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Theoretical Computer Science
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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 unary alphabet, and determines the exact number of states in DFAs needed to represent unary languages recognized by n-state UFAs: the growth rate of this function is eΘ(3√n ln2 n). The conversion of an n-state unary NFA to a UFA requires UFAs with g(n)+O(n2) = e√n ln n(1+o(1)) states, where g(n) is Landau's function. In addition, it is shown that the complement of n-state unary UFAs requires up to at least n2-o(1) states in an NFA, while the Kleene star requires up to exactly (n - 1)2 + 1 states.