On Complementing Unambiguous Automata and Graphs With Many Cliques and Cocliques

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 $(n+1) 2^n$.