Proof-relevant π-calculus: a constructive account of concurrency and causality

Abstract

We present a formalisation in Agda of the theory of concurrent transitions, residuation, and causal equivalence of traces for the π-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 π-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.

HO$$\pi $$ in Coq

“…Formalizations using de Bruijn indices have been written in Coq [22,23], Isabelle/HOL [19], and Agda [41]. Hirschkoff's work [22,23] is the closest to ours, as he formalizes a polyadic variant of the π-calculus.…”

“…While the definition of this permutation is not in the paper, it should be similar to the permut function we use (cf Section 4.2). The formalization of Perera and Cheney [41] follows a similar path in a monadic setting; their function only needs to exchange 0 and 1. Gay [19] uses a completely different strategy, as he relies on an intermediate language similar to the λ-calculus (formalized with de Bruijn indices), in which he encodes the π-calculus constructs.…”

“…If the first-order π-calculus has been formalized in various proof assistants using different representations for binders (Gay [19] and Perera and Cheney [41] list some of them), only a few recent works propose a formalization of a higherorder calculus [40,33]. The semantics of the calculus of Parrow et al [40] is based on triggers and clauses to enable the execution of transmitted processes.…”

Intrinsically-Typed Mechanized Semantics for Session Types

“…No attempt is made at formalizing the semantics or the equational theory. Perera and Cheney [2018] formalize a proof of a diamond lemma for concurrent execution in π -calculus in Agda. Their calculus is untyped, the formalization relies on de Bruijn encoding for binders and encodes the semantics using an inductive relation.…”

HOπ in Coq

“…If the first-order π -calculus has been formalized in various proof assistants, using different representations for binders (Gay [9] and Perera and Cheney [18] list some of them), only a few recent works propose a formalization of a higher-order calculus [15,17]. The semantics of the calculus of Parrow et al [17] is based on triggers and clauses to enable the execution of transmitted processes.…”