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Monoids of Upper Triangular Matrices over the Boolean Semiring

Authors Andrew Ryzhikov , Petra Wolf

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Author Details

Andrew Ryzhikov
  • Department of Computer Science, University of Oxford, UK
Petra Wolf
  • LaBRI, CNRS, Université de Bordeaux, Bordeaux INP, France

Funding

  • Ryzhikov, Andrew: Supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant agreement No. 852769, ARiAT).
  • Wolf, Petra: Supported by the French ANR, project ANR-22-CE48-0001 (TEMPOGRAL).

Acknowledgements

We thank anonymous reviewers for their comments that improved the content and presentation of the paper.

Cite AsGet BibTex

Andrew Ryzhikov and Petra Wolf. Monoids of Upper Triangular Matrices over the Boolean Semiring. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 81:1-81:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.MFCS.2024.81

Abstract

Given a finite set 𝒜 of square matrices and a square matrix B, all of the same dimension, the membership problem asks if B belongs to the monoid ℳ(𝒜) generated by 𝒜. The rank one problem asks if there is a matrix of rank one in ℳ(𝒜). We study the membership and the rank one problems in the case where all matrices are upper triangular matrices over the Boolean semiring. We characterize the computational complexity of these problems, and identify their PSPACE-complete and NP-complete special cases. We then consider, for a set 𝒜 of matrices from the same class, the problem of finding in ℳ(𝒜) a matrix of minimum rank with no zero rows. We show that the minimum rank of such matrix can be computed in linear time.We also characterize the space complexity of this problem depending on the size of 𝒜, and apply all these results to the ergodicity problem asking if ℳ(𝒜) contains a matrix with a column consisting of all ones. Finally, we show that our results give better upper bounds for the case where each row of every matrix in 𝒜 contains at most one non-zero entry than for the general case.

Subject Classification

ACM Subject Classification
  • Theory of computation → Formal languages and automata theory
Keywords
  • matrix monoids
  • membership
  • rank
  • ergodicity
  • partially ordered automata

