Taylor expansion for Call-By-Push-Value

The connection between the Call-By-Push-Value lambda-calculus introduced by Levy and Linear Logic introduced by Girard has been widely explored through a denotational view reflecting the precise ruling of resources in this language. We take a further step in this direction and apply Taylor expansion introduced by Ehrhard and Regnier. We define a resource lambda-calculus in whose terms can be used to approximate terms of Call-By-Push-Value. We show that this approximation is coherent with reduction and with the translations of Call-By-Name and Call-By-Value strategies into Call-By-Push-Value

Factorization in Call-by-Name and Call-by-Value Calculi via Linear Logic

In each variant of the λ-calculus, factorization and normalization are two key properties that show how results are computed.Instead of proving factorization/normalization for the call-by-name (CbN) and call-by-value (CbV) variants separately, we prove them only once, for the bang calculus (an extension of the λ-calculus inspired by linear logic and subsuming CbN and CbV), and then we transfer the result via translations, obtaining factorization/normalization for CbN and CbV.The approach is robust: it still holds when extending the calculi with operators and extra rules to model some additional computational features

Factorization in Call-by-Name and Call-by-Value Calculi via Linear Logic

International audienceAbstract In each variant of the $$\lambda$$ λ -calculus, factorization and normalization are two key properties that show how results are computed. Instead of proving factorization/normalization for the call-by-name (CbN) and call-by-value (CbV) variants separately, we prove them only once, for the bang calculus (an extension of the $$\lambda$$ λ -calculus inspired by linear logic and subsuming CbN and CbV), and then we transfer the result via translations, obtaining factorization/normalization for CbN and CbV. The approach is robust: it still holds when extending the calculi with operators and extra rules to model some additional computational features

Encoding Tight Typing in a Unified Framework

This paper explores how the intersection type theories of call-by-name (CBN) and call-by-value (CBV) can be unified in a more general framework provided by call-by-push-value (CBPV). Indeed, we propose tight type systems for CBN and CBV that can be both encoded in a unique tight type system for CBPV. All such systems are quantitative, i.e. they provide exact information about the length of normalization sequences to normal form as well as the size of these normal forms. Moreover, the length of reduction sequences are discriminated according to their multiplicative and exponential nature, a concept inherited from linear logic. Last but not least, it is possible to extract quantitative measures for CBN and CBV from their corresponding encodings in CBPV

The Bang Calculus Revisited

Call-by-Push-Value (CBPV) is a programming paradigm subsuming both Call-by-Name (CBN) and Call-by-Value (CBV) semantics. The paradigm was recently modelled by means of the Bang Calculus, a term language connecting CBPV and Linear Logic. This paper presents a revisited version of the Bang Calculus, called $\lambda !$, enjoying some important properties missing in the original system. Indeed, the new calculus integrates commutative conversions to unblock value redexes while being confluent at the same time. A second contribution is related to non-idempotent types. We provide a quantitative type system for our $\lambda !$-calculus, and we show that the length of the (weak) reduction of a typed term to its normal form \emph{plus} the size of this normal form is bounded by the size of its type derivation. We also explore the properties of this type system with respect to CBN/CBV translations. We keep the original CBN translation from $\lambda$-calculus to the Bang Calculus, which preserves normal forms and is sound and complete with respect to the (quantitative) type system for CBN. However, in the case of CBV, we reformulate both the translation and the type system to restore two main properties: preservation of normal forms and completeness. Last but not least, the quantitative system is refined to a \emph{tight} one, which transforms the previous upper bound on the length of reduction to normal form plus its size into two independent \emph{exact} measures for them

Finitary Simulation of Infinitary β\beta-Reduction via Taylor Expansion, and Applications

Originating in Girard's Linear logic, Ehrhard and Regnier's Taylor expansion of $\lambda$-terms has been broadly used as a tool to approximate the terms of several variants of the $\lambda$-calculus. Many results arise from a Commutation theorem relating the normal form of the Taylor expansion of a term to its B\"ohm tree. This led us to consider extending this formalism to the infinitary $\lambda$-calculus, since the $\Lambda_{\infty}^{001}$ version of this calculus has B\"ohm trees as normal forms and seems to be the ideal framework to reformulate the Commutation theorem. We give a (co-)inductive presentation of $\Lambda_{\infty}^{001}$. We define a Taylor expansion on this calculus, and state that the infinitary $\beta$-reduction can be simulated through this Taylor expansion. The target language is the usual resource calculus, and in particular the resource reduction remains finite, confluent and terminating. Finally, we state the generalised Commutation theorem and use our results to provide simple proofs of some normalisation and confluence properties in the infinitary $\lambda$-calculus

Foundations of Software Science and Computation Structures

This open access book constitutes the proceedings of the 24th International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2021, which was held during March 27 until April 1, 2021, as part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2021. The conference was planned to take place in Luxembourg and changed to an online format due to the COVID-19 pandemic. The 28 regular papers presented in this volume were carefully reviewed and selected from 88 submissions. They deal with research on theories and methods to support the analysis, integration, synthesis, transformation, and verification of programs and software systems

