Varieties of Data Languages

TL;DR

This paper establishes a novel Eilenberg-type correspondence for data languages over infinite alphabets, linking varieties of such languages with pseudovarieties of orbit-finite nominal monoids using category theory and duality principles.

Contribution

It introduces the first Eilenberg-type correspondence for data languages recognized by orbit-finite nominal monoids, utilizing nominal Stone duality and categorical generalizations.

Findings

Bijective correspondence between language varieties and monoid pseudovarieties.

Axiomatic characterization of weak pseudovarieties via nominal equations.

First such result for data languages over infinite alphabets.

Abstract

We establish an Eilenberg-type correspondence for data languages, i.e. languages over an infinite alphabet. More precisely, we prove that there is a bijective correspondence between varieties of languages recognized by orbit-finite nominal monoids and pseudovarieties of such monoids. This is the first result of this kind for data languages. Our approach makes use of nominal Stone duality and a recent category theoretic generalization of Birkhoff-type HSP theorems that we instantiate here for the category of nominal sets. In addition, we prove an axiomatic characterization of weak pseudovarieties as those classes of orbit-finite monoids that can be specified by sequences of nominal equations, which provides a nominal version of a classical theorem of Eilenberg and Sch\"utzenberger.