Separating Exponentially Ambiguous Finite Automata from Polynomially Ambiguous Finite Automata

We resolve an open problem raised by Ravikumar and Ibarra [SIAM J. Comput., 18 (1989), pp. 1263–1282] on the succinctness of representations relating to the types of ambiguity of finite automata. We show that there exists a family of nondeterministic finite automata {An} over a two-letter alphabet such that, for any positive integer n, An is exponentially ambiguous and has n states, whereas the smallest equivalent deterministic finite automaton has 2 states, and any smallest equivalent polynomially ambiguous finite automaton has 2 − 1 states.