Linear Logic, the π-calculus, and their Metatheory: A Recipe for Proofs as Processes
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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 π-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 π-calculus. What remains is reconstructing the expected metatheory of session types and the π-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 πLL, a new process calculus rooted in linear logic. The key novelty of πLL is that it comes with a carefully formulated design recipe, based on a dialgebraic view of labelled transition systems. Thanks to our recipe, πLL offers the expected transition systems of session types, which we use to establish session fidelity. We use πLL 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 πLL and our recipe form a robust basis for the further exploration of Proofs as Processes, by considering different features: polymorphism, process mobility, and recursion.
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