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

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${}^{\mathrm{\infty }}$, 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.