A Quadratic Upper Bound on the Size of a Synchronizing Word in One-Cluster Automata

  • Marie-Pierre Béal
  • Dominique Perrin
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5583)

Abstract

Černý’s conjecture asserts the existence of a synchronizing word of length at most (n − 1)2 for any synchronized n-state deterministic automaton. We prove a quadratic upper bound on the length of a synchronizing word for any synchronized n-state deterministic automaton satisfying the following additional property: there is a letter a such that for any pair of states p,q, one has p ·ar = q ·as for some integers r,s (for a state p and a word w, we denote by p ·w the state reached from p by the path labeled w). As a consequence, we show that for any finite synchronized prefix code with an n-state decoder, there is a synchronizing word of length O(n2). This applies in particular to Huffman codes.

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References

  1. 1.
    Béal, M.-P.: A note on Černý’s conjecture and rational series. preprint IGM 2003-05 (unpublished, 2003)Google Scholar
  2. 2.
    Béal, M.-P., Perrin, D.: A quadratic algorithm for road coloring. CoRR, abs/0803.0726 (2008)Google Scholar
  3. 3.
    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
  4. 4.
    Carpi, A., D’Alessandro, F.: The synchronization problem for strongly transitive automata. In: Developments in Language Theory, pp. 240–251 (2008)Google Scholar
  5. 5.
    Černý, J., Poznámka, K.: Homogénnym experimentom s konecnými automatmi. Mat. fyz. čas SAV 14, 208–215 (1964)Google Scholar
  6. 6.
    Dubuc, L.: Sur les automates circulaires et la conjecture de Černý. RAIRO Inform. Théor. Appl. 32, 21–34 (1998)MathSciNetGoogle Scholar
  7. 7.
    Eilenberg, S.: Automata, languages, and machines, vol. A. Academic Press, A subsidiary of Harcourt Brace Jovanovich, Publishers, New York (1974); Pure and Applied Mathematics, vol. 58MATHGoogle Scholar
  8. 8.
    Freiling, C.F., Jungreis, D.S., Théberge, F., Zeger, K.: Almost all complete binary prefix codes have a self-synchronizing string. IEEE Transactions on Information Theory 49, 2219–2225 (2003)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Kari, J.: A counter example to a conjecture concerning synchronizing words in finite automata. EATCS Bulletin 73, 146 (2001)MathSciNetMATHGoogle Scholar
  10. 10.
    Kari, J.: Synchronizing finite automata on eulerian digraphs. In: Sgall, J., Pultr, A., Kolman, P. (eds.) MFCS 2001. LNCS, vol. 2136, pp. 432–438. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  11. 11.
    Perrin, D., Schützenberger, M.-P.: Synchronizing words and automata and the road coloring problem, in Symbolic Dynamics and its Applications. In: Walters, P. (ed.) American Mathematical Society, vol. 135, pp. 295–318. Contemporary Mathematics (1992)Google Scholar
  12. 12.
    Pin, J.-E.: Le problème de la synchronisation et la conjecture de Černý, thèse de 3ème cycle, Université Paris VI (1978)Google Scholar
  13. 13.
    Pin, J.-E.: Sur un cas particulier de la conjecture de Černý. In: Ausiello, G., Böhm, C. (eds.) ICALP 1978. LNCS, vol. 62, Springer, Heidelberg (1978)Google Scholar
  14. 14.
    Pin, J.-E.: On two combinatorial problems arising from automata theory. Annals of Discrete Mathematics, vol. 17, pp. 535–548 (1983)Google Scholar
  15. 15.
    Sakarovitch, J.: Éléments de théorie des automates, Éditions Vuibert (2003)Google Scholar
  16. 16.
    Schützenberger, M.-P.: On synchronizing prefix codes. Inform. and Control 11, 396–401 (1967)CrossRefMATHGoogle Scholar
  17. 17.
    Trahtman, A.N.: Synchronization of some DFA. In: Cai, J.-Y., Cooper, S.B., Zhu, H. (eds.) TAMC 2007. LNCS, vol. 4484, pp. 234–243. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  18. 18.
    Trahtman, A.N.: The road coloring problem. Israel J. Math (to appear) (2008)Google Scholar
  19. 19.
    Trakhtman, A.: Some aspects of synchronization of DFA. J. Comput. Sci. Technol. 23, 719–727 (2008)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Volkov, M.V.: Synchronizing automata preserving a chain of partial orders. In: Holub, J., Žďárek, J. (eds.) CIAA 2007. LNCS, vol. 4783, pp. 27–37. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  21. 21.
    Volkov, M.V.: Synchronizing automata and the Černy 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 2009

Authors and Affiliations

  • Marie-Pierre Béal
    • 1
  • Dominique Perrin
    • 1
  1. 1.Université Paris-Est, LIGM CNRSMarne-la-Vallée Cedex 2France

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