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Answer type polymorphism in callbyname continuation passing
 In Proc. European Symposium on Programming, Springer Lecture Notes in Comput. Sci
, 2004
"... Abstract. This paper studies continuations by means of a polymorphic type system. The traditional callbyname continuation passing style transform admits a typing in which some answer types are polymorphic, even in the presence of firstclass control operators. By building on this polymorphic typin ..."
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Abstract. This paper studies continuations by means of a polymorphic type system. The traditional callbyname continuation passing style transform admits a typing in which some answer types are polymorphic, even in the presence of firstclass control operators. By building on this polymorphic typing, and using parametricity reasoning, we show that the callbyname transform satisfies the etalaw, and is in fact isomorphic to the more recent CPS transform defined by Streicher. 1
Resource modalities in tensor logic
"... The description of resources in game semantics has never achieved the simplicity and precision of linear logic, because of the misleading conception that linear logic is more primitive than game semantics. Here, we defend the opposite view, and thus advocate that game semantics is conceptually more ..."
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The description of resources in game semantics has never achieved the simplicity and precision of linear logic, because of the misleading conception that linear logic is more primitive than game semantics. Here, we defend the opposite view, and thus advocate that game semantics is conceptually more primitive than linear logic. This revised point of view leads us to introduce tensor logic, a primitive variant of linear logic where negation is not involutive. After formulating its categorical semantics, we interpret tensor logic in a model based on Conway games equipped with a notion of payoff, in order to reflect the various resource policies of the logic: linear, affine, relevant or exponential.
On various negative translations
 Proceedings of CL&C 2010, volume 47 of EPTCS
, 2010
"... Several proof translations of classical mathematics into intuitionistic mathematics have been proposed in the literature over the past century. These are normally referred to as negative translations or doublenegation translations. Among those, the most commonly cited are translations due to Kolm ..."
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Several proof translations of classical mathematics into intuitionistic mathematics have been proposed in the literature over the past century. These are normally referred to as negative translations or doublenegation translations. Among those, the most commonly cited are translations due to Kolmogorov, Gödel, Gentzen, Kuroda and Krivine (in chronological order). In this paper we propose a framework for explaining how these different translations are related to each other. More precisely, we define a notion of a (modular) simplification starting from Kolmogorov translation, which leads to a partial order between different negative translations. In this derived ordering, Kuroda and Krivine are minimal elements. Two new minimal translations are introduced, with Gödel and Gentzen translations sitting in between Kolmogorov and one of these new translations. 1
Polarity and the Logic of Delimited Continuations
"... Abstract—Polarized logic is the logic of values and continuations, and their interaction through continuationpassing style. The main limitations of this logic are the limitations of CPS: that continuations cannot be composed, and that programs are fully sequentialized. Delimited control operators w ..."
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Abstract—Polarized logic is the logic of values and continuations, and their interaction through continuationpassing style. The main limitations of this logic are the limitations of CPS: that continuations cannot be composed, and that programs are fully sequentialized. Delimited control operators were invented in response to the limitations of classical continuationpassing. That suggests the question: what is the logic of delimited continuations? We offer a simple account of delimited control, through a natural generalization of the classical notion of polarity. This amounts to breaking the perfect symmetry between positive and negative polarity in the following way: answer types are positive. Despite this asymmetry, we retain all of the classical polarized connectives, and can explain “intuitionistic polarity ” (e.g., in systems like CBPV) as a restriction on the use of connectives, i.e., as a logical fragment. Our analysis complements and generalizes existing accounts of delimited control operators, while giving us a rich logical language through which to understand the interaction of control with monadic effects. I.
Differential Linear Logic and Polarization
"... We study an extension of EhrhardRegnier's differential linear logic along the lines of Laurent's polarization. We show that a particular object of the wellknown relational model of linear logic provides a denotational semantics for this new system, which canonically extends the semanti ..."
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We study an extension of EhrhardRegnier's differential linear logic along the lines of Laurent's polarization. We show that a particular object of the wellknown relational model of linear logic provides a denotational semantics for this new system, which canonically extends the semantics of both differential and polarized linear logics: this justifies our choice of cut elimination rules. Then we show this new system models the recently introduced convolution _*ucalculus, the same as linear logic decomposes calculus.
Jump from parallel to sequential proofs: exponentials,” 2011, to appear in the special number Di?erential Linear Logic, Nets, and other quantitative approaches to ProofTheory of MSCS
"... In previous works, by importing ideas from game semantics (notably FaggianMaurelCurien’s ludics nets), we defined a new class of multiplicative/additive polarized proof nets, called Jproof nets. The distinctive feature of Jproof nets with respect to other proof net syntaxes, is the possibility o ..."
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In previous works, by importing ideas from game semantics (notably FaggianMaurelCurien’s ludics nets), we defined a new class of multiplicative/additive polarized proof nets, called Jproof nets. The distinctive feature of Jproof nets with respect to other proof net syntaxes, is the possibility of representing proof nets which are partially sequentialized, by using jumps (that is, untyped extra edges) as sequentiality constraints. Starting from this result, in the present work we extend Jproof nets to the multiplicative/exponential fragment, in order to take into account structural rules: more precisely, we replace the familiar linear logic notion of exponential box with a less restricting one (called cone) defined by means of jumps. As a consequence, we get a syntax for polarized nets where, instead of a structure of boxes nested one into the other, we have one of cones which can be partially overlapping. Moreover, we define cutelimination for exponential Jproof nets, proving, by a variant of Gandy’s method, that even in case of “superposed ” cones, reduction enjoys confluence and strong normalization.
The duality of computation under focus
, 2009
"... We use the infrastructure of CurienHerbelin’s λµ˜µcalculus to provide a syntax with patterns for a (weakly) focalised version of LK, which we call LKQ given its close relation with DanosJoinetSchellinx’s LKQ. Exploiting a simple game of patterns and counterpatterns, we then move on to a fully f ..."
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We use the infrastructure of CurienHerbelin’s λµ˜µcalculus to provide a syntax with patterns for a (weakly) focalised version of LK, which we call LKQ given its close relation with DanosJoinetSchellinx’s LKQ. Exploiting a simple game of patterns and counterpatterns, we then move on to a fully focalised syntax close to Girard’s ludics and Zeilberger’s CU. We also provide a version of our syntax closer to Girard’s original LC (the first constructivisation of classical logic) and show its usefulness as the target of a translation of a mixed callbyname/callbyvalue language.