The call-by-value Lambda-Calculus with generalized applications

The lambda-calculus with generalized applications is the Curry-Howard counterpart to the system of natural deduction with generalized elimination rules for intuitionistic implicational logic. In this paper we identify a call-by-value variant of the system and prove confluence, strong normalization, and standardization. In the end, we show that the cbn and cbv variants of the system simulate each other via mappings based on extensions of the "protecting-by-a-lambda" compilation technique.FCT -Fundação para a Ciência e a Tecnologia(UID/MAT/00013/2013

Un Lambda-Calcul Atomique

International audienceNous introduisons un lambda-calcul avec partage explicite, le lambda-calcul atomique, dans lequel la duplication des sous-termes est faite pas à pas en fonction des constructeurs. Nous donnons une fonction de dénotation du lambda-calcul atomique dans le lambda-calcul et montrons que le lambda-calcul atomique simule la -réduction et préserve la normalisation forte. Nous donnons aussi un système de type pour le lambda-calcul atomique et montrons que la réduction préserve le type

Head reduction and normalization in a call-by-value lambda-calculus

Recently, a standardization theorem has been proven for a variant of Plotkin\u27s call-by-value lambda-calculus extended by means of two commutation rules (sigma-reductions): this result was based on a partitioning between head and internal reductions. We study the head normalization for this call-by-value calculus with sigma-reductions and we relate it to the weak evaluation of original Plotkin\u27s call-by-value lambda-calculus. We give also a (non-deterministic) normalization strategy for the call-by-value lambda-calculus with sigma-reductions

Extensional proofs in a propositional logic modulo isomorphisms

System I is a proof language for a fragment of propositional logic where isomorphic propositions, such as $A\wedge B$ and $B\wedge A$, or $A\Rightarrow(B\wedge C)$ and $(A\Rightarrow B)\wedge(A\Rightarrow C)$ are made equal. System I enjoys the strong normalisation property. This is sufficient to prove the existence of empty types, but not to prove the introduction property (every closed term in normal form is an introduction). Moreover, a severe restriction had to be made on the types of the variables in order to obtain the existence of empty types. We show here that adding $\eta$-expansion rules to System I permits to drop this restriction, and yields a strongly normalising calculus with enjoying the full introduction property.Comment: 15 pages plus references and appendi

: De la réduction linéaire de tête à l'évaluation paresseuse

National audienceÀ partir de la réduction linéaire de tête, nous dérivons de manière systématique un calcul en appel par nécessité. L'introduction d'un calcul pour la réduction linéaire de tête, basée sur une analyse fine de la notion de radicaux premiers de Danos et Regnier, nous permet de construire pas à pas un lambda-calcul en appel par nécessité que l'on compare aux calculs présents dans la littérature

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

Normal Form Bisimulations By Value

Normal form bisimilarities are a natural form of program equivalence resting on open terms, first introduced by Sangiorgi in call-by-name. The literature contains a normal form bisimilarity for Plotkin's call-by-value $\lambda$-calculus, Lassen's \emph{enf bisimilarity}, which validates all of Moggi's monadic laws and can be extended to validate $\eta$. It does not validate, however, other relevant principles, such as the identification of meaningless terms -- validated instead by Sangiorgi's bisimilarity -- or the commutation of \letexps. These shortcomings are due to issues with open terms of Plotkin's calculus. We introduce a new call-by-value normal form bisimilarity, deemed \emph{net bisimilarity}, closer in spirit to Sangiorgi's and satisfying the additional principles. We develop it on top of an existing formalism designed for dealing with open terms in call-by-value. It turns out that enf and net bisimilarities are \emph{incomparable}, as net bisimilarity does not validate Moggi's laws nor $\eta$. Moreover, there is no easy way to merge them. To better understand the situation, we provide an analysis of the rich range of possible call-by-value normal form bisimilarities, relating them to Ehrhard's relational model.Comment: Rewritten version (deleted toy similarity and explained proof method on naive similarity) -- Submitted to POPL2