## Proof-nets and the Hilbert space (1995)

Venue: | Advances in Linear Logic |

Citations: | 50 - 3 self |

### BibTeX

@INPROCEEDINGS{Danos95proof-netsand,

author = {V. Danos and L. Regnier},

title = {Proof-nets and the Hilbert space},

booktitle = {Advances in Linear Logic},

year = {1995},

pages = {307--328},

publisher = {Cambridge University Press}

}

### OpenURL

### Abstract

Girard's execution formula (given in [Gir88a]) is a decomposition of usual fi-reduction (or cut-elimination) in reversible, local and asynchronous elementary moves. It can easily be presented, when applied to a -term or a net, as the sum of maximal paths on the -term/net that are not cancelled by the algebra L (as was done in [Dan90, Reg92]). It is then natural to ask for a characterization of those paths, that would be only of geometric nature. We prove here that they are exactly those paths that have residuals in any reduct of the -term/net. Remarkably, the proof puts to use for the first time the interpretation of -terms/nets as operators on the Hilbert space. 1 Presentation -Calculus is simple but not completely convincing as a real machine-language. Real machine instructions have a fixed run-time; a fi-reduction step does not. Some implementations do map fi-reductions into sequences of real elementary steps (as in environment machines for example) but they use a global time t...

### Citations

321 |
λµ-calculus: an algorithmic interpretation of classic natural deduction
- Parigot
- 1992
(Show Context)
Citation Context ...us directly inside nets. In the same way, the classical embeddings, e.g., mapping A ! B to !?A (?B or !A (?!B (see [DJS94]), allow to study "classical -calculi" (for example, the -calculus o=-=f Parigot [Par92]-=-). A second reason for choosing nets is that they provide a syntax that allows easy description of geometric entities like paths. Everything here works for LL proof-nets as well as for any known kind ... |

132 | The structure of multiplicatives - Danos, Regnier - 1989 |

101 | Rductions correctes et optimales dans le lambda-calcul - Levy - 1978 |

100 |
Geometry of Interaction I: interpretation of system F
- Girard
- 1989
(Show Context)
Citation Context ...per. A persistent path is a path on a -term which survives the action of any reduction (defined in [Reg92]). A regular path is a path which is not cancelled by Girard's algebraic device L (defined in =-=[Gir88a]-=-). The paper mainly aims at proving the equivalence of these two definitions and proceeds in the following order: first the geometric viewpoint on reduction is given, yielding a V. Danos & L. Regnier ... |

53 | Linear logic without boxes
- Gonthier, Abadi, et al.
- 1992
(Show Context)
Citation Context ...st use of the linear structure of the Hilbert space in GOI. In another paper [ADLR94], together with A. Asperti and C. Laneve, we prove that persistence can be defined in two other ways: consistency (=-=[AGL92]-=-) and legality ([AL93]). Persistence is a most natural concept and these different ways to define it make it inescapable. --- Paths and execution. It is fairly obvious that persistent paths describe t... |

35 | H.: LKQ and LKT: sequent calculi for second order logic based upon dual linear decompositions of the classical implication
- Danos, Joinet, et al.
- 1995
(Show Context)
Citation Context ...tionistic embeddings, e.g., mapping A ! B to !A ( B or !A (!B, allow to study -calculus directly inside nets. In the same way, the classical embeddings, e.g., mapping A ! B to !?A (?B or !A (?!B (see =-=[DJS94]), allow t-=-o study "classical -calculi" (for example, the -calculus of Parigot [Par92]). A second reason for choosing nets is that they provide a syntax that allows easy description of geometric entiti... |

21 |
Local and asynchronous beta-reduction
- Danos, Regnier
- 1993
(Show Context)
Citation Context ...ght functor w. The theory L R is defined by the following generators and relations: L R is a free involutive monoid, i.e., there is a mapping u \Gamma! u in L R such that: (u ) = u and (uv) = v u ; 4 =-=[DR93]-=- was our reaction against this frightening bound which is reached by the family (\Delta\Delta\Delta(I I) \Delta \Delta \Delta I) where I denotes the term x:x. V. Danos & L. Regnier for any u and v. No... |

15 |
Geometry of interaction II: deadlock free algorithms
- Girard
- 1989
(Show Context)
Citation Context .... The interest of this new model is that we are going to use our remark about the linearity of the commutation relation by superposing "small" solutions; another interest may be understood b=-=y reading [Gir88b]-=- or [MR91]. The basic structure. B(H) is clearly an involutive monoid (w.r.t. composition and adjunction) wherein there is a zero and a one. It even is a C -algebra. The shift morphism. Takes: H\Omega... |

13 |
Paths in the lambda-calculus
- Asperti, Danos, et al.
- 1994
(Show Context)
Citation Context ...ain result proved here was already known (see [Reg92]), but the method is new and leads to a simpler proof which is the first use of the linear structure of the Hilbert space in GOI. In another paper =-=[ADLR94]-=-, together with A. Asperti and C. Laneve, we prove that persistence can be defined in two other ways: consistency ([AGL92]) and legality ([AL93]). Persistence is a most natural concept and these diffe... |

13 | Une application de la logique lineaire a l' etude des processus de normalisation (principalement de -calcul - Danos - 1990 |

10 |
Some results on the interpretation of *-calculus in operator algebras
- Malacaria, Regnier
- 1991
(Show Context)
Citation Context ...st of this new model is that we are going to use our remark about the linearity of the commutation relation by superposing "small" solutions; another interest may be understood by reading [G=-=ir88b] or [MR91]-=-. The basic structure. B(H) is clearly an involutive monoid (w.r.t. composition and adjunction) wherein there is a zero and a one. It even is a C -algebra. The shift morphism. Takes: H\Omega H \Gamma!... |

6 |
Geometry of interaction III: accomodating the additives
- Girard
- 1995
(Show Context)
Citation Context ...nce; second the algebraic viewpoint, yielding a definition of regularity; and last the announced equivalence is proved. Recently, Girard extended the definition of regularity to accomodate additives (=-=[Gir95]-=-). It would be interesting to investigate whether this matching between persistence and regularity still holds in the full case. --- Context of this paper. The main result proved here was already know... |

1 |
computations and labels in lambda-calculus
- Paths
- 1993
(Show Context)
Citation Context ...ructure of the Hilbert space in GOI. In another paper [ADLR94], together with A. Asperti and C. Laneve, we prove that persistence can be defined in two other ways: consistency ([AGL92]) and legality (=-=[AL93]-=-). Persistence is a most natural concept and these different ways to define it make it inescapable. --- Paths and execution. It is fairly obvious that persistent paths describe the reduction. Intuitiv... |