On Reachability Problems for Low-Dimensional Matrix Semigroups

Authors Thomas Colcombet , Joël Ouaknine , Pavel Semukhin , James Worrell

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Thomas Colcombet
  • IRIF, CNRS, Université Paris Diderot, France
Joël Ouaknine
  • The Max Planck Institute for Software Systems, Saarbrücken, Germany
  • Department of Computer Science, University of Oxford, United Kingdom
Pavel Semukhin
  • Department of Computer Science, University of Oxford, United Kingdom
James Worrell
  • Department of Computer Science, University of Oxford, United Kingdom

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Thomas Colcombet, Joël Ouaknine, Pavel Semukhin, and James Worrell. On Reachability Problems for Low-Dimensional Matrix Semigroups. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 44:1-44:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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 GL(2,Z) and of the Heisenberg group over rational numbers.

Subject Classification

ACM Subject Classification
  • Theory of computation → Formal languages and automata theory
  • Computing methodologies → Symbolic and algebraic algorithms
  • membership problem
  • half-space reachability problem
  • matrix semigroups
  • Heisenberg group
  • general linear group


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