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References

  1. László Babai, Robert Beals, Jin-yi Cai, Gábor Ivanyos, and Eugene M. Luks. Multiplicative equations over commuting matrices. In Proceedings of the Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '96, pages 498-507, USA, 1996. Society for Industrial and Applied Mathematics. Google Scholar
  2. Jørgen Bang-Jensen and Gregory Z. Gutin. Digraphs: theory, algorithms and applications. Springer Science & Business Media, 2008. Google Scholar
  3. R. B. Bapat and T. E. S. Raghavan. Nonnegative Matrices and Applications. Encyclopedia of Mathematics and its Applications. Cambridge University Press, 1997. Google Scholar
  4. Marie-Pierre Béal, Eugen Czeizler, Jarkko Kari, and Dominique Perrin. Unambiguous automata. Mathematics in Computer Science, 1:625-638, 2008. URL: https://doi.org/10.1007/s11786-007-0027-1.
  5. Marie-Pierre Béal and Dominique Perrin. Synchronised automata. In Valérie Berthé and Michel Rigo, editors, Combinatorics, Words and Symbolic Dynamics, Encyclopedia of Mathematics and its Applications, pages 213-240. Cambridge University Press, 2016. URL: https://doi.org/10.1017/CBO9781139924733.008.
  6. M. Beaudry, P. McKenzie, and D. Thérien. The membership problem in aperiodic transformation monoids. Journal of the ACM, 39(3):599-616, 1992. Google Scholar
  7. Martin Beaudry. Membership testing in commutative transformation semigroups. Information and Computation, 79(1):84-93, 1988. URL: https://doi.org/10.1016/0890-5401(88)90018-1.
  8. Martin Beaudry. Membership testing in transformation monoids. PhD thesis, McGill University, Montreal, Quebec, 1988. Google Scholar
  9. Martin Beaudry. Membership testing in threshold one transformation monoids. Information and Computation., 113(1):1-25, 1994. URL: https://doi.org/10.1006/INCO.1994.1062.
  10. Mikhail V. Berlinkov. On two algorithmic problems about synchronizing automata - (short paper). In Arseny M. Shur and Mikhail V. Volkov, editors, Developments in Language Theory - 18th International Conference, DLT 2014, Ekaterinburg, Russia, August 26-29, 2014. Proceedings, volume 8633 of Lecture Notes in Computer Science, pages 61-67. Springer, 2014. URL: https://doi.org/10.1007/978-3-319-09698-8_6.
  11. Mikhail V. Berlinkov, Robert Ferens, Andrew Ryzhikov, and Marek Szykuła. Synchronizing Strongly Connected Partial DFAs. In Markus Bläser and Benjamin Monmege, editors, 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021), volume 187 of Leibniz International Proceedings in Informatics (LIPIcs), pages 12:1-12:16, Dagstuhl, Germany, 2021. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.STACS.2021.12.
  12. Jean Berstel, Dominique Perrin, and Christophe Reutenauer. Codes and automata, volume 129 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, 2010. Google Scholar
  13. Michael Blondin, Andreas Krebs, and Pierre McKenzie. The complexity of intersecting finite automata having few final states. computational complexity, 25(4):775-814, 2016. URL: https://doi.org/10.1007/s00037-014-0089-9.
  14. Arturo Carpi. On synchronizing unambiguous automata. Theoretical Computer Science, 60:285-296, 1988. URL: https://doi.org/10.1016/0304-3975(88)90114-4.
  15. Pierre-Yves Chevalier, Vladimir V. Gusev, Raphaël M. Jungers, and Julien M. Hendrickx. Sets of stochastic matrices with converging products: Bounds and complexity. CoRR, abs/1712.02614, 2017. URL: https://arxiv.org/abs/1712.02614.
  16. Pierre-Yves Chevalier, Julien M. Hendrickx, and Raphaël M. Jungers. Reachability of consensus and synchronizing automata. In 54th IEEE Conference on Decision and Control, CDC 2015, Osaka, Japan, December 15-18, 2015, pages 4139-4144. IEEE, 2015. URL: https://doi.org/10.1109/CDC.2015.7402864.
  17. V. Froidure. Ranks of binary relations. Semigroup Forum, 54:381-401, 1997. URL: https://doi.org/10.1007/BF02676619.
  18. Pavel Goralčík and Václav Koubek. Rank problems for composite transformations. International Journal of Algebra and Computation, 05(03):309-316, 1995. URL: https://doi.org/10.1142/S0218196795000185.
  19. Raymond Greenlaw, H. James Hoover, and Walter L. Ruzzo. Limits to parallel computation: P-completeness theory. Oxford University Press, 1995. Google Scholar
  20. Balázs Imreh and Magnus Steinby. Directable nondeterministic automata. Acta Cybernetica, 14(1):105-115, 1999. Google Scholar
  21. Masami Ito and Kayoko Shikishima-Tsuji. Shortest directing words of nondeterministic directable automata. Discrete Mathematics, 308(21):4900-4905, 2008. URL: https://doi.org/10.1016/J.DISC.2007.09.010.
  22. R. Kannan and R. J. Lipton. Polynomial-time algorithm for the orbit problem. Journal of the ACM, 33(4):808-821, 1986. URL: https://doi.org/10.1145/6490.6496.
