Decidability of the Membership Problem for $2\times 2$ integer matrices

TL;DR

This paper proves that it is possible to algorithmically determine whether a given $2\times 2$ nonsingular integer matrix can be generated by a finite set of such matrices, using a novel approach translating the problem into combinatorial and language membership problems.

Contribution

It introduces the first decidable algorithm for the membership problem in the semigroup of $2\times 2$ nonsingular integer matrices, combining algebraic and combinatorial techniques.

Findings

Decidability of the membership problem established.

Algorithm reduces matrix problem to regular language membership.

Uses algebraic properties of subgroups in $\mathrm{GL}(2,\mathbb{Z})$.

Abstract

The main result of this paper is the decidability of the membership problem for $2\times 2$ nonsingular integer matrices. Namely, we will construct the first algorithm that for any nonsingular $2\times 2$ integer matrices $M_1,\dots,M_n$ and $M$ decides whether $M$ belongs to the semigroup generated by $\{M_1,\dots,M_n\}$. Our algorithm relies on a translation of the numerical problem on matrices into combinatorial problems on words. It also makes use of some algebraical properties of well-known subgroups of $\mathrm{GL}(2,\mathbb{Z})$ and various new techniques and constructions that help to limit an infinite number of possibilities by reducing them to the membership problem for regular languages.