Logic Meets Algebra: the Case of Regular Languages

TL;DR

This paper explores the deep connections between algebraic structures and logical definability of regular languages, highlighting how algebraic characterizations inform automata theory and circuit complexity.

Contribution

It surveys key results linking logical expressibility of regular languages with algebraic properties of automata, emphasizing the role of pseudovarieties and block-products.

Findings

Many results share common underlying mechanics.

Logical substitutions relate to block-products of pseudovarieties.

Connections impact circuit complexity theory.

Abstract

The study of finite automata and regular languages is a privileged meeting point of algebra and logic. Since the work of Buchi, regular languages have been classified according to their descriptive complexity, i.e. the type of logical formalism required to define them. The algebraic point of view on automata is an essential complement of this classification: by providing alternative, algebraic characterizations for the classes, it often yields the only opportunity for the design of algorithms that decide expressibility in some logical fragment. We survey the existing results relating the expressibility of regular languages in logical fragments of MSO[S] with algebraic properties of their minimal automata. In particular, we show that many of the best known results in this area share the same underlying mechanics and rely on a very strong relation between logical substitutions and block-products of pseudovarieties of monoid. We also explain the impact of these connections on circuit complexity theory.