Coupled similarity: the first 32 years


Coupled similarity is an equivalence on (labeled) transition systems; its distinguishing power lies between (weak) bisimilarity and (may) testing equivalence. Its main feature, compared to weak bisimilarity, is an additional \(\tau \)-law that abstracts from the atomicity of internal choices among several possible branches, thus allowing for gradual commitments. The need for this \(\tau \)-law in applications was motivated by van Glabbeek and Vaandrager in 1988. Parrow and Sjödin coined the term coupled simulation in 1992 as a coinductive proof technique for the comparison of distributed (not “just” concurrent) systems, heavily exploiting gradual commitments. Over the years, coupled similarity also gained significance for the definition and analysis of the correctness of encodings, of action refinement and contraction, and of branching-time semantics for various process calculi. In this paper, we compare variants and formalizations of coupled similarity and highlight its relevance.

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  1. 1.

    Parrow and Sjödin [39] and Sangiorgi [48] here just speak of “simulation,” not “weak simulation.” But, from their examples and lemmas, it seems reasonable to read a “weak” into their definitions. Nestmann and Pierce [38] explicitly requires weak simulations.

  2. 2.

    Parrow and Sjödin [39] and Sangiorgi [48] claim a variant of this to hold without the assumption of shared stability. The Isabelle proof in Bisping [7] (following the construction from [48]) however needs this extra assumption and it is not obvious how to dispose of it.

  3. 3.

    Also, later completeness proofs for the stronger weak congruences [1, 13] turned out neater.

  4. 4.

    On \(\textsf {CCS} \) descendants, without the expressive power of \(+\), \(\equiv _{\mathrm{CS}}\) usually is a congruence. One important example is the Asynchronous Pi-Calculus [38, Prop. 2.4.4].

  5. 5.

    An asynchronous simulation\({\mathscr {S}}\) for (PQ) requires the usual (strong and weak) simulation game for \(\tau \)- and output transitions, but instead of doing the same for input transitions, the requirement becomes \((\; \overline{a} \langle z \rangle | P \;,\; \overline{a} \langle z \rangle | Q \;) \in {\mathscr {S}}\) for arbitrary messages \( \overline{a} \langle z \rangle \). Thus, inputs are not observed directly, but only indirectly via potential observable behavior after receptions.


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Bisping, B., Nestmann, U. & Peters, K. Coupled similarity: the first 32 years. Acta Informatica 57, 439–463 (2020).

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