

exercise 1.9 Each of the following two procedures defines a method for adding two positive integers in terms of the procedures inc, which increments its argument by 1, and dec, which decrements its argument by 1.

(define (+ a b) (if (= a 0) b (inc (+ (dec a) b)))) (define (+ a b) (if (= a 0) b (+ (dec a) (inc b))))

Using the substitution model, illustrate the process generated by each procedure in evaluating (+ 4 5). Are these processes iterative or recursive?

The first is iterative

(+ 4 5) (inc (+ (dec 4) 5)) (inc (+ 3 5)) (inc (inc (+ (dec 3) 5))) (inc (inc (+ 2 5))) (inc (inc (inc (+ (dec 2) 5)))) (inc (inc (inc (+ 1 5)))) (inc (inc (inc (inc (+ (dec 1) 5))))) (inc (inc (inc (inc (+ 0 5))))) (inc (inc (inc (inc 5)))) (inc (inc (inc 6))) (inc (inc 7)) (inc 8) 9

The second is recursive

(+ 4 5) (+ (dec 4) (inc 5)) (+ 3 6) (+ (dec 3) (inc 6)) (+ 2 7) (+ (dec 2) (inc 7)) (+ 1 8) (+ (dec 1) (inc 8)) (+ 0 9) 9

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) (A 2 4) (A 3 3)

1 ]=> (A 1 10) ;Value: 1024 1 ]=> (A 2 4) ;Value: 65536 1 ]=> (A 3 3) ;Value: 65536

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 5n².

f(n) = 2n g(n) = 2ⁿ h(n) = (2ⁿ)ⁿ