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Forgetting Event Order in Higher-Dimensional Automata

Author Safa Zouari

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LIPIcs.LICS.2026.84.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Concurrency
  • Theory of computation → Automata extensions
Keywords
  • higher-dimensional automata
  • true concurrency
  • pomsets with interfaces
  • interval pomsets
  • pomsets
  • presheaf semantics
  • bisimulation
  • event order
  • ST-trace

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Abstract

Higher-dimensional automata (HDAs) provide a geometric model of true concurrency, yet their standard formulation encodes an artificial total order on events. This representational artifact causes a fundamental mismatch between the combinatorial structure of HDAs and their observable behaviour, leading to logical asymmetries and complicating the application of categorical tools. In this paper, we resolve this tension by developing a semantics for HDAs that is independent of event order, based on interval ipomsets (partially ordered multisets with interfaces) that preserve only precedence and concurrency. We prove that for any HDA, the traditional ST–trace of an execution path corresponds precisely to its associated interval ipomset. On the structural side, we show that the presheaf-theoretic presentation with an unordered base and the combinatorial presentation of symmetric HDAs are categorically isomorphic. Finally, by characterizing ST- and hereditary history-preserving (hhp) bisimulation via ipomset isomorphism, we provide a unified, order-free foundation for HDA semantics. Our results resolve several ambiguities in the literature: they provide the necessary path-category structure to canonically apply the Open Maps framework, eliminate representational artifacts in temporal and modal logics, and bridge systematic mismatches between HDAs and other models of concurrency such as Petri nets.

Cite As Get BibTex

Safa Zouari. Forgetting Event Order in Higher-Dimensional Automata. In 41st Annual Symposium on Logic in Computer Science (LICS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 380, pp. 84:1-84:26, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.LICS.2026.84

Author Details

Safa Zouari
  • Norwegian University of Science and Technology, Gjøvik, Norway

Funding

  • Zouari, Safa: Supported by a PhD fellowship at the Department of Information Security and Communication Technology (IIK), Norwegian University of Science and Technology (NTNU).

Acknowledgements

I thank Krzysztof Ziemiański and Rob van Glabbeek for helpful comments and insightful discussions. Any remaining errors are mine alone.