A geometry of calculus

L’informatique fondamentale et de la théorie de la démonstration. Deux approches sont menées : la première consiste à examiner les mécanismes d’approximation multilinéaires dans des systèmes issus du λ-calcul et de la Logique Linéaire. La seconde consiste à étudier les modèles topologiques pour les systèmes distribués et à les adapter aux algorithmes probabilistes. On étudie d’abord le développement de Taylor des réseaux de preuve de la Logique Linéaire. On introduit des méthodes de démonstration qui utilisent la géométrie de l’élimination des coupures des réseaux multiplicatifs, et qui permettent de manipuler des sommes infinies de réseaux de façon sûre et correcte, pour en extraire des propriétés sur les réductions qui sont à l’œuvre. Ensuite, nous introduisons un langage permettant de définir le développement de Taylor syntaxique pour l’Appel-Par-Pousse-Valeur (Call-By-Push-Value), en capturant certaines propriétés de la sémantique dénotationelle liées aux morphismes de coalgèbres. Puis nous nous intéressons aux systèmes distribués (à mémoire partagée, tolérants aux pannes), et au problème du Consensus. On utilise un modèle topologique qui permet d’interpréter la communication dans les complexes simpliciaux, eton l’adapte de façon à transformer les résultats d’impossibilité bien connus en résultats de borne inférieure de probabilité pour des algorithmes probabilistesThis Phd thesis presents a quantitative study of various computation models of fundamental computer science and proof theory, in two principad directions :the first consists in the examination of mecanismis of multilinear approximations in systems related to λ-calculus and Linear Logic. The second consists in a study of topological models for asynchronous distributed systems, and probabilistic algorithms. We first study Taylor expansion in Linear Logic proof nets. We introduce proof methods using the geometry of cut elimination in multiplicativenets, and which allow to work with infinite sums of nets in a safe and correct way,in order to extract properties about reduction. Then, we introduce a language allowing us to define Taylor expansion for Call-By-Push-Value, while capturing some properties of the denotational semantics, related to coalgebras morphisms.We focus then on fault tolerant-distributed systems with shared memory, andto Consensus problem. We use a topological model which allows to interpret communication with simplicial complexes, and we adapt in so as to transform the well-known impossibility results in lower bounds for probabilistic algorithm

Une géométrie du calcul : Réseaux de preuve, Appel-Par-Pousse-Valeur et Topologie du consensus

This Phd thesis presents a quantitative study of various computation modelsof fundamental computer science and proof theory, in two principad directions :the first consists in the examination of mecanismis of multilinear approximationsin systems related toλ-calculus and Linear Logic. The second consists in a studyof topological models for asynchronous distributed systems, and probabilisticalgorithms. We first study Taylor expansion in Linear Logic proof nets. Weintroduce proof methods using the geometry of cut elimination in multiplicativenets, and which allow to work with infinite sums of nets in a safe and correct way,in order to extract properties about reduction. Then, we introduce a languageallowing us to define Taylor expansion for Call-By-Push-Value, while capturingsome properties of the denotational semantics, related to coalgebras morphisms.We focus then on fault tolerant-distributed systems with shared memory, andto Consensus problem. We use a topological model which allows to interpretcommunication with simplicial complexes, and we adapt in so as to transformthe well-known impossibility results in lower bounds for probabilistic algorithms.Cette thèse propose une étude quantitative de plusieurs modèles de calcul de l’informatique fondamentale et de la théorie de la démonstration. Deux approches sont menées : la première consiste à examiner les mécanismes d’approximation multilinéaires dans des systèmes issus du λ-calcul et de la Logique Linéaire. La seconde consiste à étudier les modèles topologiques pour les systèmes distribués et à les adapter aux algorithmes probabilistes. On étudie d’abord le développement de Taylor des réseaux de preuve de la Logique Linéaire. On introduit des méthodes de démonstration qui utilisent la géométrie de l’élimination des coupures des réseaux multiplicatifs, et qui permettent de manipuler des sommes infinies de réseaux de façon sûre et correcte, pour en extraire des propriétés sur les réductions qui sont à l’œuvre. Ensuite, nous introduisons un langage permettant de définir le développement de Taylor syntaxique pour l’Appel-Par-Pousse-Valeur (Call-By-Push-Value), en capturant certaines propriétés de la sémantique dénotationelle liées aux morphismes de coalgèbres. Puis nous nous intéressons aux systèmes distribués (à mémoire partagée, tolérants aux pannes), et au problème du Consensus. On utilise un modèle topologique qui permet d’interpréter la communication dans les complexes simpliciaux, et on l’adapte de façon à transformer les résultats d’impossibilité bien connus en résultats de borne inférieure de probabilité pour des algorithmes probabilistes