On Reachability Problems for Low-Dimensional Matrix Semigroups

Thomas Colcombet; Joël Ouaknine; Pavel Semukhin; James Worrell

doi:10.4230/lipics.icalp.2019.39

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arXiv · 1902.09597

On Reachability Problems for Low-Dimensional Matrix Semigroups

Thomas Colcombet·Joël Ouaknine·Pavel Semukhin·James Worrell

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Abstract

We consider the Membership and the Half-Space Reachability problems for matrices in dimensions two and three. Our first main result is that the Membership Problem is decidable for finitely generated sub-semigroups of the Heisenberg group over rational numbers. Furthermore, we prove two decidability results for the Half-Space Reachability Problem. Namely, we show that this problem is decidable for sub-semigroups of $\mathrm{GL}(2,\mathbb{Z})$ and of the Heisenberg group over rational numbers.

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Thomas Colcombet, Joël Ouaknine, Pavel Semukhin, James Worrell. 2019-02-25. On Reachability Problems for Low-Dimensional Matrix Semigroups. https://doi.org/10.4230/lipics.icalp.2019.39

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