An Infinitary Proof Theory of Linear Logic Ensuring Fair Termination in the Linear $\pi$-Calculus

TL;DR

This paper develops an infinitary proof system extending linear logic to model fair termination in process calculi, ensuring that well-typed processes eventually perform all actions despite potential infinite interactions.

Contribution

It introduces a conservative extension of $ ext{MALL}^ ext{infty}$ that aligns cut elimination with fair termination, providing a behavioral type system for $ ext{piLIN}$.

Findings

Proof terms correspond to processes in $ ext{piLIN}$.

The type system guarantees fair termination for well-typed processes.

Processes may have infinite interactions but still ensure eventual action completion.

Abstract

Fair termination is the property of programs that may diverge "in principle" but that terminate "in practice", i.e. under suitable fairness assumptions concerning the resolution of non-deterministic choices. We study a conservative extension of $\mu$MALL$^\infty$, the infinitary proof system of the multiplicative additive fragment of linear logic with least and greatest fixed points, such that cut elimination corresponds to fair termination. Proof terms are processes of $\pi$LIN, a variant of the linear $\pi$-calculus with (co)recursive types into which binary and (some) multiparty sessions can be encoded. As a result we obtain a behavioral type system for $\pi$LIN (and indirectly for session calculi through their encoding into $\pi$LIN) that ensures fair termination: although well-typed processes may engage in arbitrarily long interactions, they are fairly guaranteed to eventually perform all pending actions.