Unambiguous finite automata over a unary alphabet

  • Authors:

  • Alexander Okhotin

  • Affiliations:

  • Department of Mathematics, University of Turku, Turku FI-20014, Finland

  • Venue:
  • Information and Computation
  • Year:
  • 2012

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Abstract

Nondeterministic finite automata (NFA) with at most one accepting computation on every input string are known as unambiguous finite automata (UFA). This paper considers UFAs over a one-letter alphabet, and determines the exact number of states in DFAs needed to represent unary languages recognized by n-state UFAs in terms of a new number-theoretic function g@?. The growth rate of g@?(n), and therefore of the UFA-DFA tradeoff, is estimated as e^@Q^(^n^l^n^^^2^n^3^). The conversion of an n-state unary NFA to a UFA requires UFAs with g(n)+O(n^2)=e^(^1^+^o^(^1^)^)^n^l^n^n states, where g(n) is the greatest order of a permutation of n elements, known as Landau@?s function. In addition, it is shown that representing the complement of n-state unary UFAs requires UFAs with at least n^2^-^o^(^1^) states in the worst case, while the Kleene star requires up to exactly (n-1)^2+1 states.