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10
The Differential LambdaCalculus
 Theoretical Computer Science
, 2001
"... We present an extension of the lambdacalculus with differential constructions motivated by a model of linear logic discovered by the first author and presented in [Ehr01]. We state and prove some basic results (confluence, weak normalization in the typed case), and also a theorem relating the usual ..."
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Cited by 44 (9 self)
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We present an extension of the lambdacalculus with differential constructions motivated by a model of linear logic discovered by the first author and presented in [Ehr01]. We state and prove some basic results (confluence, weak normalization in the typed case), and also a theorem relating the usual Taylor series of analysis to the linear head reduction of lambdacalculus.
Linear Continuations
 In Nineteenth ACM Symposium on Principles of Programming Languages
, 1992
"... We present a functional interpretation of classical linear logic based on the concept of linear continuations. Unlike their nonlinear counterparts, such continuations lead to a model of control that does not inherently impose any particular evaluation strategy. Instead, such additional structure i ..."
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Cited by 30 (1 self)
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We present a functional interpretation of classical linear logic based on the concept of linear continuations. Unlike their nonlinear counterparts, such continuations lead to a model of control that does not inherently impose any particular evaluation strategy. Instead, such additional structure is expressed by admitting closely controlled copying and discarding of continuations. We also emphasize the importance of classicality in obtaining computationally appealing categorical models of linear logic and propose a simple "coreflective subcategory " interpretation of the modality "!". 1 Introduction In recent years, there has been considerable interest in applications of Girard's Linear Logic (LL) [Gir87] to programming language design and implementation. Over time, various more or less mutated versions of the original system have been proposed, but they all share the same basic premise: that assumptions made in the course of a formal proof can not necessarily be used an arbitrary n...
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 30 (9 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
Nuclear and Trace Ideals in Tensored *Categories
, 1998
"... We generalize the notion of nuclear maps from functional analysis by defining nuclear ideals in tensored categories. The motivation for this study came from attempts to generalize the structure of the category of relations to handle what might be called "probabilistic relations". The compact closed ..."
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Cited by 26 (9 self)
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We generalize the notion of nuclear maps from functional analysis by defining nuclear ideals in tensored categories. The motivation for this study came from attempts to generalize the structure of the category of relations to handle what might be called "probabilistic relations". The compact closed structure associated with the category of relations does not generalize directly, instead one obtains nuclear ideals. Most tensored categories have a large class of morphisms which behave as if they were part of a compact closed category, i.e. they allow one to transfer variables between the domain and the codomain. We introduce the notion of nuclear ideals to analyze these classes of morphisms. In compact closed tensored categories, all morphisms are nuclear, and in the tensored category of Hilbert spaces, the nuclear morphisms are the HilbertSchmidt maps. We also introduce two new examples of tensored categories, in which integration plays the role of composition. In the first, mor...
Discrete Quantum Causal Dynamics
 International Journal of Theoretical Physics
, 2003
"... We give a mathematical framework to describe the evolution of an open quantum systems subjected to nitely many interactions with classical apparatuses. The systems in question may be composed of distinct, spatially separated subsystems which evolve independently but may also interact. This evolut ..."
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Cited by 9 (4 self)
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We give a mathematical framework to describe the evolution of an open quantum systems subjected to nitely many interactions with classical apparatuses. The systems in question may be composed of distinct, spatially separated subsystems which evolve independently but may also interact. This evolution, driven both by unitary operators and measurements, is coded in a precise mathematical structure in such a way that the crucial properties of causality, covariance and entanglement are faithfully represented. We show how our framework may be expressed using the language of (poly)categories and functors. Remarkably, important physical consequences  such as covariance  follow directly from the functoriality of our axioms. We establish strong links between the physical picture we propose and linear logic. Specifically we show that the rened logical connectives of linear logic can be used to describe the entanglements of subsystems in a precise way. Furthermore, we show that there is a precise correspondence between the evolution of a given system and deductions in a certain formal logical system based on the rules of linear logic. This framework generalizes and enriches both causal posets and the histories approach to quantum mechanics. 1
Probabilistic Relations
 School of Computer Science, McGill University, Montreal
, 1998
"... The notion of binary relation is fundamental in logic. What is the correct analogue of this concept in the probabilistic case? I will argue that the notion of conditional probability distribution (Markov kernel, stochastic kernel) is the correct generalization. One can define a category based on sto ..."
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Cited by 6 (0 self)
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The notion of binary relation is fundamental in logic. What is the correct analogue of this concept in the probabilistic case? I will argue that the notion of conditional probability distribution (Markov kernel, stochastic kernel) is the correct generalization. One can define a category based on stochastic kernels which has many of the formal properties of the ordinary category of relations. Using this concept I will show how to define iteration in this category and give a simple treatment of Kozen's language of while loops and probabilistic choice. I will use the concept of stochastic relation to introduce some of the ongoing joint work with Edalat and Desharnais on Labeled Markov Processes. In my talk I will assume that people do not know what partially additive categories are but that they do know basic category theory and basic notions like measure and probability. This work is mainly due to Kozen, Giry, Lawvere and others. 1 Introduction The notion of binary relation and relation...
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 3 (1 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.
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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Cited by 2 (0 self)
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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 finiteness spaces and extensional presheaves over the lawvere theory of polynomials
 Journal of Pure and Applied Algebra
"... theory of polynomials ..."
Mathematics of Multisets Apostolos Syropoulos
 In Multiset Processing
, 2001
"... This paper is an attempt to summarize most things that are related to multiset theory. We begin by describing multisets and the operations between them. Then we present hybrid sets and their operations. We continue with a categorical approach to multisets. Next, we present fuzzy multisets and their ..."
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This paper is an attempt to summarize most things that are related to multiset theory. We begin by describing multisets and the operations between them. Then we present hybrid sets and their operations. We continue with a categorical approach to multisets. Next, we present fuzzy multisets and their operations. Finally, we present partially ordered multisets. 1