Advertisement

On the Size of the Universal Automaton of a Regular Language

  • Sylvain Lombardy
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4393)

Abstract

The universal automaton of a regular language is the maximal NFA without merging states that recognizes this language. This automaton is directly inspired by the factor matrix defined by Conway thirty years ago. We prove in this paper that a tight bound on its size with respect to the size of the smallest equivalent NFA is given by Dedekind’s numbers. At the end of the paper, we deal with the unary case. Chrobak has proved that the size of the minimal deterministic automaton with respect to the smallest NFA is tightly bounded by the Landau’s function; we show that the size of the universal automaton is in this case an exponential of the Landau’s function.

Keywords

Regular languages Universal Automaton NFA Minimization. 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Câmpeanu, C., Sântean, N., Yu, S.: Mergible states in large nfa. Theoret. Comput. Sci. 330(1), 23–34 (2005)CrossRefMathSciNetzbMATHGoogle Scholar
  2. 2.
    Champarnaud, J.-M., Coulon, F.: NFA reduction algorithms by means of regular inequalities. Theoret. Comput. Sci. 327(3), 241–253 (2004)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Chrobak, M.: Finite automata and unary languages (Errata in Theoret. Comput. Sci. 302, 497–498 (2003)). Theoret. Comput. Sci. 47 47(2), 497–498 (1986)MathSciNetGoogle Scholar
  4. 4.
    Conway, J.H.: Regular algebra and finite machines. Mathematics series. Chapmann and Hall, London (1971)zbMATHGoogle Scholar
  5. 5.
    Grunsky, I., Kurganskyy, O., Potapov, I.: On a maximal nfa without mergible states. In: Grigoriev, D., Harrison, J., Hirsch, E.A. (eds.) CSR 2006. LNCS, vol. 3967, pp. 202–210. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  6. 6.
    Hopcroft, J.E.: An n log n algorithm for minimizing states in a finite automaton. In: Theory of machines and computations (Proc. Internat. Sympos., Technion, Haifa, 1971), pp. 189–196. Academic Press, New York (1971)Google Scholar
  7. 7.
    Ilie, L., Yu, S.: Reducing NFAs by invariant equivalences. Theoret. Comput. Sci. 306(1-3), 373–390 (2003)CrossRefMathSciNetzbMATHGoogle Scholar
  8. 8.
    Jiang, T., Ravikumar, B.: Minimal NFA problems are hard. SIAM J. Comput. 22(6), 1117–1141 (1993)CrossRefMathSciNetzbMATHGoogle Scholar
  9. 9.
    Korshunov, A.D.: The number of monotone Boolean functions. Problemy Kibernet. 38, 5–108 (1981)MathSciNetGoogle Scholar
  10. 10.
    Lombardy, S.: On the construction of reversible automata for reversible languages. In: Widmayer, P., et al. (eds.) ICALP 2002. LNCS, vol. 2380, pp. 170–182. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  11. 11.
    Lombardy, S., Sakarovitch, J.: Star height of reversible languages and universal automata. In: Rajsbaum, S. (ed.) LATIN 2002. LNCS, vol. 2286, pp. 76–90. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  12. 12.
    Sakarovitch, J.: Éléments de théorie des automates (English translation to appear, Cambridge University Press). In: Les classiques de l’informatique, Vuibert, Paris (2003)Google Scholar
  13. 13.
    Wiedemann, D.: A computation of the eighth Dedekind number. Order 8(1), 5–6 (1991)CrossRefMathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Berlin Heidelberg 2007

Authors and Affiliations

  • Sylvain Lombardy
    • 1
  1. 1.Université de Marne-la-Valle, Institut Gaspard-Monge, UMR CNRS 8049, 77454 Marne-la-Valle Cedex 2 

Personalised recommendations