State complexity of catenation combined with boolean operations

TL;DR

This paper explores the combined state complexity of catenation and boolean operations on regular languages, providing proofs, improvements on conjectures, and minimal witnesses with small alphabets.

Contribution

It systematically analyzes the state complexity of combined operations, advancing theoretical understanding and offering concrete examples with minimal alphabet sizes.

Findings

Proved and improved conjectures by Brzozowski.

Provided minimal witnesses with small alphabet sizes.

Established comprehensive results on combined operation complexities.

Abstract

We exhaustively investigate possible combinations of a boolean operation together with a catenation. In many cases we prove and improve some conjectures by Brzozowski. For each family of operation, we endeavour to provide a common witness with a small size alphabet.