Christian Retore. Reseaux et Sequents Ordonnes.These de Doctorat, specialite mathematiques, UniversiteParis 7, February 1993.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....define a deductive system mixing commutative and non commutative logical operators. The specific variant I was looking for is a conservative extension of multiplicative linear logic plus mix, obtained by adding a self dual non commutative connective. This problem has been studied mainly by Retore [19, 20]: he has proof nets for his logic, which is called pomset logic, and cut elimination therein, but, despite many e#orts, nobody has been able so far to provide a sequent system for that logic. My challenge was to bring pomset logic into the realm of sequent calculus, for two reasons: 1) It is ....

Christian Retore. Reseaux et Sequents Ordonnes. These de Doctorat, specialite mathematiques, Universite Paris 7, February 1993.

....times connection of S 1 , S h .Thestructure is associative and non commutative:thiscorresponds to the new logical operator, called seq,thatweadd to those of MELL . For reasons explained in [9, 10] dealing with seq involves adding the rules mix and its nullary version mix0 (see [1, 6, 14]) mix and mix0 . This has the e#ect of collapsing the multiplicative units 1 and #:wewill only have one unit common to par, times and seq. Please notice that mix and mix0 are not an artefact of the calculus of structures. But, as shown by Retorein[14] they are required when using ....

....version mix0 (see [1, 6, 14] mix and mix0 . This has the e#ect of collapsing the multiplicative units 1 and #:wewill only have one unit common to par, times and seq. Please notice that mix and mix0 are not an artefact of the calculus of structures. But, as shown by Retorein[14], they are required when using a self dual non commutative connective. 2.1 Definition There are countably many positive and negative atoms.They, positive or negative, are denoted by a, b, Structures are denoted by S, P , Q, R, T , U , V and X.Thestructures of the language NEL are generated ....

Christian Retore. Reseaux et Sequents Ordonnes.These de Doctorat, specialite mathematiques, UniversiteParis 7, February 1993.

.... Proof Nets and TAGs Sylvain Pogodalla Xerox Research Center Europe 6 chemin de Maupertuis F 38240 Meylan, France Sylvain.Pogodalla xrce.xerox.com Introduction First introduced by [Ret93], pomset linear logic can deal with linguistic aspects by inducing a partial order on words. LR95] uses this property: it defines modules (or partial proof nets) which consist in entries for words, describing both the category of the word and its behavior when interacting with other words. Then ....

....from [AFV96] is to avoid the use of trees fl such that: 9p 2 D fl #fl(p) fl(p Delta 1) and p Delta 2 62 D fl It means there is no tree that have an X labeled node whose unique leaf is also an X labeled node. 1. 2 Lexicalized Proof Nets Proof nets in linear logic have become familiar [Gir87, Ret93, Abr95]. In this paper, we refer to [Ret96] s notations of proof nets, extended to the ordered calculus [Ret97] It defines proof nets as bicolored (Red and Blue, or Regular and Bold) graphs with the five links corresponding to the axiom, the tensor ( Omega ) the before ( the par (P) and Table 1. ....

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Christian Retore. Reseaux et sequents ordonnes. PhD thesis, University of Paris VII, 1993.

....concentrerons sur le modele des espaces coherents qui est conceptuellement lie a la logique lineaire, puisqu elle en est issue. Dans ce cadre nous avons deja e tudie un connecteur noncommutatif et autodual nomme precede , et concu le calcul ordonne qui e tend la logique lineaire a ce connecteur [Ret93]. S inspirant de la definition de ce connecteur en termes d espaces coherents, on definit une modalite autoduale pour laquelle existent les morphismes canoniques suivants: A r etraction de ffi ffi A ffi isomorphisme ffi A A rien ffi 1 ffi Il n y a pas a l heure actuelle de ....

....de jAj dans jBj. 1 DE FINITION 2 Etant donnes deux espaces coherents A et B, l espace coherent A precede B, note A B, est defini par: trame: jA Bj = jAj Theta jBj coherence: a; b) a 0 ; b 0 ) A B] ssi . Gamma a a a 0 [A] et b = b 0 Delta ou b a b 0 [B] PROPOSITION 1 ([Ret93]) Ce connecteur est: non commutatif A B 6j B A, autodual (A B) j A B , associatif A (B C) j (A B) C, admet 1 pour neutre A 1 j A j 1 A, entre la disjonction et la conjonction Omega pour l implication lineaire: pour tout couple d espaces coherents A et B, on a A Omega B ( A B et ....

Christian Retore. Reseaux et Sequents Ordonnes. These de Doctorat, specialite Mathematiques, Universite Paris 7, fevrier 1993.

