Linear logic propositions as session types

Abstract

Throughout the years, several typing disciplines for the π-calculus have been proposed. Arguably, the most widespread of these typing disciplines consists of session types. Session types describe the input/output behavior of processes and traditionally provide strong guarantees about this behavior (i.e., deadlock freedom and fidelity). While these systems exploit a fundamental notion of linearity, the precise connection between linear logic and session types has not been well understood. This paper proposes a type system for the π-calculus that corresponds to a standard sequent calculus presentation of intuitionistic linear logic, interpreting linear propositions as session types and thus providing a purely logical account of all key features and properties of session types. We show the deep correspondence between linear logic and session types by exhibiting a tight operational correspondence between cut elimination steps and process reductions. We also discuss an alternative presentation of linear session types based on classical linear logic, and compare our development with other more traditional session type systems. †

Polarized Substructural Session Types

“…In addition, we also have productivity (processes will eventually perform the action prescribed by the session type) and termination if recursive processes are appropriately restricted. The proofs of these properties closely follow those in the literature for related systems [6,22,20], so we do not formally state or prove them here. …”

“…Anticipated with some analogies for some time [11,23], session types have recently been placed upon the firm foundation of linear logic via a Curry-Howard interpretation of linear propositions as types, proofs as processes, and cut reduction as communication. Variations apply for both intuitionistic [5,6] and classical [24] linear logic. This has enabled the application of proof-theoretic techniques in this domain, for example, developing logical relations [17], corecursion [22], and parametricity and behavioral polymorphism [4].…”

Behavioral Polymorphism and Parametricity in Session-Based Communication

“…rule (T∃L)) implies inputing a type and then using the session as A, agnostic to what the actual received type can be. Note that in the presence of polymorphism the identity rule (Tid) (not present in [6,7], but used in [26,21,20]) is necessary, since it is the only way of typing a session with a type variable.…”

“…As session types may describe arbitrarily complex communication protocols, our theory of polymorphic processes enables an expressive form of abstract protocol communication. As a key distinguishing feature, our developments follow naturally from the interpretation of session types as intuitionistic linear logic propositions given in [6,7]. This allows us to obtain central technical results for polymorphic, session-typed processes in a remarkably elegant way:…”

“…In the remainder of this introduction, we briefly describe the logical interpretation of [6] and illustrate the potential of our model of polymorphic sessions with an example. Our ongoing research program on logical foundations for session-based concurrency [6,26,21,7,20,8] builds upon an interpretation of intuitionistic linear logical propositions as session types, sequent proofs as π-calculus processes [25], and cut elimination as process communication. In the resulting Curry-Howard correspondence, well-typed processes enjoy strong forms of type preservation and global progress [6,7], and are strongly normalizing [20].…”