On state complexity of unions of binary factor-free languages

TL;DR

This paper disproves a 2011 conjecture on the maximum state complexity of unions of binary factor-free languages by establishing a new lower bound that exceeds the conjectured limit for languages with at least 10 states.

Contribution

The authors provide a counterexample to the conjecture and establish a new lower bound on the state complexity of unions of factor-free languages.

Findings

The new lower bound surpasses the conjectured maximum for languages with at least 10 states.

The conjecture from 2011 is false for certain binary factor-free languages.

The results refine understanding of state complexity in automata theory.

Abstract

It has been conjectured in 2011 by Brzozowski et al. that if $K$ and $L$ are factor-free regular languages over a binary alphabet having state complexity $m$ and $n$, resp, then the state complexity of $K\cup L$ is at most $mn-(m+n)+3-\min\{m,n\}$. We disprove this conjecture by giving a lower bound of $mn-(m+n)-2-\lfloor\frac{\min\{m,n\}-2}{2}\rfloor$, which exceeds the conjectured bound whenever $\min\{m,n\}\geq 10$.