State complexity of power

  • Authors:

  • Michael Domaratzki;Alexander Okhotin

  • Affiliations:

  • Department of Computer Science, University of Manitoba, Winnipeg, MB, Canada R3T 2N2;Academy of Finland, Helsinki, Finland and Department of Mathematics, University of Turku, Turku FIN-20014, Finland

  • Venue:
  • Theoretical Computer Science
  • Year:
  • 2009

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Abstract

The number of states in a deterministic finite automaton (DFA) recognizing the language L^k, where L is regular language recognized by an n-state DFA, and k=2 is a constant, is shown to be at most n2^(^k^-^1^)^n and at least (n-k)2^(^k^-^1^)^(^n^-^k^) in the worst case, for every nk and for every alphabet of at least six letters. Thus, the state complexity of L^k is @Q(n2^(^k^-^1^)^n). In the case k=3 the corresponding state complexity function for L^3 is determined as 6n-384^n-(n-1)2^n-n with the lower bound witnessed by automata over a four-letter alphabet. The nondeterministic state complexity of L^k is demonstrated to be nk. This bound is shown to be tight over a two-letter alphabet.