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On Reachability Problems for Low-Dimensional Matrix Semigroups

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

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Author Details

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

Funding

  • Colcombet, Thomas: Supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No.670624), and by the DeLTA ANR project (ANR-16-CE40-0007).
  • Ouaknine, Joël: Supported by ERC grant AVS-ISS (648701) and by DFG grant 389792660 as part of TRR 248 (see https://perspicuous-computing.science).
  • Semukhin, Pavel: Supported by ERC grant AVS-ISS (648701).
  • Worrell, James: Supported by EPSRC Fellowship EP/N008197/1.

Cite AsGet BibTex

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)
https://doi.org/10.4230/LIPIcs.ICALP.2019.44

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 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
Keywords
  • membership problem
  • half-space reachability problem
  • matrix semigroups
  • Heisenberg group
  • general linear group

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