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DOI: 10.4230/LIPIcs.ICALP.2020.115

On the Size of Finite Rational Matrix Semigroups

Authors: Georgina Bumpus, Christoph Haase, Stefan Kiefer, Paul-Ioan Stoienescu, and Jonathan Tanner

Published in: LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

Abstract

Let n be a positive integer and M a set of rational n × n-matrices such that M generates a finite multiplicative semigroup. We show that any matrix in the semigroup is a product of matrices in M whose length is at most 2^{n (2 n + 3)} g(n)^{n+1} ∈ 2^{O(n² log n)}, where g(n) is the maximum order of finite groups over rational n × n-matrices. This result implies algorithms with an elementary running time for deciding finiteness of weighted automata over the rationals and for deciding reachability in affine integer vector addition systems with states with the finite monoid property.

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Georgina Bumpus, Christoph Haase, Stefan Kiefer, Paul-Ioan Stoienescu, and Jonathan Tanner. On the Size of Finite Rational Matrix Semigroups. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 115:1-115:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bumpus_et_al:LIPIcs.ICALP.2020.115,
  author =	{Bumpus, Georgina and Haase, Christoph and Kiefer, Stefan and Stoienescu, Paul-Ioan and Tanner, Jonathan},
  title =	{{On the Size of Finite Rational Matrix Semigroups}},
  booktitle =	{47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)},
  pages =	{115:1--115:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-138-2},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{168},
  editor =	{Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.115},
  URN =		{urn:nbn:de:0030-drops-125226},
  doi =		{10.4230/LIPIcs.ICALP.2020.115},
  annote =	{Keywords: Matrix semigroups, Burnside problem, weighted automata, vector addition systems}
}

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On the Size of Finite Rational Matrix Semigroups

Abstract

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