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