Magic Numbers in the State Hierarchy of Finite Automata

  • Viliam Geffert
Conference paper

DOI: 10.1007/11821069_36

Part of the Lecture Notes in Computer Science book series (LNCS, volume 4162)
Cite this paper as:
Geffert V. (2006) Magic Numbers in the State Hierarchy of Finite Automata. In: Královič R., Urzyczyn P. (eds) Mathematical Foundations of Computer Science 2006. MFCS 2006. Lecture Notes in Computer Science, vol 4162. Springer, Berlin, Heidelberg

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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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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