  23. Jarkko Kari and Mikhail V. Volkov. Černý’s conjecture and the road colouring problem. In Handbook of Automata Theory, pages 525-565. EMS Press, 2021. URL: https://doi.org/10.4171/AUTOMATA-1/15.
  24. Stefan Kiefer and Corto N. Mascle. On nonnegative integer matrices and short killing words. SIAM Journal on Discrete Mathematics, 35(2):1252-1267, 2021. URL: https://doi.org/10.1137/19M1250893.
  25. Dexter Kozen. Lower bounds for natural proof systems. In Proceedings of the 18th Annual Symposium on Foundations of Computer Science, pages 254-266. IEEE Computer Society, 1977. URL: https://doi.org/10.1109/SFCS.1977.16.
  26. Markus Krötzsch, Tomáš Masopust, and Michaël Thomazo. Complexity of universality and related problems for partially ordered NFAs. Information and Computation, 255:177-192, 2017. URL: https://doi.org/10.1016/J.IC.2017.06.004.
  27. Douglas Lind and Brian Marcus. An introduction to symbolic dynamics and coding. Cambridge University Press, 2021. Google Scholar
  28. Alexei Lisitsa and Igor Potapov. Membership and reachability problems for row-monomial transformations. In Jiří Fiala, Václav Koubek, and Jan Kratochvíl, editors, Mathematical Foundations of Computer Science 2004, pages 623-634, Berlin, Heidelberg, 2004. Springer Berlin Heidelberg. Google Scholar
  29. Pavel Martyugin. Computational complexity of certain problems related to carefully synchronizing words for partial automata and directing words for nondeterministic automata. Theory Comput. Syst., 54(2):293-304, 2014. URL: https://doi.org/10.1007/S00224-013-9516-6.
  30. Tomáš Masopust and Markus Krötzsch. Partially Ordered Automata and Piecewise Testability. Logical Methods in Computer Science, 17(2):14:1-14:36, 2021. URL: https://doi.org/10.23638/LMCS-17(2:14)2021.
  31. Tomáš Masopust and Michaël Thomazo. On boolean combinations forming piecewise testable languages. Theor. Comput. Sci., 682:165-179, 2017. URL: https://doi.org/10.1016/J.TCS.2017.01.017.
  32. Mike Paterson. Unsolvability in 3 × 3 matrices. Studies in Applied Mathematics, 49:105-107, 1970. Google Scholar
  33. Jean-Eric Pin and Howard Straubing. Monoids of upper triangular boolean matrices. In Semigroups. Structure and Universal AIgebraic Problems, volume 39, pages 259-272, 1981. Google Scholar
  34. Igor Potapov and Pavel Semukhin. Decidability of the membership problem for 2 times 2 integer matrices. In Philip N. Klein, editor, Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017, Barcelona, Spain, Hotel Porta Fira, January 16-19, pages 170-186. SIAM, 2017. URL: https://doi.org/10.1137/1.9781611974782.12.
  35. Vladimir Yu. Protasov. Analytic methods for reachability problems. J. Comput. Syst. Sci., 120:1-13, 2021. URL: https://doi.org/10.1016/J.JCSS.2021.02.007.
  36. Andrew Ryzhikov. Mortality and synchronization of unambiguous finite automata. In Robert Mercas and Daniel Reidenbach, editors, Combinatorics on Words - 12th International Conference, WORDS 2019, Loughborough, UK, September 9-13, 2019, Proceedings, volume 11682 of Lecture Notes in Computer Science, pages 299-311. Springer, 2019. URL: https://doi.org/10.1007/978-3-030-28796-2_24.
  37. Andrew Ryzhikov. Synchronization problems in automata without non-trivial cycles. Theoretical Computer Science, 787:77-88, 2019. Google Scholar
  38. Eugene Seneta. Non-negative matrices and Markov chains. Springer Science & Business Media, 2006. Google Scholar
  39. Michael Sipser. Introduction to the Theory of Computation. Course Technology, Boston, MA, third edition, 2013. Google Scholar
  40. Seinosuka Toda. On the complexity of topological sorting. Information Processing Letters, 35(5):229-233, 1990. URL: https://doi.org/10.1016/0020-0190(90)90050-8.
  41. Mikhail V. Volkov. Synchronizing automata and the Černý conjecture. In Carlos Martín-Vide, Friedrich Otto, and Henning Fernau, editors, Language and Automata Theory and Applications, Second International Conference, LATA 2008, Tarragona, Spain, March 13-19, 2008. Revised Papers, volume 5196 of Lecture Notes in Computer Science, pages 11-27. Springer, 2008. Google Scholar
  42. Mikhail V. Volkov. Synchronization of finite automata. Russian Mathematical Surveys, 77(5):819-891, 2022. URL: https://doi.org/10.4213/rm10005e.
  43. Petra Wolf. Synchronization under dynamic constraints. In 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020), 2020. Google Scholar
  44. Yaokun Wu and Yinfeng Zhu. Primitivity and Hurwitz primitivity of nonnegative matrix tuples: A unified approach. SIAM Journal on Matrix Analysis and Applications, 44(1):196-211, 2023. URL: https://doi.org/10.1137/22M1471535.
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