A Modular Formalization of Reversibility for Concurrent Models and Languages

Causal-consistent reversibility is the reference notion of reversibility for concurrency. We introduce a modular framework for defining causal-consistent reversible extensions of concurrent models and languages. We show how our framework can be used to define reversible extensions of formalisms as different as CCS and concurrent X-machines. The generality of the approach allows for the reuse of theories and techniques in different settings.Comment: In Proceedings ICE 2016, arXiv:1608.0313

Filter Models: Non-idempotent Intersection Types, Orthogonality and Polymorphism

This paper revisits models of typed lambda calculus based on filters of intersection types: By using non-idempotent intersections, we simplify a methodology that produces modular proofs of strong normalisation based on filter models. Building such a model for some type theory shows that typed terms can be typed with intersections only, and are therefore strongly normalising. Non-idempotent intersections provide a decreasing measure proving a key termination property, simpler than the reducibility techniques used with idempotent intersections. Such filter models are shown to be captured by orthogonality techniques: we formalise an abstract notion of orthogonality model inspired by classical realisability, and express a filter model as one of its instances, along with two term-models (one of which captures a now common technique for strong normalisation). Applying the above range of model constructions to Curry-style System F describes at different levels of detail how the infinite polymorphism of System F can systematically be reduced to the finite polymorphism of intersection types

Filter models: non-idempotent intersection types, orthogonality and polymorphism - long version

This paper revisits models of typed lambda-calculus based on filters of intersection types: By using non-idempotent intersections, we simplify a methodology that produces modular proofs of strong normalisation based on filter models. Non-idempotent intersections provide a decreasing measure proving a key termination property, simpler than the reducibility techniques used with idempotent intersections. Such filter models are shown to be captured by orthogonality techniques: we formalise an abstract notion of orthogonality model inspired by classical realisability, and express a filter model as one of its instances, along with two term-models (one of which captures a now common technique for strong normalisation). Applying the above range of model constructions to Curry-style System F describes at different levels of detail how the infinite polymorphism of System F can systematically be reduced to the finite polymorphism of intersection types

A Modular Formalization of Reversibility for Concurrent Models and Languages

International audienceCausal-consistent reversibility is the reference notion of reversibility for concurrency. We introduce a modular framework for defining causal-consistent reversible extensions of concurrent models and languages. We show how our framework can be used to define reversible extensions of formalisms as different as CCS and concurrent X-machines. The generality of the approach allows for the reuse of theories and techniques in different settings

Types intersections non-idempotents pour raffiner la normalisation forte avec des informations quantitatives

We study systems of non-idempotent intersection types for different variants of the lambda-calculus and we discuss properties and applications. Besides the pure lambda-calculus itself, the variants are a λ-calculus with explicit substitutions and a lambda-calculus with constructors, matching and a fixpoint operator. The typing systems we introduce for these calculi all characterize strongly normalising terms. But we also show that, by dropping idempotency of intersections, typing a term provides quantitative information about it: a trivial measure on its typing tree gives a boundon the size of the longest beta-reduction sequence from this term to its normal form. We explore how to refine this approach to obtain finer results: some of the typing systems, under certain conditions, even provide the exact measure of this longestbeta-reduction sequence, and the type of a term gives information on the normal form of this term. Moreover, by using filters, these typing systems can be used to define a denotational semantics.Nous étudions des systèmes de typage avec des types intersections non-idempotents pour des variantes du lambda-calcul et nous discutons de leurs propriétés et de leurs applications. Outre le lambda-calcul lui-même, les variantes sont un lambda-calcul avec des substitutions explicites et un lambda-calcul avec des constructeurs, du filtrage et un opérateur de point fixe. Les sytèmes de typage que l'on présente caractérisent les termes fortement normalisables. Mais nous montrons également qu'un jugement de typage d'un terme donne des informations quantitatives : une mesure triviale sur l'arbre de typage d'unlambda-terme quelconque donne une borne sur la taille de la plus longue séquence de beta-reductions depuis ce lambda-terme jusqu'à sa forme normale. Nous raffinons cette approche pour obtenir un résultat plus précis: certains systèmes de typages, sous certaines conditions, donnent même une mesure exacte de cette plus longue séquence de beta-reductions, et le type du terme donne des informations sur la forme normale de ce terme. De plus, en utilisant des filtres, ces systèmes de typage peuvent être utilisés pour définir une sémantique dénotationnelle

A simple presentation of the effective topos

We propose for the Effective Topos an alternative construction: a realisability framework composed of two levels of abstraction. This construction simplifies the proof that the Effective Topos is a topos (equipped with natural numbers), which is the main issue that this paper addresses. In this our work can be compared to Frey’s monadic tripos-to-topos construction. However, no topos theory or even category theory is here required for the construction of the framework itself, which provides a semantics for higher-order type theories, supporting extensional equalities and the axiom of unique choice

Filter models: non-idempotent intersection types, orthogonality and polymorphism

This paper revisits models of typed λ-calculus based on filters of intersection types: By using non-idempotent intersections, we simplify a methodology that produces modular proofs of strong normalisation based on filter models. Non-idempotent intersections provide a decreasing measure proving a key termination property, simpler than the reducibility techniques used with idempotent intersections. Such filter models are shown to be captured by orthogonality techniques: we formalise an abstract notion of orthogonality model inspired by classical realisability, and express a filter model as one of its instances, along with two term-models (one of which captures a now common technique for strong normalisation). Applying the above range of model constructions to Curry-style System F describes at different levels of detail how the infinite polymorphism of System F can systematically be reduced to the finite polymorphism of intersection types

A big-step operational semantics via non-idempotent intersection types

Abstract We present a typing system of non-idempotent intersection types that characterises strongly normalising λ-terms and can been seen as a bigstep operational semantics: we prove that a strongly normalising λ-term accepts, as its type, the structure of its normal form. As a by-product of identifying such semantical components in typing trees, we are able to define a trivial measure (the number of times a typing rule is applied) that exactly captures the length of the longest β-reduction sequence starting from a given typable term.