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Revisiting Finiteness of Matrix Monoids

Authors Rida Ait El Manssour , Roland Guttenberg , Nathan Lhote , Mahsa Shirmohammadi , James Ben Worrell

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LIPIcs.ICALP.2026.158.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Automata over infinite objects
Keywords
  • Matrix Semigroups
  • Finiteness
  • Integrality
  • Bitsize Bound

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Abstract

This paper concerns decision problems related to finite monoids of rational matrices. We show that determining finiteness of a given finitely presented monoid is in PSpace, improving the known coNExp^NP bound. We also show that the membership problem for finite matrix monoids is PSpace-complete, improving the known NExp-upper bound. Our two complexity results are corollaries of a new polynomial bit-size bound on matrix entries in finite monoids. This is obtained by reduction to the case of matrix groups, using the structure theory of noncommutative algebras and of matrix monoids. Our techniques also give us a polynomial-time algorithm for deciding whether a monoid of rational matrices is conjugate to a monoid of integer matrices.

Cite As Get BibTex

Rida Ait El Manssour, Roland Guttenberg, Nathan Lhote, Mahsa Shirmohammadi, and James Ben Worrell. Revisiting Finiteness of Matrix Monoids. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 158:1-158:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ICALP.2026.158

Author Details

Rida Ait El Manssour
  • University of Oxford, UK
Roland Guttenberg
  • University of Warsaw, Poland
Nathan Lhote
  • Aix-Marseille Université, LIS, France
Mahsa Shirmohammadi
  • CNRS & IRIF, Paris, France
James Ben Worrell
  • University of Oxford, UK

Funding

  • Ait El Manssour, Rida: Supported by the EPSRC Fellowship (EP/X033813/1).
  • Guttenberg, Roland: Supported by the ERC grant INFSYS, agreement No. 950398.
  • Shirmohammadi, Mahsa: Supported by the ERC Synergy Grant VePaSS (agreement No. 101224640) and the grant VeSyAM (ANR-22-CE48-0005).
  • Worrell, James Ben: Supported by the EPSRC Fellowship (EP/X033813/1).

References

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