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arXiv · 1604.02303

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

Igor Potapov·Pavel Semukhin
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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.

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Igor Potapov, Pavel Semukhin. 2016-04-08. Decidability of the Membership Problem for $2\times 2$ integer matrices. https://arxiv.org/abs/1604.02303

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