Complexity of Atoms, Combinatorially

TL;DR

This paper provides a combinatorial proof characterizing regular languages that meet the maximum complexity bounds of their atoms, offering an alternative to previous algebraic methods.

Contribution

It introduces a purely combinatorial proof for the characterization of regular languages with maximal atom complexity, complementing prior algebraic approaches.

Findings

Alternative combinatorial proof of atom complexity characterization

Identification of languages meeting the tight bounds on atom complexity

Enhanced understanding of atom structure in regular languages

Abstract

Atoms of a (regular) language $L$ were introduced by Brzozowski and Tamm in 2011 as intersections of complemented and uncomplemented quotients of $L$. They derived tight upper bounds on the complexity of atoms in 2013. In 2014, Brzozowski and Davies characterized the regular languages meeting these bounds. To achieve these results, they used the so-called "atomaton" of a language, introduced by Brzozowski and Tamm in 2011. In this note we give an alternative proof of their characterization, via a purely combinatorial approach.