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Efficiently Computing the Minimum Rank of a Matrix in a Monoid of Zero-One Matrices

Authors Stefan Kiefer , Andrew Ryzhikov

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ACM Subject Classification
  • Theory of computation → Formal languages and automata theory
  • Computing methodologies → Symbolic and algebraic manipulation
Keywords
  • matrix monoids
  • minimum rank
  • unambiguous automata

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Abstract

A zero-one matrix is a matrix with entries from {0, 1}. We study monoids containing only such matrices. A finite set of zero-one matrices generating such a monoid can be seen as the matrix representation of an unambiguous finite automaton, an important generalisation of deterministic finite automata which shares many of their good properties.
Let 𝒜 be a finite set of n×n zero-one matrices generating a monoid of zero-one matrices, and m be the cardinality of 𝒜. We study the computational complexity of computing the minimum rank of a matrix in the monoid generated by 𝒜. By using linear-algebraic techniques, we show that this problem is in NC and can be solved in 𝒪(mn⁴) time. We also provide a combinatorial algorithm finding a matrix of minimum rank in 𝒪(n^{2 + ω} + mn⁴) time, where 2 ≤ ω ≤ 2.4 is the matrix multiplication exponent. As a byproduct, we show a very weak version of a generalisation of the Černý conjecture: there always exists a straight line program of size 𝒪(n²) describing a product resulting in a matrix of minimum rank. 
For the special case corresponding to complete DFAs (that is, for the case where all matrices have exactly one 1 in each row), the minimum rank is the size of the smallest image of the set of states under the action of a word. Our combinatorial algorithm finds a matrix of minimum rank in time 𝒪(n³ + mn²) in this case.

Cite As Get BibTex

Stefan Kiefer and Andrew Ryzhikov. Efficiently Computing the Minimum Rank of a Matrix in a Monoid of Zero-One Matrices. In 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 327, pp. 61:1-61:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.STACS.2025.61

Author Details

Stefan Kiefer
  • Department of Computer Science, University of Oxford, UK
Andrew Ryzhikov
  • Department of Computer Science, University of Oxford, UK

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).

