Lower Bounds for Unambiguous Automata via Communication Complexity

TL;DR

This paper establishes new lower bounds for unambiguous finite automata using communication complexity, demonstrating limitations in recognizing certain languages and their complements with small automata.

Contribution

It introduces novel lower bounds for UFAs on language complement, union, and separation, improving previous results and refuting a conjecture.

Findings

Complement: small UFAs need large NFAs for complements.

Union: small UFAs need large UFAs for unions.

Separation: some languages require large UFAs despite small NFAs.

Abstract

We use results from communication complexity, both new and old ones, to prove lower bounds for unambiguous finite automata (UFAs). We show three results. $\textit{Complement:}$ There is a language $L$ recognised by an $n$-state UFA such that the complement language $\overline{L}$ requires NFAs with $n^{\tilde{\Omega}(\log n)}$ states. This improves on a lower bound by Raskin. $\textit{Union:}$ There are languages $L_1$, $L_2$ recognised by $n$-state UFAs such that the union $L_1\cup L_2$ requires UFAs with $n^{\tilde{\Omega}(\log n)}$ states. $\textit{Separation:}$ There is a language $L$ such that both $L$ and $\overline{L}$ are recognised by $n$-state NFAs but such that $L$ requires UFAs with $n^{\Omega(\log n)}$ states. This refutes a conjecture by Colcombet.