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Algebraic Characterization of FO-Definable Languages of Higher-Dimensional Automata

Authors Enzo Erlich , Jérémy Ledent , Krzysztof Ziemiański

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LIPIcs.CONCUR.2026.32.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Concurrency
  • Theory of computation → Automata extensions
  • Theory of computation → Logic and verification
Keywords
  • Higher-dimensional automata
  • Pomset languages
  • McNaughton-Papert theorem
  • Counter-free HDA
  • Aperiodic category

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Abstract

Higher-dimensional automata (HDA) are a model of concurrency that models simultaneous execution of events using higher dimensional cells. HDA recognize languages of pomsets, a generalization of finite words whose letters are partially ordered. We prove a new algebraic characterization of HDA languages: a language of pomsets is regular if and only if it is the inverse image of a functor from the category of pomsets into a finite category. Furthermore, the language is definable in first-order logic exactly when it is recognized by an aperiodic category, generalizing the McNaughton-Papert theorem to HDA languages. We also investigate a notion of counter-free HDA, and show that if a language is accepted by a counter-free HDA, it must be definable in first-order logic. The converse, however, is still open.

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Enzo Erlich, Jérémy Ledent, and Krzysztof Ziemiański. Algebraic Characterization of FO-Definable Languages of Higher-Dimensional Automata. In 37th International Conference on Concurrency Theory (CONCUR 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 391, pp. 32:1-32:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.CONCUR.2026.32

Author Details

Enzo Erlich
  • Université Paris Cité, CNRS, IRIF, F-75013, Paris, France
  • EPITA Research Laboratory (LRE), Paris, France
Jérémy Ledent
  • Université Paris Cité, CNRS, IRIF, F-75013, Paris, France
Krzysztof Ziemiański
  • Institute of Mathematics, University of Warsaw, Poland

References

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