The Algebras for Automatic Relations

Author Rémi Morvan



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Rémi Morvan
  • LaBRI, Univ. Bordeaux, CNRS & Bordeaux INP, France

Acknowledgements

We thank Pablo Barceló, Mikołaj Bojańczyk, and Diego Figueira for helpful discussions, and some anonymous reviewers for valuable feedback.

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Rémi Morvan. The Algebras for Automatic Relations. In 33rd EACSL Annual Conference on Computer Science Logic (CSL 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 326, pp. 21:1-21:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.CSL.2025.21

Abstract

We introduce "synchronous algebras", an algebraic structure tailored to recognize automatic relations (a.k.a. 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. Since this construction is effective, this implies that the membership problem is decidable for (infinitely) many natural classes of automatic relations. 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 theory 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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Algebraic language theory
Keywords
  • synchronous automata
  • automatic relations
  • regular relations
  • transductions
  • synchronous algebras
  • Eilenberg correspondence
  • pseudovarieties
  • algebraic characterizations

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