Lengths of Words Accepted by Nondeterministic Finite Automata

We consider two natural problems about nondeterministic finite automata. First, given such an automaton M of n states, and a length l, does M accept a word of length l? We show that the classic problem of triangle-free graph recognition reduces to this problem, and give an O(n^{\omega} (log n)^{1+{\epsilon}} log l)-time algorithm to solve it, where {\omega} is the optimal exponent for matrix multiplication. Second, provided L(M) is finite, we consider the problem of listing the lengths of all words accepted by M. Although this problem seems like it might be significantly harder, we show that this problem can be solved in O(n^{\omega}(log n)^{2+{\epsilon}}) time. Finally, we give a connection between NFA acceptance and the strong exponential-time hypothesis.