Howe's Method for Contextual Semantics

International audienceWe show how to use Howe's method to prove that context bisimilarity is a congruence for process calculi equipped with their usual semantics. We apply the method to two extensions of HOπ, with passivation and with join patterns, illustrating different proof techniques

Proving termination of evaluation for System F with control operators

We present new proofs of termination of evaluation in reduction semantics (i.e., a small-step operational semantics with explicit representation of evaluation contexts) for System F with control operators. We introduce a modified version of Girard's proof method based on reducibility candidates, where the reducibility predicates are defined on values and on evaluation contexts as prescribed by the reduction semantics format. We address both abortive control operators (callcc) and delimited-control operators (shift and reset) for which we introduce novel polymorphic type systems, and we consider both the call-by-value and call-by-name evaluation strategies.Comment: In Proceedings COS 2013, arXiv:1309.092

Bisimulations for Delimited-Control Operators

We present a comprehensive study of the behavioral theory of an untyped $\lambda$-calculus extended with the delimited-control operators shift and reset. To that end, we define a contextual equivalence for this calculus, that we then aim to characterize with coinductively defined relations, called bisimilarities. We consider different styles of bisimilarities (namely applicative, normal-form, and environmental) within a unifying framework, and we give several examples to illustrate their respective strengths and weaknesses. We also discuss how to extend this work to other delimited-control operators

Characterizing contextual equivalence in calculi with passivation

AbstractWe study the problem of characterizing contextual equivalence in higher-order languages with passivation. To overcome the difficulties arising in the proof of congruence of candidate bisimilarities, we introduce a new form of labeled transition semantics together with its associated notion of bisimulation, which we call complementary semantics. Complementary semantics allows to apply the well-known Howeʼs method for proving the congruence of bisimilarities in a higher-order setting, even in the presence of an early form of bisimulation. We use complementary semantics to provide a coinductive characterization of contextual equivalence in the HOπP calculus, an extension of the higher-order π-calculus with passivation, obtaining the first result of this kind. We then study the problem of defining a more effective variant of bisimilarity that still characterizes contextual equivalence, along the lines of Sangiorgiʼs notion of normal bisimilarity. We provide partial results on this difficult problem: we show that a large class of test processes cannot be used to derive a normal bisimilarity in HOπP, but we show that a form of normal bisimilarity can be defined for HOπP without restriction

Applicative Bisimilarities for Call-by-Name and Call-by-Value λμ-Calculus

International audienceWe propose the first sound and complete bisimilarities for the call-by-name and call-by-value untyped λµ-calculus, defined in the applicative style. We give equivalence examples to illustrate how our relations can be used; in particular, we prove David and Py's counter-example, which cannot be proved with Lassen's preexisting normal form bisimilarities for the λµ-calculus

Sound and Complete Bisimilarities for Call-by-Name and Call-by-Value Lambda-mu Calculus

We propose the first sound and complete bisimilarities for the call-by-name and call-by-value untyped lambda-mu calculus. We define applicative bisimilarities for both semantics and environmental bisimilarity for call-by-name. We give equivalence examples to illustrate how our relations can be used; in particular, we prove David and Py's counter-example, which cannot be proved with Lassen's preexisting normal form bisimilarities for the lambda-mu calculus.Nous proposons les premières définitions de bisimilarités correctes et complètes pour le lambda-mu calcul non typé en appel par nom et en appel par valeur. Nous définissons une bisimilarité applicative pour chacune des sémantiques, et une bisimilarité environnementale en appel par nom. Nous donnons des examples d'équivalences pour montrer comment ces relations peuvent être utilisées ; en particulier, nous prouvons le contre-exemple de David et Py, qui ne peut être démontré avec la bisimilarité de forme normale définie auparavant par Lassen

Environmental Bisimulations for Delimited-Control Operators

International audienceWe present a theory of environmental bisimilarity for the delimited-control operators shift and reset. We consider two different notions of contextual equivalence: one that does not require the presence of a top-level control delimiter when executing tested terms, and another one, fully compatible with the original CPS semantics of shift and reset, that does. For each of them, we develop sound and complete environmental bisimilarities, and we discuss up-to techniques

Diacritical Companions

International audienceCoinductive reasoning in terms of bisimulations is in practice routinely supported by carefully crafted up-to techniques that can greatly simplify proofs. However, designing and proving such bisimulation enhancements sound can be challenging, especially when striving for modularity. In this article, we present a theory of up-to techniques that builds on the notion of companion introduced by Pous and that extends our previous work which allows for powerful up-to techniques defined in terms of diacritical progress of relations. The theory of diacritical companion that we put forward works in any complete lattice and makes it possible to modularly prove soundness of up-to techniques which rely on the distinction between passive and active progresses, such as up to context in λ-calculi with control operators and extensionality

A Complete Normal-Form Bisimilarity for State

We present a sound and complete bisimilarity for an untyped $\lambda$-calculus with higher-order local references. Our relation compares values by applying them to a fresh variable, like normal-form bisimilarity, and it uses environments to account for the evolving store. We achieve completeness by a careful treatment of evaluation contexts comprising open stuck terms. This work improves over Stovring and Lassen’s incomplete environment-based normal-form bisimilarity for the $\lambda \rho$-calculus, and confirms, in relatively elementary terms, Jaber and Tabareau’s result, that the state construct is discriminative enough to be characterized with a bisimilarity without any quantification over testing arguments.Nous définissons une bisimilarité correcte et complète pour un λ-calcul non typé avec des références locales d’ordre supérieur. Notre relation compare les valeurs en leur passant comme argument une variable fraîche, comme la bisimilarité de forme normale, et utilise des environnements pour prendre en compte l’ évolution de la mémoire. Nous obtenons la complétude par un traîtement méticuleux des contextes d’ évaluation qui englobent les termes bloqués

Proving Soundness of Extensional Normal-Form Bisimilarities

International audienceNormal-form bisimilarity is a simple, easy-to-use behavioral equivalence that relates terms in λ-calculi by decomposing their normal forms into bisimilar subterms. Besides, they allow for powerful up-to techniques, such as bisimulation up to context, which simplify bisimulation proofs even further. However, proving soundness of these relations becomes complicated in the presence of η-expansion and usually relies on ad-hoc proof methods which depend on the language. In this paper, we propose a more systematic proof method to show that an extensional normal-form bisimilarity along with its corresponding bisimulation up to context are sound. We illustrate our technique with the call-by-value λ-calculus, before applying it to a call-by-value λ-calculus with the delimited-control operators shift and reset. In both cases, there was previously no sound bisimulation up to context validating the η-law. Our results have been formalized in the Coq proof assistant