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On Finite Monoids over Nonnegative Integer Matrices and Short Killing Words

Authors Stefan Kiefer, Corto Mascle

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Subject Classification

ACM Subject Classification
  • Theory of computation → Formal languages and automata theory
Keywords
  • matrix semigroups
  • unambiguous automata
  • codes
  • Restivo’s conjecture

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Abstract

Let n be a natural number and M a set of n x n-matrices over the nonnegative integers such that M generates a finite multiplicative monoid. We show that if the zero matrix 0 is a product of matrices in M, then there are M_1, ..., M_{n^5} in M with M_1 *s M_{n^5} = 0. This result has applications in automata theory and the theory of codes. Specifically, if X subset Sigma^* is a finite incomplete code, then there exists a word w in Sigma^* of length polynomial in sum_{x in X} |x| such that w is not a factor of any word in X^*. This proves a weak version of Restivo’s conjecture.

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Stefan Kiefer and Corto Mascle. On Finite Monoids over Nonnegative Integer Matrices and Short Killing Words. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 43:1-43:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.STACS.2019.43

Author Details

Stefan Kiefer
  • University of Oxford, UK
Corto Mascle
  • ENS Paris-Saclay, France

Funding

  • Kiefer, Stefan: Work supported by a Royal Society University Research Fellowship.

References

  1. P.C. Bell, M. Hirvensalo, and I. Potapov. Mortality for 2 times2 Matrices Is NP-Hard. In Proceedings of Mathematical Foundations of Computer Science (MFCS), pages 148-159. Springer, 2012. Google Scholar
  2. J. Berstel and D. Perrin. Trends in the theory of codes. Bulletin of the EATCS, 29:84-95, 1986. Google Scholar
  3. J. Berstel, D. Perrin, and C. Reutenauer. Codes and Automata. Cambridge University Press, 2009. Google Scholar
  4. A. Carpi. On synchronizing unambiguous automata. Theoretical Computer Science, 60(3):285-296, 1988. Google Scholar
  5. A. Carpi and F. D'Alessandro. On incomplete and synchronizing finite sets. Theoretical Computer Science, 664:67-77, 2017. Google Scholar
  6. P. Goralčík, Z. Hedrlín, V. Koubek, and J. Ryšlinková. A game of composing binary relations. R.A.I.R.O. Informatique théorique, 16(4):365-369, 1982. Google Scholar
  7. V.V. Gusev and E.V. Pribavkina. On Non-complete Sets and Restivo’s Conjecture. In Proceedings of Developments in Language Theory (DLT), pages 239-250. Springer, 2011. Google Scholar
  8. V. Halava, T. Harju, and M. Hirvensalo. Undecidability Bounds for Integer Matrices Using Claus Instances. International Journal of Foundations of Computer Science, 18(5):931-948, 2007. Google Scholar
  9. R.A. Horn and C.R. Johnson. Matrix analysis. Cambridge University Press, 2nd edition, 2013. Google Scholar
  10. S. Julia, A. Malapert, and J. Provillard. A Synergic Approach to the Minimal Uncompletable Words Problem. Journal of Automata, Languages and Combinatorics, 22(4):271-286, 2017. Google Scholar
  11. R.M. Jungers, V. Protasov, and V.D. Blondel. Efficient algorithms for deciding the type of growth of products of integer matrices. Linear Algebra and its Applications, 428(10):2296-2311, 2008. Google Scholar
  12. J.-Y. Kao, N. Rampersad, and J. Shallit. On NFAs where all states are final, initial, or both. Theoretical Computer Science, 410(47):5010-5021, 2009. Google Scholar
  13. P.V. Martugin. A series of slowly synchronizing automata with a zero state over a small alphabet. Information and Computation, 206(9-10):1197-1203, 2008. Google Scholar
  14. M.S. Paterson. Unsolvability in 3 times 3 Matrices. Studies in Applied Mathematics, 49(1):105-107, 1970. Google Scholar
  15. I. Potapov and P. Semukhin. Decidability of the Membership Problem for 2 times 2 integer matrices. In Proceedings of the Symposium on Discrete Algorithms (SODA), pages 170-186. SIAM, 2017. Google Scholar
  16. A. Restivo. Some remarks on complete subsets of a free monoid. In Quaderni de "La Ricerca Scientifica". Non-Commutative Structures in Algebra and Geometric Combinatorics, volume 109, pages 19-25. Consiglio Nazionale Delle Ricerche, 1981. Google Scholar
  17. A. Ryzhikov and M. Szykuła. Finding Short Synchronizing Words for Prefix Codes. In Proceedings of Mathematical Foundations of Computer Science (MFCS), volume 117 of LIPIcs, pages 21:1-21:14, 2018. Google Scholar
  18. M.-P. Schützenberger. On the definition of a family of automata. Information and Control, 4:245-270, 1961. Google Scholar
  19. W. Tzeng. A Polynomial-Time Algorithm for the Equivalence of Probabilistic Automata. SIAM Journal on Computing, 21(2):216-227, 1992. Google Scholar
  20. M.V. Volkov. Synchronizing Automata and the Černý Conjecture. In Proceedings of Language and Automata Theory and Applications (LATA), pages 11-27. Springer, 2008. Google Scholar
  21. A. Weber and H. Seidl. On finitely generated monoids of matrices with entries in ℕ. Informatique Théorique et Applications, 25(1):19-38, 1991. Google Scholar
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