Synchronizing Automata with Extremal Properties

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9234)


We present a few classes of synchronizing automata exhibiting certain extremal properties with regard to synchronization. The first is a series of automata with subsets whose shortest extending words are of length \(\varTheta (n^2)\), where n is the number of states of the automaton. This disproves a conjecture that every subset in a strongly connected synchronizing automaton is cn-extendable, for some constant c, and in particular, shows that the cubic upper bound on the length of the shortest reset words cannot be improved generally by means of the extension method. A detailed analysis shows that the automata in the series have subsets that require words as long as \(n^2/4+O(n)\) in order to be extended by at least one element.

We also discuss possible relaxations of the conjecture, and propose the image-extension conjecture, which would lead to a quadratic upper bound on the length of the shortest reset words. In this regard we present another class of automata, which turn out to be counterexamples to a key claim in a recent attempt to improve the Pin-Frankl bound for reset words.

Finally, we present two new series of slowly irreducibly synchronizing automata over a ternary alphabet, whose lengths of the shortest reset words are \(n^2-3n+3\) and \(n^2-3n+2\), respectively. These are the first examples of such series of automata for alphabets of size larger than two.


  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.
    Ananichev, D.S., Volkov, M.V., Gusev, V.V.: Primitive digraphs with large exponents and slowly synchronizing automata. J. Math. Sci. 192(3), 263–278 (2013)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Ananichev, D.S., Volkov, M.V., Zaks, Y.I.: Synchronizing automata with a letter of deficiency 2. In: Ibarra, O.H., Dang, Z. (eds.) DLT 2006. LNCS, vol. 4036, pp. 433–442. Springer, Heidelberg (2006) CrossRefGoogle Scholar
  4. 4.
    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. Foundations Comput. Sci. 22(2), 277–288 (2011)CrossRefMATHGoogle Scholar
  5. 5.
    Berlinkov, M., Szykuła, M.: Algebraic synchronization criterion and computing reset words. In: Italiano, G.F., et al (eds.) MFCS 2015. Lecture Notes in Computer Science, vol. 9234, pp. 103–115 (2015)Google Scholar
  6. 6.
    Berlinkov, M.V.: On a conjecture by Carpi and D’Alessandro. Int. J. Foundations Comput. Sci. 22(7), 1565–1576 (2011)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Berlinkov, M.V.: Synchronizing quasi-eulerian and quasi-one-cluster automata. Int. J. Foundations Comput. Sci. 24(6), 729–745 (2013)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Černý, J.: Poznámka k homogénnym eksperimentom s konečnými automatami. Matematicko-fyzikálny Čas. Slovenskej Akad. Vied 14(3), 208–216 (1964). in SlovakMATHGoogle Scholar
  9. 9.
    Dubuc, L.: Sur les automates circulaires et la conjecture de C̆erný. Informatique théorique et Appl. 32, 21–34 (1998). in FrenchMathSciNetGoogle Scholar
  10. 10.
    Gonze, F., Jungers, R.M., Trahtman, A.N.: A note on a recent attempt to improve the Pin-Frankl bound. Discrete Math. Theoret. Comput. Sci. 17(1), 307–308 (2015)MathSciNetGoogle Scholar
  11. 11.
    Gusev, V.V., Pribavkina, E.V.: Reset thresholds of automata with two cycle lengths. In: Holzer, M., Kutrib, M. (eds.) CIAA 2014. LNCS, vol. 8587, pp. 200–210. Springer, Heidelberg (2014) Google Scholar
  12. 12.
    Kari, J.: Synchronizing finite automata on Eulerian digraphs. Theoret. Comput. Sci. 295(1–3), 223–232 (2003)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Kari, J., Volkov, M.V.: Černý’s conjecture and the road coloring problem. In: Handbook of Automata. European Science Foundation (2013, to appear)Google Scholar
  14. 14.
    Kisielewicz, A., Kowalski, J., Szykuła, M.: Computing the shortest reset words of synchronizing automata. J. Comb. Optim. 29(1), 88–124 (2015)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Roman, A.: A note on Černý conjecture for automata over 3-letter alphabet. J. Automata Lang. Comb. 13(2), 141–143 (2008)MathSciNetMATHGoogle Scholar
  16. 16.
    Rystsov, I.K.: Quasioptimal bound for the length of reset words for regular automata. Acta Cybernetica 12(2), 145–152 (1995)MathSciNetMATHGoogle Scholar
  17. 17.
    Steinberg, B.: The averaging trick and the Černý conjecture. Int. J. Foundations Comput. Sci. 22(7), 1697–1706 (2011)CrossRefMATHGoogle Scholar
  18. 18.
    Steinberg, B.: The Černý conjecture for one-cluster automata with prime length cycle. Theoret. Comput. Sci. 412(39), 5487–5491 (2011)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Trahtman, A.N.: An efficient algorithm finds noticeable trends and examples concerning the Černy conjecture. In: Královič, R., Urzyczyn, P. (eds.) MFCS 2006. LNCS, vol. 4162, pp. 789–800. Springer, Heidelberg (2006) CrossRefGoogle Scholar
  20. 20.
    Trahtman, A.N.: Modifying the upper bound on the length of minimal synchronizing word. In: Owe, O., Steffen, M., Telle, J.A. (eds.) FCT 2011. LNCS, vol. 6914, pp. 173–180. Springer, Heidelberg (2011) CrossRefGoogle Scholar
  21. 21.
    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.Department of Mathematics and Computer ScienceUniversity of WrocławWrocławPoland

Personalised recommendations