On the Axiomatisability of Parallel Composition: A Journey in the Spectrum

Authors Luca Aceto , Valentina Castiglioni , Anna Ingólfsdóttir , Bas Luttik , Mathias Ruggaard Pedersen

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Luca Aceto
  • Reykjavik University, Iceland
  • Gran Sasso Science Institute, L'Aquila, Italy
Valentina Castiglioni
  • Reykjavik University, Iceland
Anna Ingólfsdóttir
  • Reykjavik University, Iceland
Bas Luttik
  • Eindhoven University of Technology, The Netherlands
Mathias Ruggaard Pedersen
  • Reykjavik University, Iceland


We thank the anonymous reviewers for their valuable comments, and Rob van Glabbeek for a fruitful discussion on the axiomatisability of failures equivalence.

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Luca Aceto, Valentina Castiglioni, Anna Ingólfsdóttir, Bas Luttik, and Mathias Ruggaard Pedersen. On the Axiomatisability of Parallel Composition: A Journey in the Spectrum. In 31st International Conference on Concurrency Theory (CONCUR 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 171, pp. 18:1-18:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


This paper studies the existence of finite equational axiomatisations of the interleaving parallel composition operator modulo the behavioural equivalences in van Glabbeek’s linear time-branching time spectrum. In the setting of the process algebra BCCSP over a finite set of actions, we provide finite, ground-complete axiomatisations for various simulation and (decorated) trace semantics. On the other hand, we show that no congruence over that language that includes bisimilarity and is included in possible futures equivalence has a finite, ground-complete axiomatisation. This negative result applies to all the nested trace and nested simulation semantics.

Subject Classification

ACM Subject Classification
  • Theory of computation → Equational logic and rewriting
  • Axiomatisation
  • Parallel composition
  • Linear time-branching time spectrum


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