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 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 $2^{O((n \log n)<sup>{1/3})}$. The previous upper bound on time was $2^{O((n \log n)<sup>{1/2})}$ 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 $$2^{\Theta((n \log² n)<sup>{1/3})}$$ 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