Slowly Synchronizing Automata and Digraphs

  • Dmitry Ananichev
  • Vladimir Gusev
  • Mikhail Volkov
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6281)

Abstract

We present several infinite series of synchronizing automatafor which the minimum length of reset words is close to the square of the number of states. These automata are closely related to primitive digraphs with large exponent.

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References

  1. 1.
    Adler, R.L., Goodwyn, L.W., Weiss, B.: Equivalence of topological Markov shifts. Israel J. Math. 27, 49–63 (1977)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Almeida, J., Steinberg, B.: Matrix mortality and the Černý–Pin conjecture. In: Diekert, V., Nowotka, D. (eds.) Developments in Language Theory. LNCS, vol. 5583, pp. 67–80. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  3. 3.
    Almeida, M., Moreira, N., Reis, R.: Enumeration and generation with a string automata representation. Theor. Comput. Sci. 387, 93–102 (2007)MATHMathSciNetGoogle Scholar
  4. 4.
    Ananichev, D.S., Volkov, M.V., Zaks, Y.I.: Synchronizing automata with a letter of deficiency 2. Theor. Comput. Sci. 376, 30–41 (2007)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Černý, J.: Poznámka k homogénnym eksperimentom s konečnými automatami. Matematicko-fyzikalny Časopis Slovensk. Akad. Vied 14(3), 208–216 (1964) (in Slovak)MATHGoogle Scholar
  6. 6.
    Dubuc, L.: Sur les automates circulaires et la conjecture de Černý. RAIRO Inform. Théor. Appl. 32, 21–34 (1998) (in French)MathSciNetGoogle Scholar
  7. 7.
    Dulmage, A.L., Mendelsohn, N.S.: The exponent of a primitive matrix. Can. Math. Bull. 5, 241–244 (1962)MATHMathSciNetGoogle Scholar
  8. 8.
    Dulmage, A.L., Mendelsohn, N.S.: Gaps in the exponent set of primitive matrices. Ill. J. Math. 8, 642–656 (1964)MATHMathSciNetGoogle Scholar
  9. 9.
    Eppstein, D.: Reset sequences for monotonic automata. SIAM J. Comput. 19, 500–510 (1990)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Higgins, P.M.: The range order of a product of i transformations from a finite full transformation semigroup. Semigroup Forum 37, 31–36 (1988)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Kari, J.: Synchronizing finite automata on Eulerian digraphs. Theoret. Comput. Sci. 295, 223–232 (2003)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Pin, J.-E.: Le problème de la synchronization et la conjecture de Černý. Thèse de 3ème cycle. Université de Paris 6 (1978) (in French)Google Scholar
  13. 13.
    Pin, J.-E.: On two combinatorial problems arising from automata theory. Ann. Discrete Math. 17, 535–548 (1983)MATHGoogle Scholar
  14. 14.
    Ramírez Alfonsín, J.L.: The diophantine Frobenius problem. Oxford University Press, Oxford (2005)MATHCrossRefGoogle Scholar
  15. 15.
    Sandberg, S.: Homing and synchronizing sequences. In: Broy, M., et al. (eds.) Model-Based Testing of Reactive Systems. LNCS, vol. 3472, pp. 5–33. Springer, Heidelberg (2005)Google Scholar
  16. 16.
    Skvortsov, E., Yu, Z.: Synchronizing random automata. In: Rigo, M. (ed.) AutoMathA 2009, Université de Liège (2009) (submitted)Google Scholar
  17. 17.
    Trahtman, A.N.: An efficient algorithm finds noticeable trends and examples concerning the Černý conjecture. In: Královič, R., Urzyczyn, P. (eds.) MFCS 2006. LNCS, vol. 4162, pp. 789–800. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  18. 18.
    Trahtman, A.N.: Notable trends concerning the synchronization of graphs and automata. Electr. Notes Discrete Math. 25, 173–175 (2006)CrossRefMathSciNetGoogle Scholar
  19. 19.
    Trahtman, A.N.: The Černý conjecture for aperiodic automata. Discrete Math. Theor. Comput. Sci. 9(2), 3–10 (2007)MATHMathSciNetGoogle Scholar
  20. 20.
    Trahtman, A.N.: Some aspects of synchronization of DFA. J. Comput. Sci. Technol. 23, 719–727 (2008)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Trahtman, A.N.: The Road Coloring Problem. Israel J. Math. 172, 51–60 (2009)MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    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
  23. 23.
    Volkov, M.V.: Synchronizing automata preserving a chain of partial orders. Theoret. Comput. Sci. 410, 2992–2998 (2009)MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Wielandt, H.: Unzerlegbare, nicht negative Matrizen. Math. Z. 52, 642–648 (1950) (in German)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Dmitry Ananichev
    • 1
  • Vladimir Gusev
    • 1
  • Mikhail Volkov
    • 1
  1. 1.Department of Mathematics and MechanicsUral State UniversityEkaterinburgRussia

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