# 1.10

## 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 (- 1 1) (A 1 (- 10 1)))
(A 0 (A 1 9))
(A 0 (A (- 1 1) (A 1 (- 9 1))))
(A 0 (A 0 (A 1 8)))
(A 0 (A 0 (A (- 1 1) (A 1 (- 8 1)))))
(A 0 (A 0 (A 0 (A 1 7))))
(A 0 (A 0 (A 0 (A (- 1 1) (A 1 (- 7 1))))))
(A 0 (A 0 (A 0 (A 0 (A 1 6)))))
(A 0 (A 0 (A 0 (A 0 (A (- 1 1) (A 1 (- 6 1)))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 1 5))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A (- 1 1) (A 1 (- 5 1))))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 4)))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A (- 1 1) (A 1 (- 4 1)))))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 3))))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A (- 1 1) (A 1 (- 3 1))))))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 2)))))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A (- 1 1) (A 1 (- 2 1)))))))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 1))))))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 2)))))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (* 2 2)))))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 4))))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (* 2 4))))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 8)))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (* 2 8)))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 16))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (* 2 16))))))
(A 0 (A 0 (A 0 (A 0 (A 0 32)))))
(A 0 (A 0 (A 0 (A 0 (* 2 32)))))
(A 0 (A 0 (A 0 (A 0 64))))
(A 0 (A 0 (A 0 (* 2 64))))
(A 0 (A 0 (A 0 128)))
(A 0 (A 0 (* 2 128)))
(A 0 (A 0 256))
(A 0 (* 2 256))
(A 0 512)
(* 2 512)
1024

(A 2 4)


The steps under the hood:

(A 2 4)


(A (- 2 1) (A 2 (- 4 1)))
(A 1 (A 2 3))
(A 1 (A (- 2 1) (A 2 (- 3 1))))
(A 1 (A 1 (A 2 2)))
(A 1 (A 1 (A (- 2 1) (A 2 (- 2 1)))))
(A 1 (A 1 (A 1 (A 2 1))))
(A 1 (A 1 (A 1 2)))
(A 1 (A 1 (A (- 1 1) (A 1 (- 2 1)))))
(A 1 (A 1 (A 0 (A 1 1))))
(A 1 (A 1 (A 0 2)))
(A 1 (A 1 (* 2 2)))
(A 1 (A 1 4))
(A 1 (A (- 1 1) (A 1 (- 4 1))))
(A 1 (A 0 (A 1 3)))
(A 1 (A 0 (A (- 1 1) (A 1 (- 3 1)))))
(A 1 (A 0 (A 0 (A 1 2))))
(A 1 (A 0 (A 0 (A (- 1 1) (A 1 (- 2 1))))))
(A 1 (A 0 (A 0 (A 0 (A 1 1)))))
(A 1 (A 0 (A 0 (A 0 2))))
(A 1 (A 0 (A 0 (* 2 2))))
(A 1 (A 0 (A 0 4)))
(A 1 (A 0 (* 2 4)))
(A 1 (A 0 8))
(A 1 (* 2 8))
(A 1 16)
(A (- 1 1) (A 1 (- 16 1)))
(A 0 (A 1 15))
(A 0 (A (- 1 1) (A 1 (- 15 1))))
(A 0 (A 0 (A 1 14)))
(A 0 (A 0 (A (- 1 1) (A 1 (- 14 1)))))
(A 0 (A 0 (A 0 (A 1 13))))
(A 0 (A 0 (A 0 (A (- 1 1) (A 1 (- 13 1))))))
(A 0 (A 0 (A 0 (A 0 (A 1 12)))))
(A 0 (A 0 (A 0 (A 0 (A (- 1 1) (A 1 (- 12 1)))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 1 11))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A (- 1 1) (A 1 (- 11 1))))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 10)))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A (- 1 1) (A 1 (- 10 1)))))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 9))))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A (- 1 1) (A 1 (- 9 1))))))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 8)))))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A (- 1 1) (A 1 (- 8 1)))))))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 7))))))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A (- 1 1) (A 1 (- 7 1))))))))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 6)))))))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A (- 1 1) (A 1 (- 6 1)))))))))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 5))))))))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A (- 1 1) (A 1 (- 5 1))))))))))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 4)))))))))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A (- 1 1) (A 1 (- 4 1)))))))))))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 3))))))))))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A (- 1 1) (A 1 (- 3 1))))))))))))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 2)))))))))))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A (- 1 1) (A 1 (- 2 1)))))))))))))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 1))))))))))))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 2)))))))))))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (* 2 2)))))))))))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 4))))))))))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (* 2 4))))))))))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 8)))))))))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (* 2 8)))))))))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 16))))))))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (* 2 16))))))))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 32)))))))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (* 2 32)))))))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 64))))))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (* 2 64))))))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 128)))))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (* 2 128)))))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 256))))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (* 2 256))))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 512)))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (* 2 512)))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 1024))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (* 2 1024))))))
(A 0 (A 0 (A 0 (A 0 (A 0 2048)))))
(A 0 (A 0 (A 0 (A 0 (* 2 2048)))))
(A 0 (A 0 (A 0 (A 0 4096))))
(A 0 (A 0 (A 0 (* 2 4096))))
(A 0 (A 0 (A 0 8192)))
(A 0 (A 0 (* 2 8192)))
(A 0 (A 0 16384))
(A 0 (* 2 16384))
(A 0 32768)
(* 2 32768)
65536

