Algebraic Synchronization Criterion and Computing Reset Words

  • Mikhail Berlinkov
  • Marek Szykuła
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9234)


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.


Markov Chain Polynomial Algorithm Short Word Huffman Code Deterministic Finite Automaton 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 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) CrossRefGoogle Scholar
  2. 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)CrossRefzbMATHGoogle Scholar
  3. 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) CrossRefGoogle Scholar
  4. 4.
    Berlinkov, M.V.: On the probability to be synchronizable (2013).
  5. 5.
    Berlinkov, M.V.: Synchronizing quasi-eulerian and quasi-one-cluster automata. Int. J. Found. Comput. Sci. 24(6), 729–745 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 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. 7.
    Berlinkov, M.V., Szykuła, M.: Algebraic synchronization criterion and computing reset words (2014).
  8. 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) CrossRefGoogle Scholar
  9. 9.
    Biskup, M.T., Plandowski, W.: Shortest synchronizing strings for huffman codes. Theoret. Comput. Sci. 410(38–40), 3925–3941 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Carpi, A., D’Alessandro, F.: Independent sets of words and the synchronization problem. Adv. Appl. Math. 50(3), 339–355 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 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. 12.
    Černý, J.: Poznámka k homogénnym eksperimentom s konečnými automatami. Matematicko-fyzikálny Časopis Slovenskej Akadémie Vied 14(3), 208–216 (1964)zbMATHGoogle Scholar
  13. 13.
    Dubuc, L.: Sur les automates circulaires et la conjecture de C̆erný. Informatique Théorique et Applications 32, 21–34 (1998)MathSciNetGoogle Scholar
  14. 14.
    Eppstein, D.: Reset sequences for monotonic automata. SIAM J. Comput. 19, 500–510 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 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. 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) CrossRefGoogle Scholar
  17. 17.
    Jürgensen, H.: Synchronization. Inform. Comput. 206(9–10), 1033–1044 (2008)CrossRefzbMATHGoogle Scholar
  18. 18.
    Kari, J.: Synchronizing finite automata on Eulerian digraphs. Theoret. Comput. Sci. 295(1–3), 223–232 (2003)MathSciNetCrossRefGoogle Scholar
  19. 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. 20.
    Nicaud, C.: Fast synchronization of random automata (2014).
  21. 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) CrossRefGoogle Scholar
  22. 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. 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 CrossRefGoogle Scholar
  24. 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. 25.
    Steinberg, B.: The averaging trick and the Černý conjecture. Int. J. Found. Comput. Sci. 22(7), 1697–1706 (2011)CrossRefzbMATHGoogle Scholar
  26. 26.
    Steinberg, B.: The Černý conjecture for one-cluster automata with prime length cycle. Theoret. Comput. Sci. 412(39), 5487–5491 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 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) CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Institute of Mathematics and Computer ScienceUral Federal UniversityYekaterinburgRussia
  2. 2.Institute of Computer ScienceUniversity of WrocławWrocławPoland

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