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Algebraic Characterizations of Classes of Regular Languages in DynFO

Authors Corentin Barloy , Felix Tschirbs , Nils Vortmeier , Thomas Zeume

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LIPIcs.STACS.2026.9.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Complexity theory and logic
  • Theory of computation → Algebraic language theory
  • Theory of computation → Regular languages
Keywords
  • Dynamic descriptive complexity
  • formal languages
  • monoid theory

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Abstract

This paper explores the fine-grained structure of classes of regular languages maintainable in fragments of first-order logic within the dynamic descriptive complexity framework of Patnaik and Immerman. A result by Hesse states that the class of regular languages is maintainable by first-order formulas even if only unary auxiliary relations can be used. Another result by Gelade, Marquardt, and Schwentick states that the class of regular languages coincides with the class of languages maintainable by quantifier-free formulas with binary auxiliary relations.
We refine Hesse’s result and show that with unary auxiliary data ∃^*∀^*-formulas can maintain all regular languages. We then obtain precise algebraic characterizations of the classes of languages maintainable with quantifier-free formulas and positive ∃^*-formulas in the presence of unary auxiliary relations.

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Corentin Barloy, Felix Tschirbs, Nils Vortmeier, and Thomas Zeume. Algebraic Characterizations of Classes of Regular Languages in DynFO. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 9:1-9:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.STACS.2026.9

Author Details

Corentin Barloy
  • Ruhr University Bochum, Germany
Felix Tschirbs
  • Ruhr University Bochum, Germany
Nils Vortmeier
  • Ruhr University Bochum, Germany
Thomas Zeume
  • Ruhr University Bochum, Germany

Funding

All authors are supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), grant 532727578.

Acknowledgements

We thank the reviewers for their careful reading and insightful comments, which significantly improved the quality of this article.

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