On Complementing Unambiguous Automata and Graphs With Many Cliques and Cocliques

TL;DR

This paper presents a new method for complementing unambiguous finite automata with significantly fewer states, leveraging extremal graph theory to improve previous bounds and results.

Contribution

It introduces a novel approach connecting automata complementation with extremal graph theory, improving the state complexity bound for automata complement construction.

Findings

Automata complement can be achieved with fewer states using graph-theoretic insights.

Established a bound on the product of cliques and cocliques in graphs with n vertices.

Connected automata theory with extremal combinatorics to improve existing results.

Abstract

We show that for any unambiguous finite automaton with $n$ states there exists an unambiguous finite automaton with $\sqrt{n+1} \cdot 2^{n/2}$ states that recognizes the complement language. This builds and improves upon a similar result by Jir\'asek et al. [Int. J. Found. Comput. Sci. 29 (5) (2018)]. Our improvement is based on a reduction to and an analysis of a problem from extremal graph theory: we show that for any graph with $n$ vertices, the product of the number of its cliques with the number of its cocliques (independent sets) is bounded by $(n+1) 2^n$.