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Convolution ¯ λµcalculus
 of Lecture Notes in Computer Science
, 2007
"... We define an extension of Herbelin’s ¯ λµcalculus, introducing a product operation on contexts (in the sense of lists of arguments, or stacks in environment machines), similar to the convolution product of distributions. This is the computational couterpart of some new semantical constructions, ext ..."
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We define an extension of Herbelin’s ¯ λµcalculus, introducing a product operation on contexts (in the sense of lists of arguments, or stacks in environment machines), similar to the convolution product of distributions. This is the computational couterpart of some new semantical constructions, extending models of EhrhardRegnier’s differential interaction nets, along the lines of Laurent’s polarization of linear logic. We demonstrate this correspondence by providing this calculus with a denotational semantics inside a lambdamodel in the category of sets and relations. 1
The Scott model of Linear Logic is the extensional collapse of its relational model
, 2011
"... We show that the extensional collapse of the relational model of linear logic is the model of primealgebraic complete lattices, a natural extension to linear logic of the well known Scott semantics of the lambdacalculus. ..."
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We show that the extensional collapse of the relational model of linear logic is the model of primealgebraic complete lattices, a natural extension to linear logic of the well known Scott semantics of the lambdacalculus.
On probabilistic coherence spaces
, 2008
"... We introduce a probabilistic version of coherence spaces and show that these objects provide a model of linear logic. We build a model of the pure lambdacalculus in this setting and show how to interpret a probabilistic version of the functional language PCF. We give a probabilistic interpretation ..."
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We introduce a probabilistic version of coherence spaces and show that these objects provide a model of linear logic. We build a model of the pure lambdacalculus in this setting and show how to interpret a probabilistic version of the functional language PCF. We give a probabilistic interpretation of the semantics of probabilistic PCF closed terms of ground type.
On Differential Interaction Nets and the Picalculus
 Preuves, Programmes et Systèmes
, 2006
"... We propose a translation of a finitary (that is, replicationfree) version of the picalculus into promotionfree differential interaction net structures, a linear logic version of the differential lambdacalculus (or, more precisely, of a resource lambdacalculus). For the sake of simplicity only, w ..."
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We propose a translation of a finitary (that is, replicationfree) version of the picalculus into promotionfree differential interaction net structures, a linear logic version of the differential lambdacalculus (or, more precisely, of a resource lambdacalculus). For the sake of simplicity only, we restrict our attention to a monadic version of the picalculus, so that the differential interaction net structures we consider need only to have exponential cells. We prove that the nets obtained by this translation satisfy an acyclicity criterion weaker than the standard Girard (or DanosRegnier) acyclicity criterion, and we compare the operational semantics of the picalculus, presented by means of an environment machine, and the reduction of differential interaction nets. Differential interaction net structures being of a logical nature, this work provides a CurryHoward interpretation of processes.
Presentation
, 2003
"... We present an extension of the lambdacalculus with di erential constructions. We state and prove some basic results (con uence, strong normalization in the typed case), and also a theorem relating the usual Taylor series of analysis to the linear head reduction of lambdacalculus. Keywords. Lambda ..."
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We present an extension of the lambdacalculus with di erential constructions. We state and prove some basic results (con uence, strong normalization in the typed case), and also a theorem relating the usual Taylor series of analysis to the linear head reduction of lambdacalculus. Keywords. Lambdacalculus, linear logic, denotational semantics, linear head reduction. Prerequisites. This paper assumes from the reader some basic knowledge in lambdacalculus and an elementary (but not technical) knowledge of di erential calculus. Notations. Following [Kri93], we denote by (s)t the lambdacalculus application of s to t. The expression (s)t1... tn denotes the term ( · · · (s)t1 · · ·)tn when n ≥ 1, and s when n = 0. Accordingly, if A1,..., An and A are types, both expressions A1,..., An → A and A1 → · · · → An → A denote the type A1 → ( · · · (An → A) · · ·). If a1,..., an are elements of some given set S, we denote by [a1,..., an] the corresponding multiset over S. If x and y are variables, δx,y is equal to 1 if x = y and to 0 otherwise. We denote by N + the set of positive integers {1, 2,...}.
unknown title
, 2007
"... Résumé de la thèse: λcalcul di érentiel et logique classique: interactions calculatoires ..."
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Résumé de la thèse: λcalcul di érentiel et logique classique: interactions calculatoires
Kähler Categories
, 2010
"... This paper establishes a relation between the recently introduced notion of differential category and the more classic theory of Kähler differentials in commutative algebra. A codifferential category is an additive symmetric monoidal category with a monad, which is furthermore an algebra modality. A ..."
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This paper establishes a relation between the recently introduced notion of differential category and the more classic theory of Kähler differentials in commutative algebra. A codifferential category is an additive symmetric monoidal category with a monad, which is furthermore an algebra modality. An algebra modality for a monad T is a natural assignment of an associative algebra structure to each object of the
A convenient differential
, 2011
"... We show that the category of convenient vector spaces in the sense of Frölicher ..."
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We show that the category of convenient vector spaces in the sense of Frölicher
Appl Categor Struct DOI 10.1007/s1048501092410 Deep Inference and Probabilistic Coherence Spaces
, 2009
"... Abstract This paper proposes a definition of categorical model of the deep inference system BV, defined by Guglielmi. Deep inference introduces the idea of performing a deduction in the interior of a formula, at any depth. Traditional sequent calculus rules only see the roots of formulae. However in ..."
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Abstract This paper proposes a definition of categorical model of the deep inference system BV, defined by Guglielmi. Deep inference introduces the idea of performing a deduction in the interior of a formula, at any depth. Traditional sequent calculus rules only see the roots of formulae. However in these new systems, one can rewrite at any position in the formula tree. Deep inference in particular allows the syntactic description of logics for which there is no sequent calculus. One such system is BV, which extends linear logic to include a noncommutative selfdual connective. This is the logic our paper proposes to model. Our definition is based on the notion of a linear functor, due to Cockett and Seely. A BVcategory is a linearly distributive