Duality and equational theory of regular languages

Mai Gehrke (1) , Serge Grigorieff (2) , Jean-Eric Pin (2)

1 IMAPP - Institute for Mathematics, Astrophysics and Particle Physics
2 LIAFA - Laboratoire d'informatique Algorithmique : Fondements et Applications

This paper presents a new result in the equational theory of regular languages, which emerged from lively discussions between the authors about Stone and Priestley duality. Let us call lattice of languages a class of regular languages closed under finite intersection and finite union. The main results of this paper can be summarized in a nutshell as follows: A set of regular languages is a lattice of languages if and only if it can be defined by a set of profinite equations. The product on profinite words is the dual of the residuation operations on regular languages. In their more general form, our equations are of the form u --> v, where u and v are profinite words. The first result not only subsumes Eilenberg-Reiterman's theory of varieties and their subsequent extensions, but it shows for instance that any class of regular languages defined by a fragment of logic closed under conjunctions and disjunctions (first order, monadic second order, temporal, etc.) admits an equational description.

Mots clés: Stone duality; regular language; equational theory; profinite topology

Domaines: Autre [cs.OH]; Logique en informatique [cs.LO]; Mathématique discrète [cs.DM]; Topologie générale [math.GN]

Mai Gehrke, Serge Grigorieff, Jean-Eric Pin. Duality and equational theory of regular languages. ICALP 2008, Jul 2008, Reykjavik, Iceland. pp.246-257. ⟨hal-00340803⟩