Lengths of Words Accepted by Nondeterministic Finite Automata

TL;DR

This paper investigates the computational complexity of determining accepted word lengths by nondeterministic finite automata, providing efficient algorithms and connecting the problem to the strong exponential-time hypothesis.

Contribution

It introduces new algorithms for checking word lengths accepted by NFAs and links the problem to major complexity hypotheses, advancing understanding of automata length recognition.

Findings

An O(n^{} (log n)^{1+b5}} log l) algorithm for length acceptance.

An O(n^{}(log n)^{2+b5}}) algorithm for listing accepted word lengths.

Connection established between NFA acceptance problems and the strong exponential-time hypothesis.

Abstract

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.