Linear Logic, the $π$-calculus, and their Metatheory: A Recipe for Proofs as Processes

Initiated by Abramsky [1994], the Proofs as Processes agenda is to establish a solid foundation for the study of concurrent languages, by researching the connection between linear logic and the $\pi$-calculus. To date, Proofs as Processes is still a partial success. Caires and Pfenning [2010] showed that linear propositions correspond to session types, which prescribe the observable behaviour of processes. Further, Carbone et al. [2018] demonstrated that adopting devices from hypersequents [Avron 1991] shapes proofs such that they correspond to the expected syntactic structure of processes in the $\pi$-calculus. What remains is reconstructing the expected metatheory of session types and the $\pi$-calculus. In particular, the hallmark of session types, session fidelity, still has to be reconstructed: a correspondence between the observable behaviours of processes and their session types, in terms of labelled transitions. In this article, we present $\pi$LL, a new process calculus rooted in linear logic. The key novelty of $\pi$LL is that it comes with a carefully formulated design recipe, based on a dialgebraic view of labelled transition systems. Thanks to our recipe, $\pi$LL offers the expected transition systems of session types, which we use to establish session fidelity. We use $\pi$LL to carry out the first thorough investigation of the metatheoretical properties enforced by linear logic on the observable behaviour of processes, uncovering connections with similarity and bisimilarity. We also prove that $\pi$LL and our recipe form a robust basis for the further exploration of Proofs as Processes, by considering different features: polymorphism, process mobility, and recursion.