References

  1. Amazigh Amrane, Hugo Bazille, Emily Clement, and Uli Fahrenberg. Languages of higher-dimensional timed automata. In Lars Michael Kristensen and Jan Martijn van der Werf, editors, Application and Theory of Petri Nets and Concurrency, pages 197-219, Cham, 2024. Springer Nature Switzerland. URL: https://doi.org/10.1007/978-3-031-61433-0_10.
  2. Amazigh Amrane, Hugo Bazille, Emily Clement, Uli Fahrenberg, Marie Fortin, and Krzysztof Ziemiański. Büchi-elgot-trakhtenbrot theorem for higher-dimensional automata, 2025. URL: https://doi.org/10.48550/arXiv.2505.10461.
  3. Amazigh Amrane, Hugo Bazille, Uli Fahrenberg, and Marie Fortin. Logic and languages of higher-dimensional automata. In Joel D. Day and Florin Manea, editors, Developments in Language Theory, pages 51-67, Cham, 2024. Springer Nature Switzerland. URL: https://doi.org/10.1007/978-3-031-66159-4_5.
  4. Amazigh Amrane, Hugo Bazille, Uli Fahrenberg, Loïc Hélouët, and Philipp Schlehuber-Caissier. Petri nets and higher-dimensional automata. In Elvio Amparore and Łukasz Mikulski, editors, Application and Theory of Petri Nets and Concurrency, pages 18-40, Cham, 2025. Springer Nature Switzerland. URL: https://doi.org/10.1007/978-3-031-94634-9_2.
  5. Amazigh Amrane, Hugo Bazille, Uli Fahrenberg, and Krzysztof Ziemiański. Closure and decision properties for higher-dimensional automata. Theoretical Computer Science, 1036:115156, 2025. URL: https://doi.org/10.1016/j.tcs.2025.115156.
  6. Amazigh Amrane, Hugo Bazille, Timothée Fragnaud, and Philipp Schlehuber-Caissier. Towards an efficient conversion of petri nets into higher dimensional automata. In Modélisation des Systèmes Réactifs (MSR'25), pages 1-15, Reims, France, November 2025. Modélisation des Systèmes Réactifs. URL: https://hal.science/hal-05281114.
  7. Marek A. Bednarczyk. Categories of Asynchronous Systems. PhD thesis, University of Sussex, GBR, 1987. AAIDX83002. Google Scholar
  8. Emily Clement, Enzo Erlich, and Jérémy Ledent. Kamp theorem for pomset languages of higher dimensional automata, 2025. URL: https://arxiv.org/abs/2410.12493.
  9. Uli Fahrenberg, Christian Johansen, Georg Struth, and Ratan Bahadur Thapa. Generating posets beyond n. In Uli Fahrenberg, Peter Jipsen, and Michael Winter, editors, Relational and Algebraic Methods in Computer Science, pages 82-99, Cham, 2020. Springer International Publishing. URL: https://doi.org/10.1007/978-3-030-43520-2_6.
  10. Uli Fahrenberg, Christian Johansen, Georg Struth, and Krzysztof Ziemiański. Languages of higher-dimensional automata. Math. Struct. Comput. Sci., 31(5):575-613, 2021. URL: https://doi.org/10.1017/S0960129521000293.
  11. Uli Fahrenberg, Christian Johansen, Georg Struth, and Krzysztof Ziemianski. Kleene theorem for higher-dimensional automata. Log. Methods Comput. Sci., 20(4):22:1-22:50, 2024. URL: https://doi.org/10.46298/LMCS-20(4:22)2024.
  12. Peter C Fishburn. Interval orders and interval graphs: A study of partially ordered sets. (No Title), 1985. Google Scholar
  13. Jay L. Gischer. The equational theory of pomsets. Theoretical Computer Science, 61(2):199-224, 1988. URL: https://doi.org/10.1016/0304-3975(88)90124-7.
  14. Jan Grabowski. On partial languages. Fundamenta Informaticae, 4(2):427-498, 1981. Google Scholar
  15. Ryszard Janicki and Maciej Koutny. Structure of concurrency. Theoretical Computer Science, 112(1):5-52, 1993. URL: https://doi.org/10.1016/0304-3975(93)90238-O.
  16. André Joyal, Mogens Nielsen, and Glynn Winskel. Bisimulation from open maps. Information and Computation, 127(2):164-185, 1996. URL: https://doi.org/10.1006/inco.1996.0057.
  17. Thomas Kahl. On symmetric higher-dimensional automata and bisimilarity. Theoretical Computer Science, 935:47-60, 2022. URL: https://doi.org/10.1016/j.tcs.2022.07.004.
  18. Mogens Nielsen, Gordon Plotkin, and Glynn Winskel. Petri nets, event structures and domains, part i. Theoretical Computer Science, 13(1):85-108, 1981. URL: https://doi.org/10.1016/0304-3975(81)90112-2.
  19. Vaughan Pratt. Modeling concurrency with geometry. In Proceedings of the 18th ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, POPL '91, pages 311-322, New York, NY, USA, 1991. Association for Computing Machinery. URL: https://doi.org/10.1145/99583.99625.
  20. Vaughan R. Pratt. Modeling concurrency with partial orders. International Journal of Parallel Programming, 15:33-71, 1986. URL: https://doi.org/10.1007/BF01379149.
  21. Mike W Shields. Concurrent machines. The Computer Journal, 28(5):449-465, 1985. URL: https://doi.org/10.1093/COMJNL/28.5.449.
  22. Georg Struth and Krzysztof Ziemiański. Presheaf automata, 2024. URL: https://arxiv.org/abs/2409.04612.
  23. Rob J. van Glabbeek. On the expressiveness of higher dimensional automata. Theoretical Computer Science, 356(3):265-290, 2006. Expressiveness in Concurrency. URL: https://doi.org/10.1016/j.tcs.2006.02.012.
  24. Rob J. van Glabbeek and Gordon D. Plotkin. Configuration structures. In Proceedings of Tenth Annual IEEE Symposium on Logic in Computer Science, pages 199-209, San Deigo, CA, USA, June 1995. IEEE. URL: https://doi.org/10.1109/LICS.1995.523257.
  25. Safa Zouari, Krzysztof Ziemiański, and Uli Fahrenberg. Bisimulations and logics for higher-dimensional automata. In Chutiporn Anutariya and Marcello M. Bonsangue, editors, Theoretical Aspects of Computing - ICTAC 2024, pages 132-150, Cham, 2025. Springer Nature Switzerland. Google Scholar
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