Olivier Laurent. Polarized proof-nets: proof-nets for LC (extended abstract). In Jean-Yves Girard, editor, Typed Lambda Calculi and 30 Applications '99, volume 1581 of Lecture Notes in Computer Science, pages 213--227. Springer, April 1999.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....by proposition 2.4.7 is still valid for R 0 and R 0 0 . We can then deduce R 0 = R 0 0 , i.e. R R 0 . We now de ne the notion of weakly polarized formula , which is related to the one of polarized formula , widely studied in the last 10 years (see [Gir91] DJS97] QTdF96] TdF97] [Lau99], LQTdF00] 4.1.3. Definition. Weakly polarized formulas) A propositional formula P (resp. N) of LL is weakly positive (resp. weakly negative) when it is built in the following way (where X is an atomic formula) P : X j P P j P N j N P N : X j N N j P N j N P We will ....

Olivier Laurent. Polarized proof-nets: proof-nets for LC (extended abstract). In Jean-Yves Girard, editor, Typed Lambda Calculi and Applications '99, volume 1581 of Lecture Notes in Computer Science, pages 213-227. Springer, April 1999.

....of [Gir87] are removed, and a correctness criterion using a 2 notion of boolean weight is given. A sequentialization theorem for MALL is proven, but only a lazy cut elimination procedure is de ned (not every cut link is eliminated) This correctness criterion has been drastically simpli ed in [Lau99] in the framework of polarized proof nets. In fact, for these proof nets, the notion of positive tree (coming from [Lau99] allowed us to give a complete solution to the problems of con uence and strong normalization (see [LQTdF01] Despite the fact that classical logic can be translated by ....

....for MALL is proven, but only a lazy cut elimination procedure is de ned (not every cut link is eliminated) This correctness criterion has been drastically simpli ed in [Lau99] in the framework of polarized proof nets. In fact, for these proof nets, the notion of positive tree (coming from [Lau99]) allowed us to give a complete solution to the problems of con uence and strong normalization (see [LQTdF01] Despite the fact that classical logic can be translated by means of polarized proof nets (see [LQTdF01] not all the notable fragments of LL are polarized: for example the system of ....

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Olivier Laurent. Polarized proof-nets: proof-nets for LC (extended abstract). In Jean-Yves Girard, editor, Typed Lambda Calculi and Applications '99, volume 1581 of Lecture Notes in Computer Science, pages 213-227. Springer, April 1999.

....which yields a more satisfactory notion of normal form, but only a weak Church Rosser property is proven. Recently, it appeared that the sequentialization proof of [Girard 95a] could be simpli ed for a signi cant fragment of LL (containing of course the additives) the polarized fragment (see [Laurent 99] The notion of positive tree of [Laurent 99] allows to give in [LQT 00] a very simple proof of strong normalization and con uence for polarized proof nets. This paper is a contribution to the study of proof net normalization of full LL (without constants) in the traditional syntax of ....

....form, but only a weak Church Rosser property is proven. Recently, it appeared that the sequentialization proof of [Girard 95a] could be simpli ed for a signi cant fragment of LL (containing of course the additives) the polarized fragment (see [Laurent 99] The notion of positive tree of [Laurent 99] allows to give in [LQT 00] a very simple proof of strong normalization and con uence for polarized proof nets. This paper is a contribution to the study of proof net normalization of full LL (without constants) in the traditional syntax of [Girard 87] Despite the lack of con uence, we have ....

Laurent O., Polarized proof-nets: proof-nets for LC, extended abstract, Proceedings of the conference \Typed LambdaCalculus and Applications '99", L'Aquila Italy, april

....idea of introducing the notion of jumps for weakenings is not new: as far as we know it is an idea of J. Y. Girard and has been explicitely mentioned in the appendix of [Girard 95a] However, the works making a precise use of this idea are very rare. Indeed, we are only aware of the recent work [Laurent 99] which makes use of a notion of jump in the restricted polarized case, mainly for a sequentialization purpose. 2.10. Definition. Let R be a proof net and B an additive box of R. Let l 1 and l 2 (resp. a 1 and a 2 , B 1 and B 2 ) two links (resp. edges, boxes) of R. We shall say that B separates ....

....you reach a normal form forget the jump function. iii) The theorem seems to be the best result one can obtain, unless one decides to change the syntax of the additives. In [Girard 95a] boxes for the additives are removed thanks to a weight function on the edges of the proof net (see also [Laurent 99] for a more elegant version in the restricted polarized case) but the problem of the additive commutative e.r.s. is not solved. In chapter 5 of [Tortora 00] we introduce the notion of multiboite 43 (an additive box with several front doors, with the terminology of definition 2.2) which ....

Laurent Olivier, Polarized Proof-Nets: Proof-Nets for LC (Extended Abstract). In Jean-Yves Girard, editor, Typed Lambda Calculi and Applications '99, volume 1581 of LNCS, pages 213227. Springer-Verlag

....( more canonical) solutions have been proposed in [9] and [15] Recently, a new fragment of LL appeared to have a great interest: in [6] and [3] the polarized fragment of LL is shown to be enough to translate faithfully classical logic. A study of proof nets for such a fragment was undertaken in [13], and the the notion of [9] drastically simpli ed. In [14] a proof of strong normalization and con uence of the cut elimination procedure is given for polarized LL, using the syntax of [5] notice that for full LL con uence is wrong and strong normalization is still not completely proven) ....

