Exercise 1.10 Ackerman's function

题目：

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)
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*n.

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根据题意可知:

　　A(N, 0)=0

　　A(0,N)=2*N

　　A(N, 1)=2

　　A(X,Y)=A(X-1, A(X, Y-1))

所以:

1. F(N)=A(0,N)=2*N

2. G(N)=A(1,N)

　　　　=A(0, A(1, N-1)

　　　　=2*A(1, N-1)

　　　　=2*G(N-1)

　而G(0)=A(1,0)=0, G(1)=A(1,1)=2

  所以 G(N)=2^N 当N>=1时

3. H(N)=A(2,N)

　　　　=A(1, A(2, N-1))

　　　　=2^(A(2,N-1))
　　　　=2^H(N-1)

　　A(2,0)=1

　　A(2,1)=2

　　所以 H(N)=2^H(N-1) 当N>=1时