Exercises

Exercise 1.9

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

(defun add (a b)
  (if (== a 0)
      b
      (inc (+ (dec a) b))))

(defun add (a b)
  (if (== a 0)
      b
      (+ (dec a) (inc b))))

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

Exercise 1.10

The Ackermann function is defined as

$$A(m, n) = \begin{cases} n+1 & \mbox{if } m = 0 \\ A(m-1, 1) & \mbox{if } m > 0 \mbox{ and } n = 0 \\ A(m-1, A(m, n-1)) & \mbox{if } m > 0 \mbox{ and } n > 0. \end{cases}$$

Here is the function in LFE:

(defun ackermann
  ((0 n) (+ n 1))
  ((m 0) (ackermann (- m 1) 1))
  ((m n) (ackermann (- m 1) (ackermann m (- n 1)))))

What are the values of the following expressions?

(ackermann 1 10)
(ackermann 2 4)
(ackermann 3 3)

Consider the following functions, where ackermann is the function defined above:

(defun f (n) (ackermann 0 n))

(defun g (n) (ackermann 1 n))

(defun h (n) (ackermann 2 n))

(defun k (n) (* 5 n n))

Give concise mathematical definitions for the functions computed by the functions f, g, and h for positive integer values of n. For example, (k n) computes $5n^2$.