Unsolvable Problems

Abstract

This chapter discusses the halting problem, semi-Thue processes, semigroups and groups, other combinatorial problems, and diophantine equations. Many important mathematical problems take the form: find an algorithm by means of which it can be determined for each element of a given set, whether or not the element possesses some given property. The solution of such a decision problem is then to consist of actually exhibiting an algorithm and providing a proof that the algorithm does what is required of it. A standard example from the elementary theory of numbers is to give an algorithm by means of which it can be determined for a given ordered triple ( a, b, c ) of integers, whether or not the equation ax + by = c has a solution in integers. In this case, a solution is: find the largest natural number d that simultaneously divides a and b and then test whether or not d is a divisor of c .