Languages given by finite automata over the unary alphabet

Abstract

This paper studies the complexity of operations on finite automata and the complexity of their decision problems when the alphabet is unary. Let n denote the number of states of the input automata considered. The following main results are obtained:

(1) Equality and inclusion of NFAs can be decided within time $2^{O\left({(n\log n)}^{1/3}\right)}$. The previous upper bound $2^{O\left({(n\log n)}^{1/2}\right)}$ was by Chrobak (1986).

(2) One can determine a UFA (unambiguous finite automata) for complement of another UFA or union of two UFAs using at most quasipolynomial number of states. However, for concatenation of two n-state UFAs, the worst case is a UFA having $2^{\Theta \left({(n{\log }^{2}n)}^{1/3}\right)}$ states.

(3) Results when an infinite ω-word given by a UFA or an NFA is a member of a given regular ω-language are obtained.