## Proof-nets: The parallel syntax for proof-theory (1996)

Venue: | Logic and Algebra |

Citations: | 88 - 1 self |

### BibTeX

@INPROCEEDINGS{Girard96proof-nets:the,

author = {Jean-yves Girard},

title = {Proof-nets: The parallel syntax for proof-theory},

booktitle = {Logic and Algebra},

year = {1996},

pages = {97--124},

publisher = {Marcel Dekker}

}

### Years of Citing Articles

### OpenURL

### Abstract

The paper is mainly concerned with the extension of proof-nets to additives, for which the best known solution is presented. It proposes two cut-elimination procedures, the lazy one being in linear time. The solution is shown to be compatible with quantifiers, and the structural rules of exponentials are also accommodated. Traditional proof-theory deals with cut-elimination; these results are usually obtained by means of sequent calculi, with the consequence that 75 % of a cutelimination proof is devoted to endless commutations of rules. It is hard to be happy with this, mainly because: ◮ the structure of the proof is blurred by all these cases; ◮ whole forests have been destroyed in order to print the same routine lemmas; ◮ this is not extremely elegant. However old-fashioned proof-theory, which is concerned with the ritual question: “is-that-theory-consistent? ” never really cared. The situation changed when subtle algorithmic aspects of cut-elimination became prominent: typically

### Citations

100 |
Towards a geometry of interaction
- Girard
(Show Context)
Citation Context ... characterization of sequentialisable proof-structures. The answer (a notion of proof-net) to this problem is important because we want to carry our program of geometrization of proofs, introduced in =-=[G87A]-=- under the name of geometry of interaction. Remember that geometry of interaction is issued from the analysis of multiplicative proof-nets in terms of permutations, [G86A]. In fact the redaction of th... |

44 |
L.: The structure of multiplicatives. Archive for Mathematical Logic 28
- Danos, Regnier
- 1989
(Show Context)
Citation Context ...sation proof was extended to quantifiers in [G88, G90] ; the second version, the only satisfactory one, made a significant use of simplifications of the method of [G86] discovered by Danos & Regnier, =-=[DR88]-=-. The problem of the extension to full linear logic, and especially to the additives (let us say to the fragment MALL of multiplicative-additive linear logic) has remained open for years. In fact two ... |

11 |
Geometry of interaction III: The general case
- Girard
- 1995
(Show Context)
Citation Context ...is of multiplicative proof-nets in terms of permutations, [G86A]. In fact the redaction of this paper was postponed until a geometry of interaction for additives was found, which is now the case, see =-=[G94]-=-. 1.3 A wrong answer : slicing Definition 6 Let ϕ be a valuation for Θ, i.e. a function from the set of eigenweights of Θ into the boolean algebra {0, 1}, which induces a function (still denoted ϕ) fr... |

5 |
Multiplicatives' Rendiconti del Seminario Matematico dell' Universit'a e Politecnico Torino
- Girard
- 1987
(Show Context)
Citation Context ...ion of proofs, introduced in [G87A] under the name of geometry of interaction. Remember that geometry of interaction is issued from the analysis of multiplicative proof-nets in terms of permutations, =-=[G86A]-=-. In fact the redaction of this paper was postponed until a geometry of interaction for additives was found, which is now the case, see [G94]. 1.3 A wrong answer : slicing Definition 6 Let ϕ be a valu... |

5 |
Quantifiers in linear logic II. In Nuovi problemi della logica e della filosofia della scienza
- Girard
- 1991
(Show Context)
Citation Context ...t of MALL. ◮ To prove sequentialisation : the case of quantifiers was considered as a preliminary case, but the solution found in [G88] was too acrobatic to be extended. This solution was improved in =-=[G90]-=- by means of a new kind of switchings, determined by certain formal dependencies. The solution immediately extends to additives, provided certain weight dependencies are forbidden (this is the depende... |

2 |
Quantifiers in linear logic. In Temi e prospettivi della logica e della filosofia della szienca contemporanee
- Girard
- 1987
(Show Context)
Citation Context ...fferent proof-nets cannot receive a satisfactory answer, especially out of MALL. ◮ To prove sequentialisation : the case of quantifiers was considered as a preliminary case, but the solution found in =-=[G88]-=- was too acrobatic to be extended. This solution was improved in [G90] by means of a new kind of switchings, determined by certain formal dependencies. The solution immediately extends to additives, p... |