Circular Proofs as Session-Typed Processes: A Local Validity Condition (1908.01909v5)

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.

• The paper provides a computational interpretation of circular proofs in subsingleton logic as session-typed processes and introduces a local, compositional validity criterion for mutually recursive processes.
• It leverages the Curry-Howard isomorphism to relate logical concepts like fixed points and proof reduction to process concepts like session types, recursion, and computation.
• The introduced local validity criterion is decidable, implies a stronger property (strong progress), and guarantees cut elimination, offering benefits for modular programming language design compared to previous conditions.

This paper explores the connection between circular proofs in substructural logic and session-typed processes. It focuses on a fragment of intuitionistic linear logic called subsingleton logic, extended with least and greatest fixed points. The core contribution is to provide a computational interpretation of this infinitary proof system as session-typed processes and to introduce a local and compositional criterion for validating mutually recursive processes, ensuring they correspond to valid circular proofs. This criterion is stricter than the guard condition proposed by Fortier and Santocanale but offers advantages in terms of locality and composability, making it more suitable for programming language design.

The authors build on the Curry-Howard isomorphism, specifically its extension to linear logic and session types, to relate logical propositions to session types, proofs to concurrent processes, and proof reduction to computation. They leverage the concept of circular proofs, represented as finite trees with loops, to model recursive process definitions.

The paper introduces a synchronous computational semantics for cut reduction in subsingleton logic with fixed points, adapted to the context of session types. The key concept is the use of priorities associated with fixed points. These priorities guide a novel algorithm to check the validity of mutually recursive process definitions.

The algorithm ensures that every valid configuration of processes exhibits a strong progress property, meaning that the configuration will either be empty or attempt communication along external channels within a finite number of steps. The algorithm operates locally, checking the validity of each process definition separately, making it suitable for modular verification of programs. The paper also proves that validity, as determined by their algorithm, implies the guard condition of Fortier and Santocanale, thus guaranteeing cut elimination.

The authors emphasize the symmetry between least and greatest fixed points in their framework. They illustrate that, in the context of session types, least and greatest fixed points are treated in a balanced manner, unlike traditional type theories where inductive reasoning (associated with least fixed points) is often favored. They address the undecidability issues related to general equirecursive types by focusing on isorecursive fixed points and providing an effectively decidable validity criterion for a subset of possible Turing machines encoded as session-typed processes.