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72
On Köthe sequence spaces and linear logic
 Mathematical Structures in Computer Science
, 2001
"... We present a category of locally convex topological vector spaces which is a model of propositional classical linear logic, based on the standard concept of Kothe sequence spaces. In this setting, the spaces interpreting the exponential have a quite simple structure of commutative Hopf algebra. The ..."
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Cited by 39 (10 self)
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We present a category of locally convex topological vector spaces which is a model of propositional classical linear logic, based on the standard concept of Kothe sequence spaces. In this setting, the spaces interpreting the exponential have a quite simple structure of commutative Hopf algebra. The coKleisli category of this linear category is a cartesian closed category of entire mappings. This work provides a simple setting where typed calculus and dierential calculus can be combined; we give a few examples of computations. 1
Differential categories
, 2005
"... Following... After establishing the basic axioms, we give a number of examples. The most important example arises from a general construction, a comonad S1 on the category of vector spaces. This comonad and associated differential operators fully capture the usual notion of derivatives of smooth map ..."
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Cited by 20 (2 self)
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Following... After establishing the basic axioms, we give a number of examples. The most important example arises from a general construction, a comonad S1 on the category of vector spaces. This comonad and associated differential operators fully capture the usual notion of derivatives of smooth maps. Finally, we derive additional properties of differential categories in certain special cases, especially when the differential comonad is a storage modality, as in linear logic. In particular, we introduce the notion of categorical model of the differential calculus, and show that it captures the notnecessarilyclosed fragment of EhrhardRegnier differential *calculus.
The Algebraic LambdaCalculus
 UNDER CONSIDERATION FOR PUBLICATION IN MATH. STRUCT. IN COMP. SCIENCE
, 2009
"... We introduce an extension of the pure lambdacalculus 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 value ..."
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Cited by 19 (2 self)
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We introduce an extension of the pure lambdacalculus 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 betareduction in this setting: we prove it is confluent, then discuss consistency and conservativity over the ordinary lambdacalculus. We also provide normalization results for a simple type system.
Category theory for linear logicians
 Linear Logic in Computer Science
, 2004
"... This paper presents an introduction to category theory with an emphasis on those aspects relevant to the analysis of the model theory of linear logic. With this in mind, we focus on the basic definitions of category theory and categorical logic. An analysis of cartesian and cartesian closed categori ..."
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Cited by 13 (2 self)
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This paper presents an introduction to category theory with an emphasis on those aspects relevant to the analysis of the model theory of linear logic. With this in mind, we focus on the basic definitions of category theory and categorical logic. An analysis of cartesian and cartesian closed categories and their relation to intuitionistic logic is followed by a consideration of symmetric monoidal closed, linearly distributive and ∗autonomous categories and their relation to multiplicative linear logic. We examine nonsymmetric monoidal categories, and consider them as models of noncommutative linear logic. We introduce traced monoidal categories, and discuss their relation to the geometry of interaction. The necessary aspects of the theory of monads is introduced in order to describe the categorical modelling of the exponentials. We conclude by briefly describing the notion of full completeness, a strong form of categorical completeness, which originated in the categorical model theory of linear logic. No knowledge of category theory is assumed, but we do assume knowledge of linear logic sequent calculus and the standard models of linear logic, and modest familiarity with typed lambda calculus. 0
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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Cited by 12 (5 self)
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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
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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Cited by 11 (1 self)
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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.
Differential Structure in Models of Multiplicative Biadditive Intuitionistic Linear Logic (Extended Abstract)
"... Abstract. In the first part of the paper I investigate categorical models of multiplicative biadditive intuitionistic linear logic, and note that in them some surprising coherence laws arise. The thesis for the second part of the paper is that these models provide the right framework for investigati ..."
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Abstract. In the first part of the paper I investigate categorical models of multiplicative biadditive intuitionistic linear logic, and note that in them some surprising coherence laws arise. The thesis for the second part of the paper is that these models provide the right framework for investigating differential structure in the context of linear logic. Consequently, within this setting, I introduce a notion of creation operator (as considered by physicists for bosonic Fock space in the context of quantum field theory), provide an equivalent description of creation operators in terms of creation maps, and show that they induce a differential operator satisfying all the basic laws of differentiation (the product and chain rules, the commutation relations, etc.). 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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Cited by 10 (3 self)
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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.
CARTESIAN DIFFERENTIAL CATEGORIES
"... a comonad (a “coalgebra modality”) and a differential combinator. The morphisms of a differential category should be thought of as the linear maps; the differentiable or smooth maps would then be morphisms of the coKleisli category. The purpose of the present paper is to directly axiomatize differen ..."
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Cited by 9 (1 self)
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a comonad (a “coalgebra modality”) and a differential combinator. The morphisms of a differential category should be thought of as the linear maps; the differentiable or smooth maps would then be morphisms of the coKleisli category. The purpose of the present paper is to directly axiomatize differentiable maps and thus to move the emphasis from the linear notion to structures resembling the coKleisli category. The result is a setting with a more evident and intuitive relationship to the familiar notion of calculus on smooth maps. Indeed a primary example is the category whose objects are Euclidean spaces and whose morphisms are smooth maps.