The Identity Correspondence Problem and its Applications

TL;DR

This paper introduces the undecidable Identity Correspondence Problem (ICP) for words and matrices, proving its undecidability and applying it to show that certain matrix semigroup membership problems are also undecidable.

Contribution

The paper defines ICP, proves its undecidability, and applies this result to establish the undecidability of matrix identity membership problems in higher dimensions.

Findings

ICP is undecidable via reduction from Post's Correspondence Problem.

Decidability of identity membership in matrix semigroups is false for dimension four.

Finite sets of matrices do not always generate a group, undecidably.

Abstract

In this paper we study several closely related fundamental problems for words and matrices. First, we introduce the Identity Correspondence Problem (ICP): whether a finite set of pairs of words (over a group alphabet) can generate an identity pair by a sequence of concatenations. We prove that ICP is undecidable by a reduction of Post's Correspondence Problem via several new encoding techniques. In the second part of the paper we use ICP to answer a long standing open problem concerning matrix semigroups: "Is it decidable for a finitely generated semigroup S of square integral matrices whether or not the identity matrix belongs to S?". We show that the problem is undecidable starting from dimension four even when the number of matrices in the generator is 48. From this fact, we can immediately derive that the fundamental problem of whether a finite set of matrices generates a group is also undecidable. We also answer several question for matrices over different number fields. Apart from the application to matrix problems, we believe that the Identity Correspondence Problem will also be useful in identifying new areas of undecidable problems in abstract algebra, computational questions in logic and combinatorics on words.