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A Kleene Theorem for Higher-Dimensional Automata

Authors Uli Fahrenberg, Christian Johansen, Georg Struth, Krzysztof Ziemiański

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Author Details

Uli Fahrenberg
  • EPITA Research and Development Laboratory (LRDE), Paris, France
Christian Johansen
  • NTNU Gjøvik, Norway
Georg Struth
  • University of Sheffield, UK
  • Collegium de Lyon, France
Krzysztof Ziemiański
  • University of Warsaw, Poland

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Uli Fahrenberg, Christian Johansen, Georg Struth, and Krzysztof Ziemiański. A Kleene Theorem for Higher-Dimensional Automata. In 33rd International Conference on Concurrency Theory (CONCUR 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 243, pp. 29:1-29:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)


We prove a Kleene theorem for higher-dimensional automata (HDAs). It states that the languages they recognise are precisely the rational subsumption-closed sets of interval pomsets. The rational operations include a gluing composition, for which we equip pomsets with interfaces. For our proof, we introduce HDAs with interfaces as presheaves over labelled precube categories and use tools inspired by algebraic topology, such as cylinders and (co)fibrations. HDAs are a general model of non-interleaving concurrency, which subsumes many other models in this field. Interval orders are used as models for concurrent or distributed systems where events extend in time. Our tools and techniques may therefore yield templates for Kleene theorems in various models and applications.

Subject Classification

ACM Subject Classification
  • Theory of computation → Automata extensions
  • higher-dimensional automata
  • interval posets
  • Kleene theorem
  • concurrency theory
  • labelled precube categories


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