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 eΘ(n ln2 n3). The conversion of an n-state unary NFA to a UFA requires UFAs with g(n)+O( n2 )=e( 1+o(1))nlnn 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 n2 -o(1) states in the worst case, while the Kleene star requires up to exactly ( n-1)2 +1 states.
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