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Categorical models for simply typed resource calculi
 ENTCS
"... We introduce the notion of differential λcategory as an extension of BluteCockettSeely’s differential Cartesian categories. We prove that differential λcategories can be used to model the simply typed versions of: (i) the differential λcalculus, a λcalculus extended with a syntactic derivative ..."
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We introduce the notion of differential λcategory as an extension of BluteCockettSeely’s differential Cartesian categories. We prove that differential λcategories can be used to model the simply typed versions of: (i) the differential λcalculus, a λcalculus extended with a syntactic derivative operator; (ii) the resource calculus, a nonlazy axiomatisation of Boudol’s λcalculus with multiplicities. Finally, we provide two
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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Cited by 5 (1 self)
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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
DIFFERENTIAL RESTRICTION CATEGORIES
"... Abstract. We combine two recent ideas: cartesian differential categories, and restriction categories. The result is a new structure which axiomatizes the category of smooth maps defined on open subsets of Rn in a way that is completely algebraic. We also give other models for the resulting structure ..."
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Abstract. We combine two recent ideas: cartesian differential categories, and restriction categories. The result is a new structure which axiomatizes the category of smooth maps defined on open subsets of Rn in a way that is completely algebraic. We also give other models for the resulting structure, discuss what it means for a partial map to be additive or linear, and show that differential restriction structure can be lifted through various completion operations.
FORMS AND EXTERIOR DIFFERENTIATION IN CARTESIAN DIFFERENTIAL CATEGORIES
"... Abstract. Cartesian differential categories abstractly capture the notion of a differentiation operation. In this paper, we develop some of the theory of such categories by defining differential forms and exterior differentiation in this setting. We show that this exterior derivative, as expected, p ..."
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Abstract. Cartesian differential categories abstractly capture the notion of a differentiation operation. In this paper, we develop some of the theory of such categories by defining differential forms and exterior differentiation in this setting. We show that this exterior derivative, as expected, produces a cochain complex. 1.