Incomputable Problems

Abstract

After the basic notions of computation had been formalized, a close link between computational problems—in particular, decision problems—and sets was discovered. This was important because the notion of the the set was finally settled, and sets made it possible to apply diagonalization, a proof method already discovered by Cantor. Diagonalization, combined with self-reference, made it possible to discover the first incomputable problem, i.e., a decision problem called the Halting Problem, for which there is no single algorithm capable of solving every instance of the problem. This was simultaneously and independently discovered by Church and Turing. After this, Computability Theory blossomed, so that in the second half of the 1930s one of the main questions became, “Which computational problems are computable and which ones are not?” Indeed, using various proof methods, many incomputable problems were discovered in different fields of science. This showed that incomputability is a constituent part of reality.

A problem is unsolvable if there is no single procedure that can construct a solution for an arbitrary instance of the problem.