On the Size of Finite Rational Matrix Semigroups

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

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Author Details

Georgina Bumpus
  • University of Oxford, UK
Christoph Haase
  • University College London, UK
Stefan Kiefer
  • University of Oxford, UK
Paul-Ioan Stoienescu
  • University of Oxford, UK
Jonathan Tanner
  • University of Oxford, UK


The authors thank James Worrell for discussing [E. Hrushovski et al., 2018] with them.

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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)


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.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Discrete mathematics
  • Theory of computation → Algebraic language theory
  • Theory of computation → Quantitative automata
  • Matrix semigroups
  • Burnside problem
  • weighted automata
  • vector addition systems


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