The Algebras for Automatic Relations

TL;DR

This paper introduces synchronous algebras, an algebraic framework for recognizing automatic relations, extending algebraic language theory to typed, dependency-aware structures that generalize monoids for regular languages.

Contribution

It defines synchronous algebras, enabling algebraic characterization and lifting of pseudovarieties from regular languages to automatic relations, including group relations.

Findings

Synchronous algebras effectively recognize automatic relations.

Finite synchronous algebras characterize synchronous relations.

Pseudovarieties of synchronous relations correspond to those of synchronous algebras.

Abstract

We introduce "synchronous algebras", an algebraic structure tailored to recognize automatic relations (aka. synchronous relations, or regular relations). They are the equivalent of monoids for regular languages, however they conceptually differ in two points: first, they are typed and second, they are equipped with a dependency relation expressing constraints between elements of different types. The interest of the proposed definition is that it allows to lift, in an effective way, pseudovarieties of regular languages to that of synchronous relations, and we show how algebraic characterizations of pseudovarieties of regular languages can be lifted to the pseudovarieties of synchronous relations that they induce. A typical example of such a pseudovariety is the class of "group relations", defined as the relations recognized by finite-state synchronous permutation automata. In order to prove this result, we adapt two pillars of algebraic language to synchronous algebras: (a) any relation admits a syntactic synchronous algebra recognizing it, and moreover, the relation is synchronous if, and only if, its syntactic algebra is finite and (b) classes of synchronous relations with desirable closure properties (i.e. pseudovarieties) correspond to pseudovarieties of synchronous algebras.