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

A Complete Normal-Form Bisimilarity for State

International audienceWe present a sound and complete bisimilarity for an untyped $λ-calculus$ 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 Støvring and Lassen's incomplete environment-based normal-form bisimilarity for the $λρ-calculus$, 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

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

Effectful Normal Form Bisimulation

International audienceNormal form bisimulation, also known as open bisimulation, is a coinductive technique for higher-order program equivalence in which programs are compared by looking at their essentially infinitary tree-like normal forms, i.e. at their Böhm or Lévy-Longo trees. The technique has been shown to be useful not only when proving metatheorems about λ-calculi and their semantics, but also when looking at concrete examples of terms. In this paper, we show that there is a way to generalise normal form bisimulation to calculi with algebraic effects, à la Plotkin and Power. We show that some mild conditions on monads and relators, which have already been shown to guarantee effectful applicative bisimi-larity to be a congruence relation, are enough to prove that the obtained notion of bisimilarity, which we call effectful normal form bisimilarity, is a congruence relation, and thus sound for contextual equivalence. Additionally , contrary to applicative bisimilarity, normal form bisimilarity allows for enhancements of the bisimulation proof method, hence proving a powerful reasoning principle for effectful programming languages

Proving Soundness of Extensional Normal-Form Bisimilarities

Normal-form bisimilarity is a simple, easy-to-use behavioral equivalence that relates terms in $\lambda$-calculi by decomposing their normal forms into bisimilar subterms. Moreover, it typically allows 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 $\eta$-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 up to context technique are sound. We illustrate our technique with three calculi: the call-by-value $\lambda$-calculus, the call-by-value $\lambda$-calculus with the delimited-control operators shift and reset, and the call-by-value $\lambda$-calculus with the abortive control operators call/cc and abort. In the first two cases, there was previously no sound up to context technique validating the $\eta$-law, whereas no theory of normal-form bisimulations for a calculus with call/cc and abort has been presented before. Our results have been fully formalized in the Coq proof assistant

Proving Soundness of Extensional Normal-Form Bisimilarities

International audienceNormal-form bisimilarity is a simple, easy-to-use behavioral equivalence that relates terms in lambda-calculi by decomposing their normal forms into bisimilar subterms. Moreover, it typically allows 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 eta-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 up to context technique are sound. We illustrate our technique with three calculi: the call-by-value lambda-calculus, the call-by-value lambda-calculus with the delimited-control operators shift and reset, and the call-by-value lambda-calculus with the abortive control operators call/cc and abort. In the first two cases, there was previously no sound up to context technique validating the eta-law, whereas no theory of normal-form bisimulations for a calculus with call/cc and abort has been presented before. Our results have been fully formalized in the Coq proof assistant

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

Foundations of Software Science and Computation Structures

This open access book constitutes the proceedings of the 22nd International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2019, which took place in Prague, Czech Republic, in April 2019, held as part of the European Joint Conference on Theory and Practice of Software, ETAPS 2019. The 29 papers presented in this volume were carefully reviewed and selected from 85 submissions. They deal with foundational research with a clear significance for software science