....coming from [17] are also introduced (injective 1 experiments) to be used later in section 7. Section 4 is devoted to de ne and study the notion of correct sliced proof structure (or sliced proof net) The polarization constraints allow to extend to our framework the correctness criterion of [13]. We de ne a sliced cut elimination procedure (de nition 12) we prove that correctness is preserved by our sliced cut elimination steps (theorem 1) and that our semantical interpretation is sound (theorem 2) Our sliced proof nets are thus proven to be computational objetcs. In section 5, we ....

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Olivier Laurent. Polarized proof-nets: proof-nets for LC (extended abstract). In Jean-Yves Girard, editor, Typed Lambda Calculi and Applications '99, volume 1581 of Lecture Notes in Computer Science, pages 213-227. Springer, April 1999.

.... strategies for modeling terms (and is therefore incomplete) In this paper we build a game model of the polarized fragment LLP of LL and introduce a polarization of formulas games in the spirit of Girard s polarities [7] As LLP is expressive enough to encode LC, the calculus, [15, 16, 17], we therefore get a model for various classical deduction systems. Moreover these encodings are de nable for both call by name and call by value evaluations giving a uniform setting for interpreting them. Negative games are the usual intuitionistic ones, but we add their dual: positive games ....

....s; t i 2 id A ; i = 1; 2g : A 1 P A 2 A. Weakening. wA is the strategy on A de ned by wA = f g [ fm j m 2 M i A g. 4 Linear Logic with Polarities (exponential case) LLP is a linear system for classical logic. The main deterministic classical systems have translations into LLP [15, 16, 17]. Moreover LLP is expressive enough to interpret both call by name and call by value classical logics by pointing out negative or positive formulas. This corresponds to the duality between control and co control categories [19] 4.1 LLP Polarized formulas. We replace the lifted formulas by the ....

Olivier Laurent. Polarized proof-nets: proof-nets for LC (extended abstract). In 11 Jean-Yves Girard, editor, Typed Lambda Calculi and Applications '99, volume 1581 of Lecture Notes in Computer Science, pages 213-227. Springer, April 1999.

....due to a better understanding of LL itself. Most notably, the works [AP91, Gir91a, DJS97] pointed out two crucial properties of LL: focalization and reversion. In [QTdF96] these two properties gave birth to the classical system LK ; pol , for which some results are stated (but not proven) In [Lau99b], the same two properties were studied in the framework of linear logic proof nets. The starting point of the present work was to realize that these two papers are in fact tightly linked. We put together our knowledge on proof nets [Lau99b, Lau99a, TdF00a, TdF00b] on denotational semantics ....

...., for which some results are stated (but not proven) In [Lau99b] the same two properties were studied in the framework of linear logic proof nets. The starting point of the present work was to realize that these two papers are in fact tightly linked. We put together our knowledge on proof nets [Lau99b, Lau99a, TdF00a, TdF00b], on denotational semantics [Qua96] on the linear logic approach to classical logic [JSTdF98, TdF97, QTdF96] and focused on reversion and focalization. In this paper, we study these two properties both in a classical and in a linear framework. This leads us to consider two couples of ....

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Olivier Laurent. Polarized proof-nets: proof-nets for LC (extended abstract). In Jean-Yves Girard, editor, Typed Lambda Calculi and Applications '99, volume 1581 of Lecture Notes in Computer Science, pages 213-227. Springer, April 1999.

.... the sequential information by translating it as exponential boxes (typically the t translation in [2] de nes the encoding of the classical arrow by: A B A ( B) In this paper we set up and study a translation of calculus into polarized proof nets (PPN) for the fragment LLP of linear logic [7, 8]. LLP is a subsystem of LL dealing with polarized formulas where polarities are de ned as a linear version of LC s polarities. Polarized proof nets allow a ner use of exponential boxes: structural rules, which are reserved to formulas of the shape A in LL, are now applied to any negative formula ....

....pairing may be encoded by the connective, Selinger s disjunctive connective may be encoded by the P connective. As in the calculus, we can use linear rst and second order quanti ers to encode the classical ones. The proof net technology needed for all these extensions has been developed in [7, 8]. Also, in the last section we use the same trick as in the calculus for applying our results to the untyped calculus. 1 In fact the t translation may be factorized through ours: rst use the LLP translation, then use the translation of LLP into LL shown in section 1.4. 2 DRAFT Note that ....

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Olivier Laurent. Polarized proof-nets: proof-nets for LC (extended abstract). In Jean-Yves Girard, editor, Typed Lambda Calculi and Applications '99, volume 1581 of Lecture Notes in Computer Science, pages 213-227. Springer, April 1999. 22 DRAFT

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Olivier Laurent. Polarized proof-nets: proof-nets for LC (extended abstract). In Jean-Yves Girard, editor, Typed Lambda Calculi and 30 Applications '99, volume 1581 of Lecture Notes in Computer Science, pages 213--227. Springer, April 1999.

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