# Exercise 1.10 of SICP

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)
1024
> (A 2 4)
65536
> (A 3 3)
65536


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

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

$$f(n) = 2n$$

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

(g 3)
(A 1 3)
(A 0 (A 1 2))
(* 2 (A 0 (A 1 1)))
(* 2 (* 2 2)
8


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

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

(h 3)
(A 2 3)
(A 1 (A 2 2))


From the previous result I know that (h 3) will look like $$2^{A(2,2)}$$

(A 2 2)
(A 1 (A 2 1))


From the previous result I know that (h 2) will look like $$2^{A(2,1)}$$

$$h(3) = 2^{2^{A(2,1)}} = 2^{2^2}$$

$$h(n) = 2^{2^{\cdot^{\cdot^{2^2}}}}$$

Where the exponent repeats n times. This operation is called tetration.

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

$$f(n) = 5n^2$$