....rules. The key point in these plugging rules is that they preserve a very simple correctness criterion which states that objects we construct correspond to proofs in the underlying logical system. We first give an overview of the whole logical system, called POMSET logic, introduced in Retore (1993) see Retore (1997) for an updated presentation in English. Roughly speaking, this logical calculus is based on Multiplicative Linear Logic enriched with the non commutative connective before, and deals with Partially Ordered Multi sets instead of ordinary multi sets of formulae. Then, we ....

....the de Morgan laws. Exactly like the best way to represent proofs in Intuitionistic Logic consists in using Natural Deduction (where proofs look like (pseudo )trees) the natural syntax for Linear Logic (LL) is proof net syntax that was introduced in the original paper (Girard, 1987) As shown in Retore (1993) Pomset logic very simply extends the usual proof net syntax and our presentation will follow Retore (1997) It is based on the notion of R B graph. Let us call a R B graph an edge bicoloured graph, the two colours of the edges being B(bold, blue) and R(regular, red) The B edges are undirected ....

Retore, Christian. 1993. "Reseaux et Sequents. Ordonnes". These de Doctorat, specialite Mathematiques, Universite Paris 7.

....modality also exists in the category of hypercoherences that Thomas Ehrhard introduced in[Ehr93] This is encouraging since hypercoherences, which may be viewed as a refinement of coherence spaces, are a different semantics, also very close to linear logic. In our previous work on pomset logic [Ret93, Ret95], we studied a self dual connective before , together with partially ordered multisets of formulae. This lead us to the modality to be described. It is a functor, it is self dual, and it enjoys both left and right contraction with respect to before , and A is a retract of A. Fortunately it ....

....Myriam Quatrini did in [Qua95] for the logical calculus LC of [Gir91] INRIA A self dual modality for before 5 2 Preliminary remarks 2.1 Before We refer the reader to [Gir87, Tro92] for the definition of coherence spaces. Let us simply recall the multiplicative connective before studied in [Ret93, Ret95], written A B: Definition 1 Given two coherence spaces A and B, the coherence space A B is defined by: web jA Bj = jAj Theta jBj coherence (a; b) a 0 ; b 0 ) A B] whenever (a a a 0 [A] and b = b 0 ) or b a b 0 [B] From [Ret93] we know that the following easy proposition ....

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Christian Retore. Reseaux et Sequents Ordonnes. These de Doctorat, specialite Mathematiques, Universite Paris 7, February 1993.

....a new coherence space A B from two already built coherence spaces A and B. Therefore, an interpretation associates to each formulae a coherence space. A multiplicative (binary) connective is a connective such that the web jA Bj is jAj Theta jBj, and they are exactly three such connectives [Ret93b, Ret95]: Omega ; which are all associative, while only the two first ones are commutative. x; y) a (x 0 ; y 0 ) A Omega B] iff x a x 0 [A] and y a y 0 [B] x; y) a (x 0 ; y 0 ) AB] iff x a x 0 [A] or y a y 0 [B] x; y) a (x 0 ; y 0 ) A B] iff (x a x 0 [A] ....

....simply for proof nets ffl we are working with the mix rule ffl we use a correctness criterion a la Danos Regnier The reader should not worry about that: Girard s original idea straightforwardly applies. Moreover, the proof for an even bigger calculus, implying theorem 1, is taken up again from [Ret93b, Ret95] in appendix. 1.3 Contents of the paper When proving the previous theorem, the argument makes such an intensive use of the correctness criterion, that we start thinking the converse is true. Noticing that: ffl coherence spaces naturally interprets the mix rule ffl experiments could be defined ....

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Christian Retore. Reseaux et Sequents Ordonnes. These de Doctorat, specialite Mathematiques, Universite Paris 7, February 1993.

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Christian Retore. Reseaux et Sequents Ordonnes.These de Doctorat, specialite mathematiques, UniversiteParis 7, February 1993.

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Christian Retore. Reseaux et Sequents Ordonnes.These de Doctorat, specialite mathematiques, UniversiteParis 7, February 1993.

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Christian Retore. Reseaux et Sequents Ordonnes. These de Doctorat, specialite mathematiques, Universite Paris 7, February 1993.

No context found.

Christian Retore. Reseaux et Sequents Ordonnes. These de Doctorat, specialite mathematiques, Universite Paris 7, February 1993.

No context found.

Christian Retore. Reseaux et Sequents Ordonnes. These de Doctorat, specialite mathematiques, Universite Paris 7, February 1993.

No context found.

Christian Retore. Reseaux et Sequents Ordonnes. These de Doctorat, specialite mathematiques, Universite Paris 7, February 1993.

No context found.

Christian Retore. Reseaux et Sequents Ordonnes.These de Doctorat, specialite mathematiques, UniversiteParis 7, February 1993.

No context found.

Christian Retore. Reseaux et Sequents Ordonnes.These de Doctorat, specialite mathematiques, UniversiteParis 7, February 1993.

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