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HO\(\pi \) in Coq

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Abstract

We present a formalization of HO\(\pi \) in Coq, a process calculus where messages carry processes. Such a higher-order calculus features two very different kinds of binder: process input, similar to \(\lambda \)-abstraction, and name restriction, whose scope can be expanded by communication. For the latter, we compare four approaches to represent binders: locally nameless, de Bruijn indices, nominal, and Higher-Order Abstract Syntax. In each case, we formalize strong context bisimilarity and prove it is compatible, i.e., closed under every context, using Howe’s method, based on several proof schemes we developed in a previous paper.

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Notes

  1. In the code, we define coercions from respectively

    figure bs

    and

    figure bt

    to

    figure bu

  2. Charguéraud [13] denotes this property as locally closed, but we prefer to use a different term, as our notion of closed process refers to process variables and not names.

  3. Regular HOAS, which relies on functions from terms to terms, cannot be used in Coq to define the syntax of some object language L, as inductive types of the form \((L \rightarrow L) \rightarrow L\) are not allowed [18].

  4. Process functions can be seen as contexts, since they map name to terms.

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Authors and Affiliations

  1. Université de Rennes 1 - ENS Lyon, Rennes, France

    Guillaume Ambal

  2. Université de Lorraine, Nancy, France

    Sergueï Lenglet

  3. Inria, Rennes, France

    Alan Schmitt

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Appendix

Appendix

We define two processes P and Q such that \(b \in \textsf {fn}(P) \cup \textsf {fn}(Q)\), \(P \sim Q\), but renaming a into b in P and Q breaks the bisimilarity. The example has not been formalized in Coq, so we use the notations of Sect. 2 for readability.

In a calculus with a choice operator, so that

figure acm

the example would be \(P \mathrel {{\mathop {=}\limits ^{\varDelta }}}\overline{a}!(\oslash ).\oslash \parallel b?X.\oslash \) and \(Q \mathrel {{\mathop {=}\limits ^{\varDelta }}}\overline{a}!(\oslash ).b?X.\oslash + b?X.\overline{a}!(\oslash ).\oslash \). The process P can either do an output on a and then an input on b, which corresponds to the first branch in Q, or do the opposite, which corresponds to the second branch. If we rename a into b in P and Q (written \(\{ a \rightarrow b \} P\)), P can do a communication on a, a \(\xrightarrow {\tau }\)-transition that Q cannot match.

Erratum The conference version of this article then defines two HO\(\pi \) processes P and Q which mimic the above behavior without using \(+\). The example is incorrect: the P and Q given in the conference version are not strong bisimilar, some \(\tau \)-actions are not matched. In the light of previous works by Hirschkoff and Pous [24], we conjecture that we cannot find two HO\(\pi \) processes that are strongly bisimilar but are no longer bisimilar after a renaming.

However, there exist such processes if we consider weak bisimilarity instead of strong bisimilarity, and if we consider more expressive calculi (as shown with \(+\) above). Therefore we believe Definition 5 is still the right property to establish in general.

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Ambal, G., Lenglet, S. & Schmitt, A. HO\(\pi \) in Coq. J Autom Reasoning 65, 75–124 (2021). https://doi.org/10.1007/s10817-020-09553-0

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