Proof-relevant $\pi$-calculus: a constructive account of concurrency and causality

TL;DR

This paper formalizes the theory of concurrent transitions, residuation, and causal equivalence in the pi-calculus using Agda, providing proofs of key lemmas and a novel approach to trace equivalence involving binders.

Contribution

It introduces a formal Agda-based framework for concurrency in pi-calculus, including proofs of the diamond lemma and a new trace equivalence accommodating binders.

Findings

Proof of the diamond lemma for concurrent transitions.

Formal definition of trace equivalence up to permutation.

Application of braiding to relate states with different binders.

Abstract

We present a formalisation in Agda of the theory of concurrent transitions, residuation, and causal equivalence of traces for the pi-calculus. Our formalisation employs de Bruijn indices and dependently-typed syntax, and aligns the "proved transitions" proposed by Boudol and Castellani in the context of CCS with the proof terms naturally present in Agda's representation of the labelled transition relation. Our main contributions are proofs of the "diamond lemma" for the residuals of concurrent transitions and a formal definition of equivalence of traces up to permutation of transitions. In the pi-calculus transitions represent propagating binders whenever their actions involve bound names. To accommodate these cases, we require a more general diamond lemma where the target states of equivalent traces are no longer identical, but are related by a braiding that rewires the bound and free names to reflect the particular interleaving of events involving binders. Our approach may be useful for modelling concurrency in other languages where transitions carry metadata sensitive to particular interleavings, such as dynamically allocated memory addresses.