# Towards a Theory of Complexity of Regular Languages

## TL;DR

This paper surveys recent findings on the complexity measures of regular languages, including state complexity, syntactic semigroup size, and atom complexity, and examines their behavior under various operations and subclasses.

## Contribution

It provides a comprehensive overview of the interrelations among different complexity measures of regular languages and identifies 'most complex' languages within various subclasses.

## Key findings

- Identifies relationships among complexity measures.
- Highlights existence of 'most complex' languages in several subclasses.
- Analyzes complexity of common regular language operations.

## Abstract

We survey recent results concerning the complexity of regular languages represented by their minimal deterministic finite automata. In addition to the quotient complexity of the language -- which is the number of its (left) quotients, and is the same as its state complexity -- we also consider the size of its syntactic semigroup and the quotient complexity of its atoms -- basic components of every regular language. We then turn to the study of the quotient/state complexity of common operations on regular languages: reversal, (Kleene) star, product (concatenation) and boolean operations. We examine relations among these complexity measures. We discuss several subclasses of regular languages defined by convexity. In many, but not all, cases there exist "most complex" languages, languages satisfying all these complexity measures.