Languages given by Finite Automata over the Unary Alphabet

TL;DR

This paper investigates the computational complexity of various decision problems and language operations for finite automata over unary alphabets, providing new upper bounds and complexity insights.

Contribution

It improves existing bounds on automata language inclusion, union, and complement operations for unary automata, and analyzes the complexity of omega-language recognition.

Findings

Decidability of language inclusion and equality in subexponential time.

Construction of UFAs for union and complement with quasipolynomial state bounds.

Complexity results for recognizing omega-regular languages with unary automata.

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 maximum of the number of states of the input finite automata considered in the corresponding results. The following main results are obtained: (1) Given two unary NFAs recognising $L$ and $H$, respectively, one can decide whether $L \subseteq H$ as well as whether $L = H$ in time $2^{O((n \log n)^{1/3})}$. The previous upper bound on time was $2^{O((n \log n)^{1/2})}$ as given by Chrobak (1986), and this bound was not significantly improved since then. (2) Given two unary UFAs (unambiguous finite automata) recognising $L$ and $H$, respectively, one can determine a UFA recognising $L \cup H$ and a UFA recognising complement of $L$, where these output UFAs have the number of states bounded by a quasipolynomial in $n$. However, in the worst case, a UFA for recognising concatenation of languages recognised by two $n$-state UFAs, uses $2^{\Theta((n \log^2 n)^{1/3})}$ states. (3) Given a unary language $L$, if $L$ contains the word of length $k$, then let $L(k)=1$ else let $L(k)=0$. Let $\omega_L$ be the $\omega$-word $L(0)L(1)\ldots$ and let $\cal L$ be a fixed $\omega$-regular language. The last section studies how difficult it is to decide, given an $n$-state UFA or NFA