Intersection types and runtime errors in the pi-calculus

Abstract

We introduce a type system for the π-calculus which is designed to guarantee that typable processes are well-behaved, namely they never produce a run-time error and, even if they may diverge, there is always a chance for them to łfinish their workž, i.e., to reduce to an idle process. The introduced type system is based on non-idempotent intersections, and is thus very powerful as for the class of processes it can capture. Indeed, despite the fact that the underlying property is Π 0 2-complete, there is a way to show that the system is complete, i.e., that any well-behaved process is typable, although for obvious reasons infinitely many derivations need to be considered.

A tale of intersection types

“…There are, for example, type systems guaranteeing deadlock and livelock freedom of processes [55,58,64,76], type systems for checking the correctness of communications among processes, such as session types [51], and type systems for proving the termination of processes [35,82]. However, little was known about the existence of type systems able not only to guarantee but also to characterise some relevant property of concurrent processes, i.e., yielding completeness in addition to soundness, until the intersection type system in [33].…”

“…The process calculus in [33] is based on the polyadic asynchronous localised π -calculus [73] (Section 5.6), where a notion of asynchronous hyperlocalised π -process is added. This "hyperlocalisation" means that the resulting calculus is actually a fragment of the localised calculus, i.e., with further constraints on the presence of input names under input prefixes (the limitations introduced by these restrictions are discussed in [33], where it is shown that they have a modest impact on the expressiveness of the calculus) 2 .…”

“…As an example of the use of (non-idempotent) intersection types in [33] we consider the typing of parallel composition. Typing judgments are of the shape Γ ⊢ P :: ∆, where:…”

“…• Γ associates types and channel dependencies to channels used by the process P as inputs; • ∆ associates types to channels used by P in outputs. The intersection-based type system in [33] defines and characterises (by means of a completeness theorem) a safety property for π -processes, named good behaviour, which is essentially a form of deadlock-freedom and may-termination. The verification of the good-behaviour property is shown to be very hard.…”

“…The system stems from a recently introduced construction [62], which is based on the following ideas: (i) intersection types can be seen as approximations in linear logic; (ii) a programming language has an intersection-flavoured type discipline if it can be encoded in linear logic; (iii) it is known that the π -calculus itself can be translated in linear logic. Many encodings are present the literature, such as the one in [50], which is exploited in [33].…”