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An introduction to substructural logics
, 2000
"... Abstract: This is a history of relevant and substructural logics, written for the Handbook of the History and Philosophy of Logic, edited by Dov Gabbay and John Woods. 1 1 ..."
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Cited by 139 (16 self)
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Abstract: This is a history of relevant and substructural logics, written for the Handbook of the History and Philosophy of Logic, edited by Dov Gabbay and John Woods. 1 1
Simple free starautonomous categories and full coherence
, 2005
"... This paper gives a simple presentation of the free starautonomous category over a category, based on EilenbergKellyMacLane graphs and Trimble rewiring, for full coherence. ..."
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Cited by 12 (0 self)
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This paper gives a simple presentation of the free starautonomous category over a category, based on EilenbergKellyMacLane graphs and Trimble rewiring, for full coherence.
On the Relation Between Coherence Semantics and Multiplicative Proof Nets
, 1994
"... It is known that (mix) proof nets admit a coherence semantics, computed as a set of experiments. We prove here the converse: a proof structure is shown to be a proof net whenever its set of experiments is a semantical object  a clique of the corresponding coherence space. Moreover the interpretat ..."
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Cited by 6 (4 self)
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It is known that (mix) proof nets admit a coherence semantics, computed as a set of experiments. We prove here the converse: a proof structure is shown to be a proof net whenever its set of experiments is a semantical object  a clique of the corresponding coherence space. Moreover the interpretation of atomic formulae can be restricted to a given coherent space with four tokens in its web. This is done by transforming cutlinks into tensorlinks. Dealing directly with noncutfree proof structure we characterise the deadlock freeness of the proof structure. These results are especially convenient for Abramsky 's proof expressions, and are extended to the pomset calculus.
Simple multiplicative proof nets with units
, 2005
"... Abstract. This paper presents a simple notion of proof net for multiplicative linear logic with units. Cut elimination is direct and strongly normalising, in contrast to previous approaches which resorted to moving jumps (attachments) of par units during normalisation. Composition ..."
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Cited by 4 (1 self)
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Abstract. This paper presents a simple notion of proof net for multiplicative linear logic with units. Cut elimination is direct and strongly normalising, in contrast to previous approaches which resorted to moving jumps (attachments) of par units during normalisation. Composition
Digital Equipment Corporation 1991
"... The purpose of this paper is to give an exposition of material dealing with constructive logics, typed calculi, and linear logic. The first part of this paper gives an exposition of background material (with the exception of the Girardtranslation of classical logic into intuitionistic logic, which ..."
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The purpose of this paper is to give an exposition of material dealing with constructive logics, typed calculi, and linear logic. The first part of this paper gives an exposition of background material (with the exception of the Girardtranslation of classical logic into intuitionistic logic, which is new). This second part is devoted to linear logic and proof nets. Particular attention is given to the algebraic semantics (in Girard's terminology, phase semantics) of linear logic. We show how phase spaces arise as an instance of a Galois connection. We also give a direct proof of the correctness of the DanosRegnier criterion for proof nets. This proof is based on a purely graphtheoretic decomposition lemma. As a corollary, we give an O(n 2 )time algorithm for testing whether a proof net is correct. Although the existence of such an algorithm has been announced by Girard, our algorithm appears to be original. R esum e Le but de cet article est de donner une presentation d'elements...
Incremental Parsing Of Lambek Calculus Using ProofNet Interfaces
, 2003
"... The paper describes an incremental parsing algorithm for natural languages that uses normalized interfaces of modules of proofnets. This algorithm produces at each step the dierent possible partial syntactical analyses of the rst words of a sentence. Thus, it can analyze texts on the y leaving part ..."
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The paper describes an incremental parsing algorithm for natural languages that uses normalized interfaces of modules of proofnets. This algorithm produces at each step the dierent possible partial syntactical analyses of the rst words of a sentence. Thus, it can analyze texts on the y leaving partially analyzed sentences.
Canonical proof nets for classical logic
"... Abstract. Proof nets provide abstract counterparts to sequent proofs modulo rule permutations; the idea being that if two proofs have the same underlying proofnet, they are in essence the same proof. Providing a convincing proofnet counterpart to proofs in the classical sequent calculus is thus an ..."
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Abstract. Proof nets provide abstract counterparts to sequent proofs modulo rule permutations; the idea being that if two proofs have the same underlying proofnet, they are in essence the same proof. Providing a convincing proofnet counterpart to proofs in the classical sequent calculus is thus an important step in understanding classical sequent calculus proofs. By convincing, we mean that (a) there should be a canonical function from sequent proofs to proof nets, (b) it should be possible to check the correctness of a net in polynomial time, (c) every correct net should be obtainable from a sequent calculus proof, and (d) there should be a cutelimination procedure which preserves correctness. Previous attempts to give proofnetlike objects for propositional classical logic have failed at least one of the above conditions. In [23], the author presented a calculus of proof nets (expansion nets) satisfying (a) and (b); the paper defined a sequent calculus corresponding to expansion nets but gave no explicit demonstration of (c). That sequent calculus, called LK ∗ in this paper, is a novel onesided sequent calculus with both additively and multiplicatively formulated disjunction rules. In this paper (a selfcontained extended version of [23]) , we give a full proof of (c) for expansion nets with respect to LK ∗, and in addition give a cutelimination procedure internal to expansion nets – this makes expansion nets the first notion of proofnet for classical logic satisfying all four criteria. 1