Automata with Extremal Minimality Conditions

  • Antonio Restivo
  • Roberto Vaglica
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6224)

Abstract

It is well known that the minimality of a deterministic finite automaton (DFA) depends on the set of final states. In this paper we study the minimality of a strongly connected DFA by varying the set of final states. We consider, in particular, some extremal cases. A strongly connected DFA is called uniformly minimal if it is minimal, for any choice of the set of final states. It is called never-minimal if it is not minimal, for any choice of the set of final states. We show that there exists an infinite family of uniformly minimal automata and that there exists an infinite family of never-minimal automata. Some properties of these automata are investigated and, in particular, we consider the complexity of the problem to decide whether an automaton is uniformly minimal or never-minimal.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Beal, M.P., Perrin, D.: Symbolic dynamics and finite automata. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages, vol. 2, pp. 463–505. Springer, Heidelberg (1997)Google Scholar
  2. 2.
    Berstel, J., Perrin, D.: Theory of codes. Academic Press, Inc., London (1985)MATHGoogle Scholar
  3. 3.
    Brzozowski, J.: On Single-Loop Realizations of Sequential Machines. Information and Control 10(3), 292–314 (1967)MATHCrossRefGoogle Scholar
  4. 4.
    Carpi, A., D’Alessandro, F.: Strongly transitive automata and the Černý conjecture. Acta Informatica 46(8), 591–607 (2009)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Eilenberg, S.: Automata, Languages, and Machines, vol. A (1974)Google Scholar
  6. 6.
    Friedman, A.D.: Feedback in Synchronous Sequential Switching Circuits. IEEE Trans. Electronic Computers EC-15(3), 354–367 (1966)CrossRefGoogle Scholar
  7. 7.
    Gill, A., Kou, L.T.: Multiple-entry finite automata. Journal of Computer and System Sciences 9(1), 1–19 (1974)MATHMathSciNetGoogle Scholar
  8. 8.
    Goralcik, P., Koubek, V.: On the Disjunctive Set Problem. Theoretical Computer Science 204(1-2), 99–118 (1998)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Han, Y.S., Wang, Y., Wood, D.: Infix-free Regular Expressions and Languages. International Journal of Foundations of Computer Science 17(2), 379–393 (2006)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Holzer, M., Salomaa, K., Yu, S.: On the state complexity of k-entry deterministic finite automata. Journal of Automata, Languages and Combinatorics 6(4), 453–466 (2001)MATHMathSciNetGoogle Scholar
  11. 11.
    Hopcroft, J.E.: An nlogn algorithm for minimizing the states in a finite automaton. In: Kohavi, Z., Paz, A. (eds.) Theory of Machines and Computations, Proc. Internat. Sympos. Technion, Haifa, pp. 189–196. Academic Press, New York (1971)Google Scholar
  12. 12.
    Hopcroft, J., Ullman, J.: Introduction to Automata Theory, Languages, and Computation, 2nd edn. Addison-Wesley, Reading (1979)MATHGoogle Scholar
  13. 13.
    Kao, J.Y., Rampersd, N., Shallit, J.: On NFA’s where all states are final, initial, or both. TCS 410, 5010–5021 (2009)Google Scholar
  14. 14.
    Lind, D., Marcus, B.: An introduction to symbolic dynamics and coding. Cambridge University Press, New York (1995)MATHCrossRefGoogle Scholar
  15. 15.
    Moore, E.F.: Gedanken-experiments on sequential machines. The Journal of Symbolic Logic 23(1) (1958)Google Scholar
  16. 16.
    Veloso, P.A.S., Gill, A.: Some remarks on multiple-entry finite automata. Journal of Computer and System Sciences 18, 304–306 (1979)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Volkov, M.V.: Syncronizing 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

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Antonio Restivo
    • 1
  • Roberto Vaglica
    • 1
  1. 1.Dipartimento di Matematica e InformaticaUniversità degli Studi di PalermoPalermoItaly

Personalised recommendations