Advertisement

Brzozowski Algorithm Is Generically Super-Polynomial for Deterministic Automata

  • Sven De Felice
  • Cyril Nicaud
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7907)

Abstract

We study the number of states of the minimal automaton of the mirror of a rational language recognized by a random deterministic automaton with n states. We prove that, for any d > 0, the probability that this number of states is greater than n d tends to 1 as n tends to infinity. As a consequence, the generic and average complexities of Brzozowski minimization algorithm are super-polynomial for the uniform distribution on deterministic automata.

Keywords

Random Permutation Accessible State Combinatorial Explosion Average Complexity Small Cycle 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bassino, F., David, J., Nicaud, C.: Average case analysis of Moore’s state minimization algorithm. Algorithmica 63(1-2), 509–531 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Bassino, F., David, J., Sportiello, A.: Asymptotic enumeration of minimal automata. In: Dürr, Wilke (eds.) [9], pp. 88–99Google Scholar
  3. 3.
    Bassino, F., Giambruno, L., Nicaud, C.: The average state complexity of rational operations on finite languages. International Journal of Foundations of Computer Science 21(4), 495–516 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Bassino, F., Nicaud, C.: Enumeration and random generation of accessible automata. Theor. Comput. Sci. 381(1-3), 86–104 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Brzozowski, J.A.: Canonical regular expressions and minimal state graphs for definite events. In: Mathematical Theory of Automata. MRI Symposia Series, vol. 12, pp. 529–561. Polytechnic Press, Polytechnic Institute of Brooklyn, N.Y (1962)Google Scholar
  6. 6.
    Carayol, A., Nicaud, C.: Distribution of the number of accessible states in a random deterministic automaton. In: Dürr, Wilke (eds.) [9], pp. 194–205Google Scholar
  7. 7.
    Chassaing, P., Azad, E.Z.: Asymptotic behavior of some factorizations of random words (2010), arXiv:1004.4062v1Google Scholar
  8. 8.
    David, J.: Average complexity of Moore’s and Hopcroft’s algorithms. Theor. Comput. Sci. 417, 50–65 (2012)zbMATHCrossRefGoogle Scholar
  9. 9.
    Dürr, C., Wilke, T. (eds.): 29th International Symposium on Theoretical Aspects of Computer Science, STACS 2012, Paris, France, February 29 - March 3. LIPIcs, vol. 14. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2012)Google Scholar
  10. 10.
    Erdős, P., Turán, P.: On some problems of a statistical group-theory I. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 4, 175–186 (1965)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Flajolet, P., Odlyzko, A.M.: Random mapping statistics. In: Quisquater, J.-J., Vandewalle, J. (eds.) EUROCRYPT 1989. LNCS, vol. 434, pp. 329–354. Springer, Heidelberg (1990)CrossRefGoogle Scholar
  12. 12.
    Flajolet, P., Sedgewick, R.: Analytic Combinatorics. Cambridge University Press (2009)Google Scholar
  13. 13.
    Hopcroft, J.E.: An n logn algorithm for minimizing the states in a finite automaton. In: Kohavi, Z. (ed.) The Theory of Machines and Computations, pp. 189–196. Academic Press (1971)Google Scholar
  14. 14.
    Korshunov, A.: Enumeration of finite automata. Problemy Kibernetiki 34, 5–82 (1978) (in Russian)Google Scholar
  15. 15.
    Landau, E.: Handbuch der lehre von der verteilung der primzahlen, vol. 2. B. G. Teubner (1909)Google Scholar
  16. 16.
    Nicaud, C.: Average state complexity of operations on unary automata. In: Kutylowski, M., Pacholski, L., Wierzbicki, T. (eds.) MFCS 1999. LNCS, vol. 1672, pp. 231–240. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  17. 17.
    Tabakov, D., Vardi, M.Y.: Experimental evaluation of classical automata constructions. In: Sutcliffe, G., Voronkov, A. (eds.) LPAR 2005. LNCS (LNAI), vol. 3835, pp. 396–411. Springer, Heidelberg (2005)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Sven De Felice
    • 1
  • Cyril Nicaud
    • 1
  1. 1.LIGMUniversité Paris-Est & CNRSMarne-la-Vallée Cedex 2France

Personalised recommendations