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14
Sequential algorithms and strongly stable functions
 in the Linear Summer School, Azores
, 2003
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About Translations of Classical Logic into Polarized Linear Logic
 In Proceedings of the eighteenth annual IEEE symposium on Logic In Computer Science
, 2003
"... We show that the decomposition of Intuitionistic Logic into Linear Logic along the equation A ! B = !A ( B may be adapted into a decomposition of classical logic into LLP, the polarized version of Linear Logic. Firstly we build a categorical model of classical logic (a Control Category) from a categ ..."
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We show that the decomposition of Intuitionistic Logic into Linear Logic along the equation A ! B = !A ( B may be adapted into a decomposition of classical logic into LLP, the polarized version of Linear Logic. Firstly we build a categorical model of classical logic (a Control Category) from a categorical model of Linear Logic by a construction similar to the coKleisli category. Secondly we analyse two standard ContinuationPassing Style (CPS) translations, the Plotkin and the Krivine's translations, which are shown to correspond to two embeddings of LLP into LL.
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
A terminating and confluent linear lambda calculus
 PROC. OF 17TH INT. CONFERENCE RTA 2006, VOLUME 4098 OF LECTURE NOTES IN COMPUTER SCIENCE
, 2006
"... We present a rewriting system for the linear lambda calculus corresponding to the {!, ⊸}fragment of intuitionistic linear logic. This rewriting system is shown to be strongly normalizing, and ChurchRosser modulo the trivial commuting conversion. Thus it provides a simple decision method for the eq ..."
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We present a rewriting system for the linear lambda calculus corresponding to the {!, ⊸}fragment of intuitionistic linear logic. This rewriting system is shown to be strongly normalizing, and ChurchRosser modulo the trivial commuting conversion. Thus it provides a simple decision method for the equational theory of the linear lambda calculus. As an application we prove the strong normalization of the simply typed computational lambda calculus by giving a reductionpreserving translation into the linear lambda calculus.
Comparing Hierarchies of Types in Models of Linear Logic
, 2003
"... We show that two models M and N of linear logic collapse to the same extensional hierarchy of types, when (1) their monoidal categories C and D are related by a pair of monoidal functors F : C D : G and transformations Id C ) GF and Id D ) FG, and (2) their exponentials ! are related by distri ..."
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We show that two models M and N of linear logic collapse to the same extensional hierarchy of types, when (1) their monoidal categories C and D are related by a pair of monoidal functors F : C D : G and transformations Id C ) GF and Id D ) FG, and (2) their exponentials ! are related by distributive laws % : ! : ! M G ) G ! N commuting to the promotion rule. The key ingredient of the proof is a notion of backandforth translation between the hierarchies of types induced by M and N. We apply this result to compare (1) the qualitative and the quantitative hierarchies induced by the coherence (or hypercoherence) space model, (2) several paradigms of games semantics: errorfree vs. erroraware, alternated vs. nonalternated, backtracking vs. repetitive, uniform vs. nonuniform.
Whats is a categorical model of linear logic
, 2004
"... The aim of these notes is to give an outline of the categorical structure required for a model of linear logic. That means that we only try to motivate the various conditions imposed on such models; this is by no means intended to be a full account. For intuitionistic linear logic a detailed descrip ..."
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The aim of these notes is to give an outline of the categorical structure required for a model of linear logic. That means that we only try to motivate the various conditions imposed on such models; this is by no means intended to be a full account. For intuitionistic linear logic a detailed description of the process of defining such a model can be found in Gavin Bierman’s thesis [Bie94]. There are a number of exercises in these notes, they either aim to give examples for the various notions defined or they state minor results that are used subsequently and which hopefully are not too difficult to establish. If the category theory in these notes is too advanced you may want to try [BS] which does not assume any knowledge in that area at all, but proceeds fairly rapidly to the notions we use here. Originally these notes were written for my PhD students, but since I made the first version available on my webpage a number of people have told me they found them useful. As a result I felt obliged to to add details which were rather sketchy in the original account. In particular the account now contains more proofs and so I am hopeful that the various formulations of the notion of linear exponential comonad are finally correct. I would like to thank Paola Maneggia, Peter Selinger, Robin Houston and Nicola Gambino for useful feedback on these
A categorical semantics for polarized mall
 Ann. Pure Appl. Logic
"... In this paper, we present a categorical model for Multiplicative Additive Polarized Linear Logic MALLP, which is the linear fragment (without structural rules) of Olivier Laurent’s Polarized Linear Logic. Our model is based on an adjunction between reflective/coreflective full subcategories C−/C+ of ..."
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In this paper, we present a categorical model for Multiplicative Additive Polarized Linear Logic MALLP, which is the linear fragment (without structural rules) of Olivier Laurent’s Polarized Linear Logic. Our model is based on an adjunction between reflective/coreflective full subcategories C−/C+ of an ambient ∗autonomous category C (with products). Similar structures were first introduced by M. Barr in the late 1970’s in abstract duality theory and more recently in work on game semantics for linear logic. The paper has two goals: to discuss concrete models and to present various completeness theorems. As concrete examples, we present (i) a hypercoherence model, using Ehrhard’s hereditary/antihereditary objects, (ii) a Chuspace model, (iii) a double gluing model over our categorical framework, and (iv) a model based on iterated double gluing over a ∗autonomous category. For the multiplicative fragment MLLP of MALLP, we present both weakly full (Läuchlistyle) as well as full completeness theorems, using a polarized version of functorial
Dialectica and Chu Constructions: Cousins?
 In this Volume
, 2006
"... This note investigates two generic constructions used to produce categorical models of linear logic, the Chu construction and the Dialectica construction, in parallel. The constructions have the same objects, but are rather di#erent in other ways. We discuss similarities and di#erences and prove ..."
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This note investigates two generic constructions used to produce categorical models of linear logic, the Chu construction and the Dialectica construction, in parallel. The constructions have the same objects, but are rather di#erent in other ways. We discuss similarities and di#erences and prove that the dialectica construction can be done over a symmetric monoidal closed basis. We also point out several interesting open problems concerning the Dialectica construction.
Asynchronous games 1: Uniformity by group invariance
"... This article opens a series of papers on asynchronous games semantics, which aims at a concurrent and geometric account of interference and states in programming languages. In order to develop our theory, we need to reformulate arena games in a simpler algebraic vocabulary, inspired by Girard's Geo ..."
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This article opens a series of papers on asynchronous games semantics, which aims at a concurrent and geometric account of interference and states in programming languages. In order to develop our theory, we need to reformulate arena games in a simpler algebraic vocabulary, inspired by Girard's Geometry of Interaction and Abramsky, Jagadeesan and Malacaria (AJM) token games. This is precisely the task of this article, which prepares the field for the positional / homotopic account of innocence in (Mellies 2004). 1.