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Proofs nets for unitfree multiplicativeadditive linear logic
 18th IEEE Intl. Symp. Logic in Computer Science (LICS’03
, 2003
"... A cornerstone of the theory of proof nets for unitfree multiplicative linear logic (MLL) is the abstract representation of cutfree proofs modulo inessential commutations of rules. The only known extension to additives, based on monomial weights, fails to preserve this key feature: a host of cutfr ..."
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Cited by 39 (4 self)
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A cornerstone of the theory of proof nets for unitfree multiplicative linear logic (MLL) is the abstract representation of cutfree proofs modulo inessential commutations of rules. The only known extension to additives, based on monomial weights, fails to preserve this key feature: a host of cutfree monomial proof nets can correspond to the same cutfree proof. Thus the problem of finding a satisfactory notion of proof net for unitfree multiplicativeadditive linear logic (MALL) has remained open since the inception of linear logic in 1986. We present a new definition of MALL proof net which remains faithful to the cornerstone of the MLL theory. 1
Higher Dimensional Automata Revisited
 MATHEMATICAL STRUCTURES IN COMPUTER SCIENCE
, 2000
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Sequentiality vs. Concurrency in Games and Logic
 Math. Structures Comput. Sci
, 2001
"... Connections between the sequentiality/concurrency distinction and the semantics of proofs are investigated, with particular reference to games and Linear Logic. ..."
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Cited by 13 (0 self)
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Connections between the sequentiality/concurrency distinction and the semantics of proofs are investigated, with particular reference to games and Linear Logic.
Chu Spaces as a Semantic Bridge Between Linear Logic and Mathematics
 Theoretical Computer Science
, 1998
"... The motivating role of linear logic is as a "logic behind logic." We propose a sibling role for it as a logic of transformational mathematics via the selfdual category of Chu spaces, a generalization of topological spaces. These create a bridge between linear logic and mathematics by soundly interp ..."
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Cited by 12 (2 self)
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The motivating role of linear logic is as a "logic behind logic." We propose a sibling role for it as a logic of transformational mathematics via the selfdual category of Chu spaces, a generalization of topological spaces. These create a bridge between linear logic and mathematics by soundly interpreting linear logic while fully and concretely embedding a comprehensive range of concrete categories of mathematics. Our main goal is to treat each end of this bridge in expository detail. In addition we introduce the dialectic lambdacalculus, and show that dinaturality semantics is not fully complete for the Chu interpretation of linear logic. 1 Introduction Linear logic was introduced by J.Y. Girard as a "logic behind logic." It separates logical reasoning into a core linear part in which formulas are merely moved around, and an auxiliary nonlinear part in which formulas may be deleted and copied. The core, multiplicative linear logic (MLL), is a substructural logic whose basic connect...
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 7 (1 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
Chu’s Construction: A Prooftheoretic Approach
 LOGIC FOR CONCURRENCY AND SYNCHRONISATION”, KLUWER TRENDS IN LOGIC N.18, 2003, PP.93114. LAMBDA CALCULUS 37
, 2001
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Deep Inference Proof Theory Equals Categorical Proof Theory Minus Coherence
, 2004
"... This paper links deep inference proof theory, as studied by Guglielmi et al., to categorical proof theory in the sense of Lambek et al.. It observes how deep inference proof theory is categorical proof theory, minus the coherence diagrams/laws. Coherence yields a readymade and well studied notion o ..."
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Cited by 4 (1 self)
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This paper links deep inference proof theory, as studied by Guglielmi et al., to categorical proof theory in the sense of Lambek et al.. It observes how deep inference proof theory is categorical proof theory, minus the coherence diagrams/laws. Coherence yields a readymade and well studied notion of equality on deep inference proofs. The paper notes a precise correspondence between the symmetric deep inference system for multiplicative linear logic (the linear fragment of SKSg) and the presentation of #autonomous categories as symmetric linearly distributive categories with negation. Contraction and weakening in SKSg corresponds precisely to the presence of (co)monoids.
Towards Hilbert's 24th Problem: Combinatorial Proof Invariants
, 2006
"... Proofs Without Syntax [37] introduced polynomialtime checkable combinatorial proofs for classical propositional logic. This sequel approaches Hilbert’s 24 th Problem with combinatorial proofs as abstract invariants for sequent calculus proofs, analogous to homotopy groups as abstract invariants for ..."
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Cited by 3 (0 self)
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Proofs Without Syntax [37] introduced polynomialtime checkable combinatorial proofs for classical propositional logic. This sequel approaches Hilbert’s 24 th Problem with combinatorial proofs as abstract invariants for sequent calculus proofs, analogous to homotopy groups as abstract invariants for topological spaces. The paper lifts a simple, strongly normalising cut elimination from combinatorial proofs to sequent calculus, factorising away the mechanical commutations of structural rules which litter traditional syntactic cut elimination. Sequent calculus fails to be surjective onto combinatorial proofs: the paper extracts a semantically motivated closure of sequent calculus from which there is a surjection, pointing towards an abstract combinatorial refinement of Herbrand’s theorem.
The Dialectica Interpretation of FirstOrder Classical Affine Logic
 In this volume
, 2006
"... Abstract. We give a Dialecticastyle interpretation of firstorder classical affine logic. By moving to a contractionfree logic, the translation (a.k.a. Dtranslation) of a firstorder formula into a highertype ∃∀formula can be made symmetric with respect to duality, including exponentials. It tur ..."
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Cited by 2 (0 self)
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Abstract. We give a Dialecticastyle interpretation of firstorder classical affine logic. By moving to a contractionfree logic, the translation (a.k.a. Dtranslation) of a firstorder formula into a highertype ∃∀formula can be made symmetric with respect to duality, including exponentials. It turned out that the propositional part of our Dtranslation uses the same construction as de Paiva’s dialectica category GC and we show how our Dtranslation extends GC to the firstorder setting in terms of an indexed category. Furthermore the combination of Girard’s?!translation and our Dtranslation results in the essentially equivalent ∃∀formulas as the doublenegation translation and Gödel’s original Dtranslation.
Intensional double glueing, biextensional collapse, and the Chu construction
 In Mathematical Foundations of Programming Semantics
, 2004
"... The superficial similarity between the Chu construction and the HylandTan double glueing construction G has been observed widely. This paper establishes a more formal mathematical relationship between the two. We show that double glueing on relations subsumes the Chu construction on sets: we presen ..."
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Cited by 1 (1 self)
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The superficial similarity between the Chu construction and the HylandTan double glueing construction G has been observed widely. This paper establishes a more formal mathematical relationship between the two. We show that double glueing on relations subsumes the Chu construction on sets: we present a full monoidal embedding of the category chu(Set, K) of biextensional Chu spaces over K into G(Rel K), and a full monoidal embedding of the category Chu(Set, K) of Chu spaces over K into IG(Rel K), where we define IG, the intensional double glueing construction, by substituting multisets for sets in G. We define a biextensional collapse from IG to G which extends the familiar notion on Chu spaces. This yields a new interpretation of the monic specialisation implicit in G as a form of biextensionality. 1