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A Brief Guide to Linear Logic
, 1993
"... An overview of linear logic is given, including an extensive bibliography and a simple example of the close relationship between linear logic and computation. ..."
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Cited by 53 (8 self)
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An overview of linear logic is given, including an extensive bibliography and a simple example of the close relationship between linear logic and computation.
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 29 (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
Event Spaces and Their Linear Logic
 In AMAST’91: Algebraic Methodology and Software Technology, Workshops in Computing
, 1991
"... Boolean logic treats disjunction and conjunction symmetrically and algebraically. The corresponding operations for computation are respectively nondeterminism (choice) and concurrency. Petri nets treat these symmetrically but not algebraically, while event structures treat them algebraically but not ..."
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Cited by 22 (9 self)
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Boolean logic treats disjunction and conjunction symmetrically and algebraically. The corresponding operations for computation are respectively nondeterminism (choice) and concurrency. Petri nets treat these symmetrically but not algebraically, while event structures treat them algebraically but not symmetrically. Here we achieve both via the notion of an event space as a poset with all nonempty joins representing concurrence and a top representing the unreachable event. The symmetry is with the dual notion of state space, a poset with all nonempty meets representing choice and a bottom representing the start state. The algebra is that of a parallel programming language expanded to the language of full linear logic, Girard's axiomatization of which is satisfied by the event space interpretation of this language. Event spaces resemble finite dimensional vector spaces in distinguishing tensor product from direct product and in being isomorphic to their double dual, but differ from them i...
Linearizing Intuitionistic Implication
 In Proc. 6th Annual IEEE Symposium on Logic in Computer Science
, 1993
"... An embedding of the implicational propositional intuitionistic logic (iil) into the nonmodal fragment of intuitionistic linear logic (imall) is given. The embedding preserves cutfree proofs in a proof system that is a variant of iil. The embedding is efficient and provides an alternative proof of t ..."
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Cited by 16 (5 self)
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An embedding of the implicational propositional intuitionistic logic (iil) into the nonmodal fragment of intuitionistic linear logic (imall) is given. The embedding preserves cutfree proofs in a proof system that is a variant of iil. The embedding is efficient and provides an alternative proof of the pspacehardness of imall. It exploits several prooftheoretic properties of intuitionistic implication that analyze the use of resources in iil proofs. Linear logic is a refinement of classical and intuitionistic logic that provides an intrinsic and natural accounting of resources. In Girard's words [12], "linear logic is a logic behind logic." A convenient way to present linear logic is by modifying the traditional Gentzenstyle sequent calculus axiomatization of classical logic (see, e.g., [15, 22]). The modification may be briefly described in three steps. The first step is to remove two structural rules, contraction and weakening, which manipulate the use of hypotheses and conclusi...
THE CHU CONSTRUCTION
, 1996
"... We take another look at the Chu construction and show how to simplify it by looking at ..."
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Cited by 13 (1 self)
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We take another look at the Chu construction and show how to simplify it by looking at
Some Properties Of Linear Logic Proved By Semantic Methods
 Journal of Logic and Computation
, 1994
"... We construct several simple algebraic models of the multiplicative and multiplicative additive fragments of linear logic and demonstrate the value of such models by proving some unexpected prooftheoretical properties of these fragments. I. ..."
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Cited by 4 (0 self)
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We construct several simple algebraic models of the multiplicative and multiplicative additive fragments of linear logic and demonstrate the value of such models by proving some unexpected prooftheoretical properties of these fragments. I.
Categorical Multirelations, Linear Logic and Petri Nets
, 1991
"... This note presents a category of multirelations, which is, in a loose sense a generalisation of both our previous work (the categories GC, [dP'89]) and of Chu's construction ANC [Barr'79]. The main motivation for writing this note was the utilisation of the category GC by Brown and Gu ..."
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Cited by 3 (1 self)
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This note presents a category of multirelations, which is, in a loose sense a generalisation of both our previous work (the categories GC, [dP'89]) and of Chu's construction ANC [Barr'79]. The main motivation for writing this note was the utilisation of the category GC by Brown and Gurr [BG90] to model Petri Nets. We wanted to extend their work to deal with multirelations, as Petri Nets are usually modelled using multirelations pre and post. That proved easy enough and people interested mainly in concurrency theory should refer to our joint work [BGdP'91]; this note deals with the mathematics underlying [BGdP'91]. The upshot of this work is that we build a model of Intuitionistic Linear Logic (without modalities) over any symmetric monoidal closed category C with a distinguished object (N; ; ffi; e \Gammaffi)  a closed poset. Moreover, if the category C is cartesian closed with free commutative monoids, we build a model of Intuitionistic Linear Logic with a nontrivial modality `!'...
COMPACTLY ACCESSIBLE CATEGORIES AND QUANTUM KEY DISTRIBUTION
"... Abstract. Compact categories have lately seen renewed interest via applications to quantum physics. Being essentially finitedimensional, they cannot accomodate (co)limitbased constructions. For example, they cannot capture protocols such as quantum key distribution, that rely on the law of large n ..."
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Cited by 2 (2 self)
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Abstract. Compact categories have lately seen renewed interest via applications to quantum physics. Being essentially finitedimensional, they cannot accomodate (co)limitbased constructions. For example, they cannot capture protocols such as quantum key distribution, that rely on the law of large numbers. To overcome this limitation, we introduce the notion of a compactly accessible category, relying on the extra structure of a factorisation system. This notion allows for infinite dimension while retaining key properties of compact categories: the main technical result is that the choiceofduals functor on the compact
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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Cited by 1 (0 self)
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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.