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Algebraic Synchronization Criterion and Computing Reset Words

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Mathematical Foundations of Computer Science 2015 (MFCS 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9234))

Abstract

We refine results about relations between Markov chains and synchronizing automata. We express the condition that an automaton is synchronizing in terms of linear algebra, and obtain upper bounds for the reset thresholds of automata with a short word of a small rank. The results are applied to make several improvements in the area.

We improve the best general upper bound for reset thresholds of finite prefix codes (Huffman codes): we show that an n-state synchronizing decoder has a reset word of length at most \(O(n \log ^3 n)\). Also, we prove the Černý conjecture for n-state automata with a letter of rank at most \(\root 3 \of {6n-6}\). In another corollary, based on the recent results of Nicaud, we show that the probability that the Černý conjecture does not hold for a random synchronizing binary automaton is exponentially small in terms of the number of states. It follows that the expected value of the reset threshold of an n-state random synchronizing binary automaton is at most \(n^{7/4+o(1)}\).

Moreover, reset words of the lengths within our bounds are computable in polynomial time. We present suitable algorithms for this task for various classes of automata for which our results can be applied. These include (quasi-)one-cluster and (quasi-)Eulerian automata.

M. Berlinkov—Supported by the Presidential Program “Leading Scientific Schools of the Russian Federation”, project no. 5161.2014.1, the Russian Foundation for Basic Research, project no. 13-01-00852, the Ministry of Education and Science of the Russian Federation, project no. 1.1999.2014/K, and the Competitiveness Program of Ural Federal University.

M. Szykuła—Supported in part by Polish NCN grant DEC-2013/09/N/ST6/01194.

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References

  1. Ananichev, D., Gusev, V., Volkov, M.: Slowly synchronizing automata and digraphs. In: Hliněný, P., Kučera, A. (eds.) MFCS 2010. LNCS, vol. 6281, pp. 55–65. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  2. Béal, M.P., Berlinkov, M.V., Perrin, D.: A quadratic upper bound on the size of a synchronizing word in one-cluster automata. Int. J. Found. Comput. Sci. 22(2), 277–288 (2011)

    Article  MATH  Google Scholar 

  3. Béal, M.-P., Perrin, D.: A quadratic upper bound on the size of a synchronizing word in one-cluster automata. In: Diekert, V., Nowotka, D. (eds.) DLT 2009. LNCS, vol. 5583, pp. 81–90. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  4. Berlinkov, M.V.: On the probability to be synchronizable (2013). http://arxiv.org/abs/1304.5774

  5. Berlinkov, M.V.: Synchronizing quasi-eulerian and quasi-one-cluster automata. Int. J. Found. Comput. Sci. 24(6), 729–745 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  6. Berlinkov, M.V.: On two algorithmic problems about synchronizing automata. In: Shur, A.M., Volkov, M.V. (eds.) DLT 2014. LNCS, vol. 8633, pp. 61–67. Springer, Heidelberg (2014)

    Google Scholar 

  7. Berlinkov, M.V., Szykuła, M.: Algebraic synchronization criterion and computing reset words (2014). http://arxiv.org/abs/1412.8363

  8. Biskup, M.T.: Shortest synchronizing strings for huffman codes. In: Ochmański, E., Tyszkiewicz, J. (eds.) MFCS 2008. LNCS, vol. 5162, pp. 120–131. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  9. Biskup, M.T., Plandowski, W.: Shortest synchronizing strings for huffman codes. Theoret. Comput. Sci. 410(38–40), 3925–3941 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. Carpi, A., D’Alessandro, F.: Independent sets of words and the synchronization problem. Adv. Appl. Math. 50(3), 339–355 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  11. Carpi, A., D’Alessandro, F.: Černý-like problems for finite sets of words. In: Proceedings of the 15th Italian Conference on Theoretical Computer Science, Perugia, Italy, September 17–19, 2014. pp. 81–92 (2014)

    Google Scholar 

  12. Černý, J.: Poznámka k homogénnym eksperimentom s konečnými automatami. Matematicko-fyzikálny Časopis Slovenskej Akadémie Vied 14(3), 208–216 (1964)

