Unambiguous finite automata over a unary alphabet

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 eΘ(nln2n3). 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 Landauʼs 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.