Languages given by Finite Automata over the Unary Alphabet

TL;DR

This paper addresses the complexity and size growth of regular operations and decision problems for unary finite automata, focusing on NFAs and UFAs. It develops ETH-based conditional lower bounds and leverages the Chrobak Normal Form along with number-theoretic tools (notably the Prime Number Theorem) to derive tight bounds for problems such as NFA containment, UFA complementation, and ω-language membership. The key contributions include a near-optimal 2^{O((n log n)^{1/3})} time algorithm for unary NFA containment, quasipolynomial blow-ups for most UFA Boolean operations, and a matched 2^{Θ((n log^2 n)^{1/3})} bound for UFA concatenation, together with ETH-based lower bounds for ω-word membership. These results illuminate the boundary between polynomial, quasipolynomial, and exponential-type complexities in unary automata and connect automata theory with number theory and complexity assumptions. The findings have implications for the feasibility of automated reasoning over unary languages and for understanding the intrinsic difficulty of regular operations in constrained alphabets.

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 $2^{Θ((n \log^2 n)^{1/3})}$ 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 $ω_L$ be the $ω$-word $L(0)L(1)\ldots$ and let $\cal L$ be a fixed $ω$-regular language. The last section studies how difficult it is to decide, given an $n$-state UFA or NFA

Figures (2)

• Figure 1: Table of results on State Complexity. Here $c(n)=n^{\log n+O(1)}$. The bounds for union and intersection of arbitrarily many $n$-state UFAs are matching and reached when using $\Theta(n/\log n)$ many UFAs. Specific known formulas for combining $k$ arbitrary UFAs were adjusted to this case. Furthermore, let $LCM_n$ be the least common multiple of the natural numbers from $1$ to $n$. $LCM_n \in 2^{\Theta(n)}$. Pointers to results obtained by the authors in this paper are fully in blue.
• Figure 2: Table of results on Computational Complexity. Here $c(n)=n^{\log n+O(1)}$.

Theorems & Definitions (18)

• Proposition 1: Fernau and Krebs FK17, Tan Ta22
• Theorem 2
• Theorem 3
• Theorem 5
• Corollary 6: Stearns and Hunt SH85
• Proposition 7
• Corollary 8
• Proposition 9
• Proposition 10
• Proposition 11: Holzer and Kutrib HK02: (a), (b); Jirásková and Okhotin JO18: (d); Okhotin Ok12: (g); Raskin Ra18: (i); Yu, Zhuang and Salomaa YZS94: (j)