## Polarized Proof-Nets: Proof-Nets for LC (Extended Abstract) (1999)

Venue: | Typed Lambda Calculi and Applications '99 |

Citations: | 16 - 3 self |

### BibTeX

@INPROCEEDINGS{Laurent99polarizedproof-nets:,

author = {Olivier Laurent},

title = {Polarized Proof-Nets: Proof-Nets for LC (Extended Abstract)},

booktitle = {Typed Lambda Calculi and Applications '99},

year = {1999},

pages = {213--227},

publisher = {Springer}

}

### Years of Citing Articles

### OpenURL

### Abstract

) Olivier Laurent Institut de Math'ematiques de Luminy CNRS-Marseille, France olaurent@iml.univ-mrs.fr Abstract. We define a notion of polarization in linear logic (LL) coming from the polarities of Jean-Yves Girard's classical sequent calculus LC [4]. This allows us to define a translation between the two systems. Then we study the application of this polarization constraint to proofnets for full linear logic described in [7]. This yields an important simplification of the correctness criterion for polarized proof-nets. In this way we obtain a system of proof-nets for LC. The study of cut-elimination takes an important place in proof-theory. Much work is spent to deal with commutation of rules for cut-elimination in sequent calculi. The introduction of proof-nets (see [7] for instance) solves commutation problems and allows us to define a clear notion of reduction and complexity. In [4], Jean-Yves Girard defines the sequent calculus LC using polarities. LC is a refinement...

### Citations

634 | Light linear logic - Girard - 1998 |

163 |
A new constructive logic: Classical logic
- Girard
- 1991
(Show Context)
Citation Context ...Luminy CNRS-Marseille, France olaurent@iml.univ-mrs.fr Abstract. We define a notion of polarization in linear logic (LL) coming from the polarities of Jean-Yves Girard's classical sequent calculus LC =-=[4]-=-. This allows us to define a translation between the two systems. Then we study the application of this polarization constraint to proofnets for full linear logic described in [7]. This yields an impo... |

133 |
The structure of multiplicatives
- Danos, Regnier
- 1989
(Show Context)
Citation Context ...whether a proof-structure is a proof. More technically, can you inductively deconstruct a proof-structure? There exist different correctness criterions for multiplicative proof-structures like [3] or =-=[2] which lea-=-d to the criterion of [7] for the full case. We present here this general criterion. Definition 8 (Sequentialization of a proof-structure). The relation "L sequentializes R into E " is defin... |

89 | Proof-nets: The parallel syntax for proof-theory. In: Logic and Algebra
- Girard
- 1996
(Show Context)
Citation Context ... sequent calculus LC [4]. This allows us to define a translation between the two systems. Then we study the application of this polarization constraint to proofnets for full linear logic described in =-=[7]-=-. This yields an important simplification of the correctness criterion for polarized proof-nets. In this way we obtain a system of proof-nets for LC. The study of cut-elimination takes an important pl... |

17 |
Quantifiers in linear logic II
- Girard
- 1991
(Show Context)
Citation Context ...;A; 9x:A ffl ! ` !A; ?A ? ` ?!A; ?A ? ` ?B ? ; ?!A; ?A ? ` ?B ? ; ?!A; !?A ? ` ?B ? ; ?!A; 9x!?A ? 4 Proof-Nets Proof-nets have been introduced in [3] for the multiplicative case and then extended in =-=[5]-=- and [7] to full linear logic. 4.1 Proof-Structure The following definitions come from [7] with just some modifications. Definition 7 (Weight). Given a set of elementary weights, i.e. boolean variable... |

7 |
de Falco. Polarisation des preuves classiques et renversement. Compte-Rendu a l'Academie des Sciences
- Quatrini, Tortora
- 1996
(Show Context)
Citation Context ...le rules with structural ones so we have no loss of provability in LC rev . A study of these commutations of reversible rules has been done in a similar case by M. Quatrini and L. Tortora de Falco in =-=[9]-=- for translation of LK j;ae pol into LL. 3.1 LC rev ! LLPc Definition 5. The translation G 7! G ffl from LC rev into LLP c is defined on formulas by: A ffl = !A (:P ) ffl = P ffl ? V ffl = 1 F ffl = 0... |

6 | From proof nets to games (extended abstract
- Lamarche
- 1996
(Show Context)
Citation Context ...lows us to forget the notion of switch and then also the notion of slice. The idea of orientation linked to polarization in proof-nets has already been used. For example Francois Lamarche proposed in =-=[8]-=- a criterion for proof-nets for intuitionistic linear logic with Danos-Regnier polarities. We define a new orientation on proof-structures, the orientation of polarizations(or p-orientation): positive... |

3 | Computational isomorphisms in classical logic (extended abstract
- Danos, Joinet, et al.
- 1996
(Show Context)
Citation Context ...s to add the constraints of LC rev to LC as we have done, but another one is to introduce cuts for the translation of these rules. This has been done with linear isomorphisms in Danos-JoinetSchellinx =-=[1]-=-. 3.2 LLPc ! LC rev Definition 6. The translation G 7! G from LLP c into LC rev is defined on strictly polarized formulas by: (!A) = A (!N ) = N 1 = V 0 = F (P\Omega Q) = PsQ (P \Phi Q) = PsQ (P\Omega... |