    MATH  Google Scholar 

  13. Dubuc, L.: Sur les automates circulaires et la conjecture de C̆erný. Informatique Théorique et Applications 32, 21–34 (1998)

    MathSciNet  Google Scholar 

  14. Eppstein, D.: Reset sequences for monotonic automata. SIAM J. Comput. 19, 500–510 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  15. Gawrychowski, P., Straszak, D.: Strong inapproximability of the shortest reset word. In: Italiano, G.F., et al. (eds.) MFCS 2015, Part I, LNCS 9234, pp. 243–255. Springer, Heidelberg (2015)

    Google Scholar 

  16. Gerbush, M., Heeringa, B.: Approximating minimum reset sequences. In: Domaratzki, M., Salomaa, K. (eds.) CIAA 2010. LNCS, vol. 6482, pp. 154–162. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  17. Jürgensen, H.: Synchronization. Inform. Comput. 206(9–10), 1033–1044 (2008)

    Article  MATH  Google Scholar 

  18. Kari, J.: Synchronizing finite automata on Eulerian digraphs. Theoret. Comput. Sci. 295(1–3), 223–232 (2003)

    Article  MathSciNet  Google Scholar 

  19. Kari, J., Volkov, M.V.: Černý’s conjecture and the road coloring problem. In: Handbook of Automata. European Science Foundation (to appear)

    Google Scholar 

  20. Nicaud, C.: Fast synchronization of random automata (2014). http://arxiv.org/abs/1404.6962

  21. Olschewski, J., Ummels, M.: The complexity of finding reset words in finite automata. In: Hliněný, P., Kučera, A. (eds.) MFCS 2010. LNCS, vol. 6281, pp. 568–579. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  22. Pin, J.E.: Utilisation de l’algèbre linéaire en théorie des automates. In: Act. Collouq. AFCET-SMF Math. Appl. II. pp. 85–92. AFCET (1978)

    Google Scholar 

  23. Pin, J.E.: Sur un cas particulier de la conjecture de Černý. Automata, Languages and Programming. LNCS, pp. 345–352. Springer, Heidelberg (1978). in French

    Chapter  Google Scholar 

  24. Pin, J.E.: On two combinatorial problems arising from automata theory. In: Proceedings of the International Colloquium on Graph Theory and Combinatorics, vol. 75, pp. 535–548. North-Holland Mathematics Studies (1983)

    Google Scholar 

  25. Steinberg, B.: The averaging trick and the Černý conjecture. Int. J. Found. Comput. Sci. 22(7), 1697–1706 (2011)

    Article  MATH  Google Scholar 

  26. Steinberg, B.: The Černý conjecture for one-cluster automata with prime length cycle. Theoret. Comput. Sci. 412(39), 5487–5491 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  27. Volkov, M.V.: Synchronizing automata and the Černý conjecture. In: Martín-Vide, C., Otto, F., Fernau, H. (eds.) LATA 2008. LNCS, vol. 5196, pp. 11–27. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics and Computer Science, Ural Federal University, Yekaterinburg, Russia

    Mikhail Berlinkov

  2. Institute of Computer Science, University of Wrocław, Wrocław, Poland

    Marek Szykuła

Authors
  1. Mikhail Berlinkov
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  2. Marek Szykuła
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Correspondence to Mikhail Berlinkov .

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Editors and Affiliations

  1. Università di Roma "Tor Vergata", Rome, Italy

    Giuseppe F Italiano

  2. Università degli Studi di Milano, Milan, Italy

    Giovanni Pighizzini

  3. University of Edinburgh, Edinburgh, United Kingdom

    Donald T. Sannella

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Berlinkov, M., Szykuła, M. (2015). Algebraic Synchronization Criterion and Computing Reset Words. In: Italiano, G., Pighizzini, G., Sannella, D. (eds) Mathematical Foundations of Computer Science 2015. MFCS 2015. Lecture Notes in Computer Science(), vol 9234. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48057-1_8

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  • DOI: https://doi.org/10.1007/978-3-662-48057-1_8

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