CS4123 Theory of Computation. B97

Prof. Carolina Ruiz
Department of Computer Science
Worcester Polytechnic Institute

Homework 2

• (Taken from Lewis and Papadimitriou '98) Show that the following functions are primitive recursive:
  1. factorial(n) = n!
  2. gcd(m,n), the greatest common divisor of m and n.
  3. the function prime(n) defined as follows: if n is a prime number then prime(n) = 1, otherwise prime(n) = 0.
  4. p(n), the n_th prime number, where p(0) = 2, p(1) = 3, and so on.
  5. the function F: N -> N defined by: F(n) = f(f(f(... f(n) ...))), where there are n function compositions and the function f: N -> N is primitive recursive.

• Show that the language L = {a^n.b^n.c^n | n >= 0} is decidable.
  (Note: "a^n" denotes "a" repeated n times, and "." denotes concatenation. That is, a^n.b^n.c^n denotes the string that consists of n repetitions of "a" followed by n repetitions of "b" followed by n repetitions of "c".)