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46
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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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
Restriction Categories I
 Categories of Partial Maps, Theoret. Comput. Sci
, 2006
"... modality”) and a differential combinator, satisfying a number of coherence conditions. In ..."
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modality”) and a differential combinator, satisfying a number of coherence conditions. In
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
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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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
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 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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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.
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.
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
On linear combinations of λterms
"... Abstract. We define an extension of λcalculus with linear combinations, endowing the set of terms with a structure of Rmodule, where R is 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 vect ..."
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Abstract. We define an extension of λcalculus with linear combinations, endowing the set of terms with a structure of Rmodule, where R is 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 extend βreduction on those algebraic λterms as follows: at + u reduces to at ′ + u as soon as term t reduces to t ′ and a is a nonzero scalar. We prove that reduction is confluent. Under the assumption that the set R of scalars is positive (i.e. a sum of scalars is zero iff all of them are zero), we show that this algebraic λcalculus is a conservative extension of ordinary λcalculus. On the other hand, we show that if R admits negative elements, then every term reduces to every other term. We investigate the causes of that collapse, and discuss some possible fixes. Preliminary definitions and notations. Recall that a rig (also known as “semiring with zero and unit”) is the same as a ring, without the condition that every element admits an opposite for addition. Let R be a rig. We write R • for R \ {0}. We denote by letters a, b, c the elements of R, and say that R is positive if, for all a, b ∈ R, a + b = 0 implies a = 0 and b = 0. An example of positive rig is N, the set of natural numbers, with usual addition and multiplication. If i, j ∈ N, we write [i; j] for the set {k ∈ N; i ≤ k ≤ j}. Also, we write application of λterms à la Krivine: (s)t denotes the application of term s to term t. 1