(A 3 3)


The steps under the hood:

(A 3 3)


(A (- 3 1) (A 3 (- 3 1)))
(A 2 (A 3 2))
(A 2 (A (- 3 1) (A 3 (- 2 1))))
(A 2 (A 2 (A 3 1)))
(A 2 (A 2 2))
(A 2 (A (- 2 1) (A 2 (- 2 1))))
(A 2 (A 1 (A 2 1)))
(A 2 (A 1 2))
(A 2 (A (- 1 1) (A 1 (- 2 1))))
(A 2 (A 0 (A 1 1)))
(A 2 (A 0 2))
(A 2 (* 2 2))
(A 2 4)
(A (- 2 1) (A 2 (- 4 1)))
(A 1 (A 2 3))
(A 1 (A (- 2 1) (A 2 (- 3 1))))
(A 1 (A 1 (A 2 2)))
(A 1 (A 1 (A (- 2 1) (A 2 (- 2 1)))))
(A 1 (A 1 (A 1 (A 2 1))))
(A 1 (A 1 (A 1 2)))
(A 1 (A 1 (A (- 1 1) (A 1 (- 2 1)))))
(A 1 (A 1 (A 0 (A 1 1))))
(A 1 (A 1 (A 0 2)))
(A 1 (A 1 (* 2 2)))
(A 1 (A 1 4))
(A 1 (A (- 1 1) (A 1 (- 4 1))))
(A 1 (A 0 (A 1 3)))
(A 1 (A 0 (A (- 1 1) (A 1 (- 3 1)))))
(A 1 (A 0 (A 0 (A 1 2))))
(A 1 (A 0 (A 0 (A (- 1 1) (A 1 (- 2 1))))))
(A 1 (A 0 (A 0 (A 0 (A 1 1)))))
(A 1 (A 0 (A 0 (A 0 2))))
(A 1 (A 0 (A 0 (* 2 2))))
(A 1 (A 0 (A 0 4)))
(A 1 (A 0 (* 2 4)))
(A 1 (A 0 8))
(A 1 (* 2 8))
(A 1 16)
(A (- 1 1) (A 1 (- 16 1)))
(A 0 (A 1 15))
(A 0 (A (- 1 1) (A 1 (- 15 1))))
(A 0 (A 0 (A 1 14)))
(A 0 (A 0 (A (- 1 1) (A 1 (- 14 1)))))
(A 0 (A 0 (A 0 (A 1 13))))
(A 0 (A 0 (A 0 (A (- 1 1) (A 1 (- 13 1))))))
(A 0 (A 0 (A 0 (A 0 (A 1 12)))))
(A 0 (A 0 (A 0 (A 0 (A (- 1 1) (A 1 (- 12 1)))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 1 11))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A (- 1 1) (A 1 (- 11 1))))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 10)))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A (- 1 1) (A 1 (- 10 1)))))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 9))))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A (- 1 1) (A 1 (- 9 1))))))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 8)))))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A (- 1 1) (A 1 (- 8 1)))))))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 7))))))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A (- 1 1) (A 1 (- 7 1))))))))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 6)))))))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A (- 1 1) (A 1 (- 6 1)))))))))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 5))))))))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A (- 1 1) (A 1 (- 5 1))))))))))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 4)))))))))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A (- 1 1) (A 1 (- 4 1)))))))))))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 3))))))))))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A (- 1 1) (A 1 (- 3 1))))))))))))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 2)))))))))))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A (- 1 1) (A 1 (- 2 1)))))))))))))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 1 1))))))))))))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 2)))))))))))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (* 2 2)))))))))))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 4))))))))))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (* 2 4))))))))))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 8)))))))))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (* 2 8)))))))))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 16))))))))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (* 2 16))))))))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 32)))))))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (* 2 32)))))))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 64))))))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (* 2 64))))))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 128)))))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (* 2 128)))))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 256))))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (* 2 256))))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (A 0 512)))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 (* 2 512)))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (A 0 1024))))))
(A 0 (A 0 (A 0 (A 0 (A 0 (* 2 1024))))))
(A 0 (A 0 (A 0 (A 0 (A 0 2048)))))
(A 0 (A 0 (A 0 (A 0 (* 2 2048)))))
(A 0 (A 0 (A 0 (A 0 4096))))
(A 0 (A 0 (A 0 (* 2 4096))))
(A 0 (A 0 (A 0 8192)))
(A 0 (A 0 (* 2 8192)))
(A 0 (A 0 16384))
(A 0 (* 2 16384))
(A 0 32768)
(* 2 32768)
65536

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

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


(* 2 n)

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


(A (- 1 1) (A 1 (- n 1)))

(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 $5n^2$.

$(f\;n) = 2n$

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

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

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

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

$...$

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