Synchronizing Automata with Random Inputs

(Short Paper)
  • Vladimir V. Gusev
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8633)


We study the problem of synchronization of automata with random inputs. We present a series of automata such that the expected number of steps until synchronization is exponential in the number of states. At the same time, we show that the expected number of letters to synchronize any pair of the famous Černý automata is at most cubic in the number of states.


Transition Function Expected Number Random Input Sink State Successive Order 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ananichev, D.S., Gusev, V.V., Volkov, M.V.: Primitive digraphs with large exponents and slowly synchronizing automata. Journal of Mathematical Sciences (US) 192(3), 263–278 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Berlinkov, M.V.: On the probability of being synchronizable (2013), ArXiv:
  3. 3.
    Černý, J.: Poznámka k homogénnym eksperimentom s konečnými automatami. Matematicko-fyzikalny Časopis Slovensk. Akad. Vied 14(3), 208–216 (1964) (in Slovak) Google Scholar
  4. 4.
    Gusev, V.V.: Lower bounds for the length of reset words in eulerian automata. Int. J. Found. Comput. Sci. 24(2), 251–262 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Jungers, R.M.: The synchronizing probability function of an automaton. SIAM J. Discret. Math. 26(1), 177–192 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Kisielewicz, A., Kowalski, J., Szykuła, M.: A Fast Algorithm Finding the Shortest Reset Words. In: Du, D.-Z., Zhang, G. (eds.) COCOON 2013. LNCS, vol. 7936, pp. 182–196. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  7. 7.
    Pin, J.-E.: On two combinatorial problems arising from automata theory. Ann. Discrete Math. 17, 535–548 (1983)zbMATHGoogle Scholar
  8. 8.
    Privault, N.: Understanding Markov Chains. Springer (2013)Google Scholar
  9. 9.
    Sandberg, S.: Homing and synchronizing sequences. In: Broy, M., Jonsson, B., Katoen, J.-P., Leucker, M., Pretschner, A. (eds.) Model-Based Testing of Reactive Systems. LNCS, vol. 3472, pp. 5–33. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  10. 10.
    Skvortsov, E.S., Zaks, Y.: Synchronizing random automata. Discr. Math. and Theor. Comp. Sci. 12(4), 95–108 (2010)zbMATHMathSciNetGoogle Scholar
  11. 11.
    Skvortsov, E., Tipikin, E.: Experimental study of the shortest reset word of random automata. In: Bouchou-Markhoff, B., Caron, P., Champarnaud, J.-M., Maurel, D. (eds.) CIAA 2011. LNCS, vol. 6807, pp. 290–298. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  12. 12.
    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
  13. 13.
    Volkov, M.V.: Synchronizing automata preserving a chain of partial orders. Theoret. Comput. Sci. 410, 2992–2998 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Zaks, Y.I., Skvortsov, E.S.: Synchronizing random automata on a 4-letter alphabet. Journal of Mathematical Sciences 192(3), 303–306 (2013)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Vladimir V. Gusev
    • 1
  1. 1.Institute of Mathematics and Computer ScienceUral Federal UniversityEkaterinburgRussia

Personalised recommendations