Acknowledgements

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

References

  1. Jorge Almeida and Benjamin Steinberg. Matrix mortality and the Černý-Pin conjecture. In Volker Diekert and Dirk Nowotka, editors, Developments in Language Theory, pages 67-80, Berlin, Heidelberg, 2009. Springer Berlin Heidelberg. URL: https://doi.org/10.1007/978-3-642-02737-6_5.
  2. Sanjeev Arora and Boaz Barak. Computational complexity: a modern approach. Cambridge University Press, 2009. Google Scholar
  3. Christel Baier, Stefan Kiefer, Joachim Klein, David Müller, and James Worrell. Markov chains and unambiguous automata. Journal of Computer and System Sciences, 136:113-134, 2023. URL: https://doi.org/10.1016/J.JCSS.2023.03.005.
  4. M.-P. Béal and D. 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.
  5. Marie-Pierre Béal, Eugen Czeizler, Jarkko Kari, and Dominique Perrin. Unambiguous automata. Math. Comput. Sci., 1(4):625-638, 2008. URL: https://doi.org/10.1007/S11786-007-0027-1.
  6. Paul C. Bell, Igor Potapov, and Pavel Semukhin. On the mortality problem: From multiplicative matrix equations to linear recurrence sequences and beyond. Information and Computation, 281:104736, 2021. URL: https://doi.org/10.1016/J.IC.2021.104736.
  7. Stuart J. Berkowitz. On computing the determinant in small parallel time using a small number of processors. Information Processing Letters, 18(3):147-150, 1984. URL: https://doi.org/10.1016/0020-0190(84)90018-8.
  8. 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.
  9. Jean Berstel, Dominique Perrin, and Christophe Reutenauer. Codes and automata, volume 129. Cambridge University Press, 2010. Google Scholar
  10. Vincent D. Blondel, Raphaël M. Jungers, and Alex Olshevsky. On primitivity of sets of matrices. Automatica, 61:80-88, 2015. URL: https://doi.org/10.1016/J.AUTOMATICA.2015.07.026.
  11. Allan Borodin, Joachim von zur Gathen, and John E. Hopcroft. Fast parallel matrix and GCD computations. Information and Control, 52(3):241-256, 1982. URL: https://doi.org/10.1016/S0019-9958(82)90766-5.
  12. Greg Budzban and Philip Feinsilver. The generalized road coloring problem and periodic digraphs. Applicable Algebra in Engineering, Communication and Computing, 22:21-35, 2011. URL: https://doi.org/10.1007/S00200-010-0135-Z.
  13. Georgina Bumpus, Christoph Haase, Stefan Kiefer, Paul-Ioan Stoienescu, and Jonathan Tanner. On the size of finite rational matrix semigroups. In Artur Czumaj, Anuj Dawar, and Emanuela Merelli, editors, 47th International Colloquium on Automata, Languages, and Programming, ICALP 2020, July 8-11, 2020, Saarbrücken, Germany (Virtual Conference), volume 168 of LIPIcs, pages 115:1-115:13. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPICS.ICALP.2020.115.
  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. Arturo Carpi and Flavio D'Alessandro. Strongly transitive automata and the Černý conjecture. Acta Informatica, 46(8):591-607, 2009. URL: https://doi.org/10.1007/S00236-009-0106-7.
  16. Julien Cassaigne, Vesa Halava, Tero Harju, and François Nicolas. Tighter undecidability bounds for matrix mortality, zero-in-the-corner problems, and more. CoRR, abs/1404.0644, 2014. URL: https://arxiv.org/abs/1404.0644.
  17. Yves Césari. Sur l'application du théorème de Suschkewitsch à l'étude des codes rationnels complets. In Jacques Loeckx, editor, Automata, Languages and Programming, 2nd Colloquium, University of Saarbrücken, Germany, July 29 - August 2, 1974, Proceedings, volume 14 of Lecture Notes in Computer Science, pages 342-350. Springer, 1974. URL: https://doi.org/10.1007/3-540-06841-4_73.
  18. Stephen A. Cook. A taxonomy of problems with fast parallel algorithms. Inf. Control., 64(1-3):2-21, 1985. URL: https://doi.org/10.1016/S0019-9958(85)80041-3.
  19. Laszlo Csanky. Fast parallel matrix inversion algorithms. SIAM Journal on Computing, 5(4):618-623, 1976. URL: https://doi.org/10.1137/0205040.
  20. David Eppstein. Reset sequences for monotonic automata. SIAM Journal on Computing, 19(3):500-510, 1990. URL: https://doi.org/10.1137/0219033.
  21. Joel Friedman. On the road coloring problem. Proceedings of the American Mathematical Society, 110(4):1133-1135, 1990. Google Scholar
  22. Balázs Gerencsér, Vladimir V. Gusev, and Raphaël M. Jungers. Primitive sets of nonnegative matrices and synchronizing automata. SIAM Journal on Matrix Analysis and Applications, 39(1):83-98, 2018. URL: https://doi.org/10.1137/16M1094099.
  23. Oded Goldreich. Computational Complexity: A Conceptual Perspective. Cambridge University Press, 2008. URL: https://doi.org/10.1017/CBO9780511804106.
  24. 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.
  25. Vladimir V. Gusev and Elena V. Pribavkina. On synchronizing colorings and the eigenvectors of digraphs. In Piotr Faliszewski, Anca Muscholl, and Rolf Niedermeier, editors, 41st International Symposium on Mathematical Foundations of Computer Science, MFCS 2016, August 22-26, 2016 - Kraków, Poland, volume 58 of LIPIcs, pages 48:1-48:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2016. URL: https://doi.org/10.4230/LIPICS.MFCS.2016.48.
