Some Remarks on Automata Minimality

  • Antonio Restivo
  • Roberto Vaglica
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6795)

Abstract

It is well known that the minimization problem of deterministic finite automata (DFAs) is related to the indistinguishability notion of states (cf. [HMU00]). Indeed, a well known technique to minimize a DFA, essentially, consists in finding pairs of states that are equivalent (or indistinguishable), namely pairs of states (p,q) such that it is impossible to assert the difference between p and q only by starting in each of the two states and asking whether or not a given input string leads to a final state. Since, in the testing states equivalence, the notion of initial state is irrelevant, some of the main techniques for the minimization of automata, such as Moore’s algorithm [Moo56] and Hopcroft’s algorithm [Hop71], do not care what is the initial state of the automaton, when applied to accessible automata (i.e. such that all states can be reached from the initial state). Therefore a natural question that arises is, for accessible automata, on what does minimality depend? Obviously, it depends on both the automata transitions and the set of final states. In this paper, our main focus is to investigate to what extent minimality depends on the particular subset of final states.

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References

  1. [GK98]
    Goralcik, P., Koubek, V.: On the disjunctive set problem. Theor. Comput. Sci. 204(1-2), 99–118 (1998)MATHCrossRefMathSciNetGoogle Scholar
  2. [HMU00]
    Hopcroft, J.E., Motwani, R., Ullman, J.D.: Introduction to Automata Theory, Languages, and Computation, 2nd edn. Addison Wesley, Reading (2000)Google Scholar
  3. [Hop71]
    Hopcroft, J.E.: An n log n algorithm for minimizing the states in a finite automaton, pp. 189–196. Academic Press, London (1971)Google Scholar
  4. [How91]
    Howie, J.M. (ed.): Automata and Languages. Oxford Science Publications. Oxford University Press, Inc., New York (1991)MATHGoogle Scholar
  5. [Moo56]
    Moore, E.F.: Gedanken experiments on sequential machines. In: Moore, E.F. (ed.) Automata Studies, pp. 129–153. Princeton U., Princeton (1956)Google Scholar
  6. [RS11]
    Rodaro, E., Silva, P.: Never minimal automata and the rainbow bipartite subgraph problem. In: Proceedings of the 15th International Conference on Developments in Language Theory. Springer, Heidelberg (2011)Google Scholar
  7. [RV10]
    Restivo, A., Vaglica, R.: Automata with extremal minimality conditions. In: Gao, Y., Lu, H., Seki, S., Yu, S. (eds.) DLT 2010. LNCS, vol. 6224, pp. 399–410. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  8. [RV11]
    Restivo, A., Vaglica, R.: Extremal minimality conditions on automata (2011) (submitted)Google Scholar
  9. [Č64]
    Černy, J.: Poznámka k homogénnym experimenton s konečnými automatmi. Mat.-Fyz. Cas. Slovensk. Akad. Vied. 14, 208–215 (1964)MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

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

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