Magic Numbers in the State Hierarchy of Finite Automata

  • Viliam Geffert
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4162)

Abstract

A number d is magic for n, if there is no regular language for which an optimal nondeterministic finite state automaton (nfa) uses exactly n states, but for which the optimal deterministic finite state automaton (dfa) uses exactly d states. We show that, in the case of unary regular languages, the state hierarchy of dfa’s, for the family of languages accepted by n-state nfa’s, is not contiguous. There are some “holes” in the hierarchy, i.e., magic numbers in between values that are not magic. This solves, for automata with a single letter input alphabet, an open problem of existence of magic numbers [7].

As an additional bonus, we get a universal lower bound for the conversion of unary d-state dfa’s into equivalent nfa’s: nondeterminism does not reduce the number of states below log2d, not even in the best case.

Keywords

Descriptional complexity finite-state automata 

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References

  1. 1.
    Bertoni, A., Mereghetti, C., Pighizzini, G.: An optimal lower bound for nonregular languages. Inform. Process. Lett. 50, 289–292 (1994); Corrigendum: ibid. 52, 339 (1994)Google Scholar
  2. 2.
    Chrobak, M.: Finite automata and unary languages. Theoret. Comput. Sci. 47, 149–158 (1986); Corrigendum: ibid. 302, 497–498 (2003)Google Scholar
  3. 3.
    Geffert, V.: (Non)determinism and the size of one-way finite automata. In: Proc. Descr. Compl. Formal Syst., pp. 23–37 (2005) (IFIP & Univ. Milano)Google Scholar
  4. 4.
    Geffert, V., Mereghetti, C., Pighizzini, G.: Converting two-way nondeterministic unary automata into simpler automata. Theoret. Comput. Sci. 295, 189–203 (2003)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Hardy, G., Wright, E.: An Introduction to the Theory of Numbers, 5th edn. Oxford University Press, Oxford (1979)MATHGoogle Scholar
  6. 6.
    Hopcroft, J., Motwani, R., Ullman, J.: Introduction to Automata Theory, Languages, and Computation. Addison-Wesley, Reading (2001)MATHGoogle Scholar
  7. 7.
    Iwama, K., Kambayashi, Y., Takaki, K.: Tight bounds on the number of states of DFA’s that are equivalent to n-state NFA’s. Theoret. Comput. Sci. 237, 485–494 (2000)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Iwama, K., Matsuura, A., Paterson, M.: A family of NFA’s which need 2n − α deterministic states. Theoret. Comput. Sci. 301, 451–462 (2003)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Jirásková, G.: Note on minimal finite automata. In: Sgall, J., Pultr, A., Kolman, P. (eds.) MFCS 2001. LNCS, vol. 2136, pp. 421–431. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  10. 10.
    Lupanov, O.B.: Uber den Vergleich zweier Typen endlicher Quellen. Probleme der Kybernetik 6, 329–335 (1966) (in German)Google Scholar
  11. 11.
    Mereghetti, C., Pighizzini, G.: Optimal simulations between unary automata. SIAM J. Comput. 30, 1976–1992 (2001)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Moore, F.: On the bounds for state-set size in the proofs of equivalence between deterministic, nondeterministic, and two-way finite automata. IEEE Trans. Comput. C-20, 1211–1214 (1971)CrossRefGoogle Scholar
  13. 13.
    Rabin, M., Scott, D.: Finite automata and their decision problems. IBM J. Res. Develop. 3, 114–125 (1959)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Sakoda, W., Sipser, M.: Nondeterminism and the size of two-way finite automata. In: Proc. ACM Symp. Theory of Comput., pp. 275–286 (1978)Google Scholar
  15. 15.
    Szalay, M.: On the maximal order in S n and \(S_n^{\ast}\). Acta Arith 37, 321–331 (1980)MATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Viliam Geffert
    • 1
  1. 1.Department of Computer ScienceP. J. Šafárik UniversityKošiceSlovakia

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