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What is a Categorical Model of the Differential and the Resource λCalculi?
"... The differential λcalculus is a paradigmatic functional programming language endowed with a syntactical differentiation operator that allows to apply a program to an argument in a linear way. One of the main features of this language is that it is resource conscious and gives the programmer suitab ..."
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The differential λcalculus is a paradigmatic functional programming language endowed with a syntactical differentiation operator that allows to apply a program to an argument in a linear way. One of the main features of this language is that it is resource conscious and gives the programmer suitable primitives to handle explicitly the resources used by a program during its execution. The differential operator also allows to write the full Taylor expansion of a program. Through this expansion every program can be decomposed into an infinite sum (representing nondeterministic choice) of ‘simpler’ programs that are strictly linear. The aim of this paper is to develop an abstract ‘model theory ’ for the untyped differential λcalculus. In particular, we investigate what should be a general categorical definition of denotational model for this calculus. Starting from the work of Blute, Cockett and Seely on differential categories we provide the notion of Cartesian closed differential category and we prove that linear reflexive objects living in such categories constitute sound models of the untyped differential λcalculus. We also give sufficient conditions for Cartesian closed differential categories to model the Taylor expansion. This entails that every model living in such categories equates all programs having the same full Taylor expansion. We then
Algebraic λcalculus
"... We introduce an extension of λcalculus by endowing the set of terms with a structure of vector space (or more generally of module), over a fixed set of scalars. Terms are moreover subject to identities similar to usual pointwise definition of linear combinations of functions with values in a vector ..."
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We introduce an extension of λcalculus by endowing the set of terms with a structure of vector space (or more generally of module), over a fixed set of scalars. Terms are moreover subject to identities similar to usual pointwise definition of linear combinations of functions with values in a vector space. We then study a natural extension of βreduction in this setting: we prove it is confluent, then discuss consistency and conservativity over ordinary λcalculus. We also provide normalization results for a simple type system.
The separation theorem for differential interaction nets
, 2007
"... Differential interaction nets (DIN) have been introduced by Thomas Ehrhard and Laurent Regnier as an extension of linear logic proofnets. We prove that DIN enjoy an internal separation property: given two different normal nets, there exists a dual net separating them, in analogy with Böhm’s theore ..."
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Differential interaction nets (DIN) have been introduced by Thomas Ehrhard and Laurent Regnier as an extension of linear logic proofnets. We prove that DIN enjoy an internal separation property: given two different normal nets, there exists a dual net separating them, in analogy with Böhm’s theorem for the λcalculus. Our result implies in particular the faithfulness of every nontrivial denotational model of DIN (such as Ehrhard’s finiteness spaces). We also observe that internal separation does not hold for linear logic proofnets: our work points out that this failure is due to the fundamental asymmetry of linear logic exponential modalities, which are instead completely symmetric in DIN.
Full Abstraction for Resource Calculus with Tests
 In CSL, Lecture Notes in Computer Science
, 2011
"... We study the semantics of a resource sensitive extension of the λcalculus in a canonical reflexive object of a category of sets and relations, a relational version of the original Scott D ∞ model of the pure λcalculus. This calculus is related to Boudol’s resource calculus and is derived from Ehrh ..."
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We study the semantics of a resource sensitive extension of the λcalculus in a canonical reflexive object of a category of sets and relations, a relational version of the original Scott D ∞ model of the pure λcalculus. This calculus is related to Boudol’s resource calculus and is derived from Ehrhard and Regnier’s differential extension of Linear Logic and of the λcalculus. We extend it with new constructions, to be understood as implementing a very simple exception mechanism, and with a “must ” parallel composition. These new operations allow to associate a context of this calculus with any point of the model and to prove full abstraction for the finite subcalculus where ordinary λcalculus application is not allowed. The result is then extended to the full calculus by means of a Taylor Expansion formula. 1998 ACM Subject Classification F.4.1 Lambda calculus and related systems
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.
Editorial Board
, 2011
"... Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available in the Internet at ..."
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Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available in the Internet at