Journal of Mathematical Sciences

, Volume 192, Issue 3, pp 263–278 | Cite as

Primitive digraphs with large exponents and slowly synchronizing automata


We present several infinite series of synchronozing automata for which the minimum length of reset words is close to the square of the number of states. All these automata are tightly related to primitive digraphs with large exponents. Bibliography: 28 titles.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    R. L. Adler, L. W. Goodwyn, and B. Weiss, “Equivalence of topological Markov shifts,” Israel J. Math., 27, 49–63 (1977).MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    M. Almeida, N. Moreira, and R. Reis, “Enumeration and generation with a string automata representation,” Theoret. Comput. Sci., 387, 93–102 (2007).MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    D. S. Ananichev, V. V. Gusev, and M. V. Volkov, “Slowly synchronizing automata and digraphs,” Lect. Notes Comput. Sci., 6281, 55–64 (2010).MathSciNetCrossRefGoogle Scholar
  4. 4.
    D. S. Ananichev, M. V. Volkov, and Yu. I. Zaks, “Synchronizing automata with a letter of deficiency 2,” Theoret. Comput. Sci., 376, 30–41 (2007).MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    M. Berlinkov, “Approximating the minimum length of synchronizing words is hard,” Lect. Notes Comput. Sci., 6072, 37–47 (2010).CrossRefGoogle Scholar
  6. 6.
    M. Berlinkov, “On a conjecture by Carpi and D'Alessandro,” Internat. J. Found. Comput. Sci., 22, 1565–1576 (2011).MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    R. Brualdi and H. Ryser, Combinatorial Matrix Theory, Cambridge Univ. Press, Cambridge (1991).MATHCrossRefGoogle Scholar
  8. 8.
    A. Carpi and F. D'Alessandro, “On the hybrid Černý–road coloring problem and Hamiltonian paths,” Lect. Notes Comput. Sci., 6224, 124–135 (2010).MathSciNetCrossRefGoogle Scholar
  9. 9.
    J. Černý, “Poznámka k homogénnym eksperimentom s konečnými automatami,” Matematicko-fyzikalny Časopis Slovensk. Akad. Vied”, 14, No. 3, 208–216 (1964).MATHGoogle Scholar
  10. 10.
    A. L. Dulmage and N. S. Mendelsohn, “The exponent of a primitive matrix,” Canad. Math. Bull., 5, 241–244 (1962).MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    A. L. Dulmage and N. S. Mendelsohn, “Gaps in the exponent set of primitive matrices,” Illinois J. Math., 8, 642–656 (1964).MathSciNetMATHGoogle Scholar
  12. 12.
    V. Gusev, “Lower bounds for the length of reset words in Eulerian automata,” Lect. Notes Comput. Sci., 6945, 180–190 (2011).CrossRefGoogle Scholar
  13. 13.
    J. Kari, “A counter example to a conjecture concerning synchronizing words in finite automata,” Bull. Eur. Assoc. Theoret. Comput. Sci., 73, 146 (2001).MathSciNetMATHGoogle Scholar
  14. 14.
    V. A. Liskovets, “The number of connected initial automata,” Kibernetika, No. 3, 16–19 (1969).Google Scholar
  15. 15.
    J. Olschewski and M. Ummels, “The complexity of finding reset words in finite automata,” Lect. Notes Comput. Sci., 6281, 568–579 (2010).MathSciNetCrossRefGoogle Scholar
  16. 16.
    J.-E. Pin, “On two combinatorial problems arising from automata theory,” Ann. Discrete Math., 17, 535–548 (1983).MATHGoogle Scholar
  17. 17.
    J. L. Ramírez Alfonsín, The Diophantine Frobenius Problem, Oxford Univ. Press, Oxford (2005).MATHCrossRefGoogle Scholar
  18. 18.
    V. N. Sachkov and V. E. Tarakanov, Combinatorics of Nonnegative Matrices, Amer. Math. Soc., Providence, Rhode Island (2002).Google Scholar
  19. 19.
    S. Sandberg, “Homing and synchronizing sequences,” Lect. Notes Comput. Sci., 3472, 5–33 (2005).CrossRefGoogle Scholar
  20. 20.
    E. Skvortsov and E. Tipikin, “Experimental study of the shortest reset word of random automata,”Lect. Notes Comput. Sci., 6807, 290–298 (2011).MathSciNetCrossRefGoogle Scholar
  21. 21.
    B. Steinberg, “The Černý conjecture for one-cluster automata with prime length cycle,” Theoret. Comput. Sci., 412, 5487–5491 (2011).MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    A. N. Trahtman, “An efficient algorithm finds noticeable trends and examples concerning the Černý conjecture,” Lect. Notes Comput. Sci., 4162, 789–800 (2006).MathSciNetCrossRefGoogle Scholar
  23. 23.
    A. N. Trahtman, “Notable trends concerning the synchronization of graphs and automata,” Electron. Notes Discrete Math., 25, 173–175 (2006).MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    A. N. Trahtman, “The road coloring problem,” Israel J. Math., 172, 51–60 (2009).MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    A. N. Trahtman, “Modifying the upper bound on the length of minimal synchronizing word,” Lect. Notes Comput. Sci, 6914, 173–180 (2011).MathSciNetCrossRefGoogle Scholar
  26. 26.
    M. V. Volkov, “Synchronizing automata and the Černý conjecture,” Lect. Notes Comput. Sci., 5196, 11–27 (2008).CrossRefGoogle Scholar
  27. 27.
    M. V. Volkov, “Synchronizing automata preserving a chain of partial orders,” Theoret. Comput. Sci., 410, 3513–3519 (2009).MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    H. Wielandt, “Unzerlegbare, nicht negative Matrizen,” Math. Z., 52, 642–648 (1950).MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Institute of Mathematics and Computer ScienceUral Federal UniversityEkaterinburgRussia

Personalised recommendations