  26. Markus Holzer and Sebastian Jakobi. On the computational complexity of problems related to distinguishability sets. Information and Computation, 259(2):225-236, 2018. URL: https://doi.org/10.1016/J.IC.2017.09.003.
  27. Oscar H. Ibarra, Shlomo Moran, and Louis E. Rosier. A note on the parallel complexity of computing the rank of order n matrices. Information Processing Letters, 11(4/5):162, 1980. URL: https://doi.org/10.1016/0020-0190(80)90042-3.
  28. Gérard Jacob. Un algorithme calculant le cardinal, fini ou infini, des demi-groupes de matrices. Theoretical Computer Science, 5(2):183-204, 1977. URL: https://doi.org/10.1016/0304-3975(77)90006-8.
  29. Jarkko Kari. A counter example to a conjecture concerning synchronizing words in finite automata. Bulletin of the EATCS, 73:146, 2001. Google Scholar
  30. Jarkko Kari, Andrew Ryzhikov, and Anton Varonka. Words of minimum rank in deterministic finite automata. In Piotrek Hofman and Michal Skrzypczak, editors, Developments in Language Theory - 23rd International Conference, DLT 2019, Warsaw, Poland, August 5-9, 2019, Proceedings, volume 11647 of Lecture Notes in Computer Science, pages 74-87. Springer, 2019. URL: https://doi.org/10.1007/978-3-030-24886-4_5.
  31. Nasim Karimi. Reaching the minimum ideal in a finite semigroup. Semigroup Forum, 94(2):390-425, 2017. Google Scholar
  32. Stefan Kiefer. Notes on equivalence and minimization of weighted automata. https://arxiv.org/abs/2009.01217, 2020. URL: https://arxiv.org/abs/arXiv:2009.01217.
  33. Stefan Kiefer, Ines Marusic, and James Worrell. Minimisation of multiplicity tree automata. Logical Methods in Computer Science, 13(1), 2017. URL: https://doi.org/10.23638/LMCS-13(1:16)2017.
  34. 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.
  35. Stefan Kiefer and Cas Widdershoven. Efficient analysis of unambiguous automata using matrix semigroup techniques. In Peter Rossmanith, Pinar Heggernes, and Joost-Pieter Katoen, editors, 44th International Symposium on Mathematical Foundations of Computer Science, MFCS 2019, August 26-30, 2019, Aachen, Germany, volume 138 of LIPIcs, pages 82:1-82:13. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/LIPICS.MFCS.2019.82.
  36. Stefan Kiefer and Cas Widdershoven. Efficient analysis of unambiguous automata using matrix semigroup techniques. CoRR, abs/1906.10093, 2019. URL: https://arxiv.org/abs/1906.10093.
  37. A.A. Klyachko, I.K. Rystsov, and M.A. Spivak. An extremal combinatorial problem associated with the bound on the length of a synchronizing word in an automaton. Cybernetics, 23(2):165-171, 1987. Google Scholar
  38. Douglas A. Lind and Brian Marcus. An introduction to symbolic dynamics and coding. Cambridge university press, 2021. Google Scholar
  39. Arnaldo Mandel and Imre Simon. On finite semigroups of matrices. Theoretical Computer Science, 5(2):101-111, 1977. URL: https://doi.org/10.1016/0304-3975(77)90001-9.
  40. Mike Paterson. Unsolvability in 3 × 3 matrices. Studies in Applied Mathematics, 49:105-107, 1970. Google Scholar
  41. Vladimir Yu. Protasov. Analytic methods for reachability problems. Journal of Computer and System Sciences, 120:1-13, 2021. URL: https://doi.org/10.1016/J.JCSS.2021.02.007.
  42. Vladimir Yu. Protasov and Andrey S. Voynov. Matrix semigroups with constant spectral radius. Linear Algebra and its Applications, 513:376-408, 2017. Google Scholar
  43. Igor Rystsov. Rank of a finite automaton. Cybernetics and Systems Analysis, 28(3):323-328, 1992. Google Scholar
  44. Igor K. Rystsov. Reset words for commutative and solvable automata. Theoretical Computer Science, 172(1-2):273-279, 1997. URL: https://doi.org/10.1016/S0304-3975(96)00136-3.
  45. Wojciech Rytter. The space complexity of the unique decipherability problem. Information Processing Letters, 23(1):1-3, 1986. URL: https://doi.org/10.1016/0020-0190(86)90121-3.
  46. 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.
  47. Andrew Ryzhikov. On shortest products for nonnegative matrix mortality. In Laura Kovács and Ana Sokolova, editors, Reachability Problems - 18th International Conference, RP 2024, Vienna, Austria, September 25-27, 2024, Proceedings, volume 15050 of Lecture Notes in Computer Science, pages 104-119. Springer, 2024. URL: https://doi.org/10.1007/978-3-031-72621-7_8.
  48. Sujin Shin and Jisang Yoo. A note on the rank of semigroups. Semigroup Forum, 81(2):335-343, 2010. Google Scholar
  49. 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
  50. Mikhail V. Volkov. Synchronization of finite automata. Russian Mathematical Surveys, 77(5):819-891, 2022. URL: https://doi.org/10.4213/rm10005e.
  51. 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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