Circular Proofs as Session-Typed Processes: A Local Validity Condition

TL;DR

This paper extends the Curry-Howard correspondence to a fragment of substructural logic with fixed points, providing a local criterion for recognizing valid circular proofs as session-typed processes, enhancing language design.

Contribution

It introduces a local, decidable criterion for validating circular proofs in session-typed processes, improving upon previous guard conditions for better language integration.

Findings

The criterion is stricter than previous guard conditions.

It is local and compositional, suitable for programming language implementation.

The approach effectively recognizes mutually recursive processes.

Abstract

Proof theory provides a foundation for studying and reasoning about programming languages, most directly based on the well-known Curry-Howard isomorphism between intuitionistic logic and the typed lambda-calculus. More recently, a correspondence between intuitionistic linear logic and the session-typed pi-calculus has been discovered. In this paper, we establish an extension of the latter correspondence for a fragment of substructural logic with least and greatest fixed points. We describe the computational interpretation of the resulting infinitary proof system as session-typed processes, and provide an effectively decidable local criterion to recognize mutually recursive processes corresponding to valid circular proofs as introduced by Fortier and Santocanale. We show that our algorithm imposes a stricter requirement than Fortier and Santocanale's guard condition, but is local and compositional and therefore more suitable as the basis for a programming language.