## A Deconstruction of Non-deterministic Classical Cut Elimination (0)

Venue: | TLCA'01, LNCS 2044, 268--282 |

Citations: | 4 - 0 self |

### BibTeX

@INPROCEEDINGS{Laird_adeconstruction,

author = {J. Laird},

title = {A Deconstruction of Non-deterministic Classical Cut Elimination},

booktitle = {TLCA'01, LNCS 2044, 268--282},

year = {},

pages = {268--282},

publisher = {Springer}

}

### OpenURL

### Abstract

This paper shows how a symmetric and non-deterministic cut elimination procedure for a classical sequent calculus can be faithfully simulated using a non-deterministic choice operator to combine different `double-negation' translations of each cut. The resulting interpretation of classical proofs in a -calculus with non-deterministic choice leads to a simple proof of termination for cut elimination.

### Citations

662 | Light linear logic
- Girard
- 1998
(Show Context)
Citation Context ...nistic behaviour of cut elimination, and this non-determinism is a formidable obstacle to proof theoretical, semantic, or computational interpretations of classical proofs. Previous analyses, such as =-=[10, 16, 11, 7]-=- have often resolved this problem by `controlling' the non-determinism of classical cuts out of existence by predetermining their behaviour, either systematically or by attaching additional informatio... |

374 |
Proofs and Types
- Girard, Lafont, et al.
- 1989
(Show Context)
Citation Context ...is necessary to choose which branch to commute the cut up first. Because of the presence of structural rules, this choice has important consequences. A well known example is the observation of Lafont =-=[12]-=- that any two proofs of the same formula can be merged (using Cut) into a proof which has cut free forms derived from both of the original proofs. Definition 2. Given proofs ;s0 ` \Gamma , formsors0 `... |

337 |
An algorithmic interpretation of classical natural deduction
- Parigot
- 1992
(Show Context)
Citation Context ...nistic behaviour of cut elimination, and this non-determinism is a formidable obstacle to proof theoretical, semantic, or computational interpretations of classical proofs. Previous analyses, such as =-=[10, 16, 11, 7]-=- have often resolved this problem by `controlling' the non-determinism of classical cuts out of existence by predetermining their behaviour, either systematically or by attaching additional informatio... |

250 | A formulae-as-types notion of control
- Griffin
- 1990
(Show Context)
Citation Context ..., either systematically or by attaching additional information to proofs. This permits an interpretation in terms of a deterministic system such as double negation translation [13], control operators =-=[14]-=- or `linear decoration' [7]. Although these interpretations have generated many key insights (ingredients of the approach described here) they seem inevitably to lose some proof-theoretic content beca... |

214 | Games and full completeness for multiplicative linear logic
- Abramsky, Jagadeesan
- 1994
(Show Context)
Citation Context ...ivalence) generated by such a cut elimination procedure must equatesands0 and hence be trivial. However, this is consistent with an idea underlying the geometry of interaction [9], and game semantics =-=[4, 1, 5]-=-, that cut elimination is a processsanalogous to computation. Non-determinism is both a standard property of computational processes and a key feature of many important algorithms and the physical sys... |

203 |
The Lambda Calculus: its Syntax and Semantics (revised ed
- Barendregt
- 1984
(Show Context)
Citation Context ...h that s i t and u i fi [[t]]. Next we prove that a form of the Church-Rosser theorem holds fors+ (using a method which is a straightforward variant of the standard proof of that theorem --- see e.g. =-=[3]-=-). Lemma 2. Suppose u is as+ term such u i v and u i fi v 0 , then there exists as+ -term w such that v i fi w and v 0 \Gamma! w. [[s]] // [[ ]]s@ @ @ @ @ @ @ @ u fi fflffl fflffl [[t]] u // // fi v f... |

169 |
A new constructive logic: classical logic
- Girard
- 1991
(Show Context)
Citation Context ...nistic behaviour of cut elimination, and this non-determinism is a formidable obstacle to proof theoretical, semantic, or computational interpretations of classical proofs. Previous analyses, such as =-=[10, 16, 11, 7]-=- have often resolved this problem by `controlling' the non-determinism of classical cuts out of existence by predetermining their behaviour, either systematically or by attaching additional informatio... |

109 | A new deconstructive logic: Linear logic
- Danos, Joinet, et al.
- 1997
(Show Context)
Citation Context |

103 |
Towards a geometry of interaction
- Girard
- 1989
(Show Context)
Citation Context ...ance, a denotational equivalence) generated by such a cut elimination procedure must equatesands0 and hence be trivial. However, this is consistent with an idea underlying the geometry of interaction =-=[9]-=-, and game semantics [4, 1, 5], that cut elimination is a processsanalogous to computation. Non-determinism is both a standard property of computational processes and a key feature of many important a... |

73 |
A symmetric lambda-calculus for classical program extraction
- Barbanera, Berardi
- 1996
(Show Context)
Citation Context ...dients of the approach described here) they seem inevitably to lose some proof-theoretic content because only a limited selection of the possible cut elimination behaviours can be pre-determined (see =-=[2, 17]-=- and the discussion in Section 1.2 below). On the other hand, more general symmetric and non-deterministic cut elimination and normalisation procedures have been described via rewriting systems for te... |

46 |
A semantics of evidence for classical arithmetic
- Coquand
- 1995
(Show Context)
Citation Context ...ivalence) generated by such a cut elimination procedure must equatesands0 and hence be trivial. However, this is consistent with an idea underlying the geometry of interaction [9], and game semantics =-=[4, 1, 5]-=-, that cut elimination is a processsanalogous to computation. Non-determinism is both a standard property of computational processes and a key feature of many important algorithms and the physical sys... |

37 | Strong normalisation of cut-elimination in classical logic, TLCA ’99
- Urban, Bierman
- 1999
(Show Context)
Citation Context ...dients of the approach described here) they seem inevitably to lose some proof-theoretic content because only a limited selection of the possible cut elimination behaviours can be pre-determined (see =-=[2, 17]-=- and the discussion in Section 1.2 below). On the other hand, more general symmetric and non-deterministic cut elimination and normalisation procedures have been described via rewriting systems for te... |

21 |
A Game Semantics for Linear Logic”, Annals of Pure and Applied Logic 56
- Blass
- 1992
(Show Context)
Citation Context ...ivalence) generated by such a cut elimination procedure must equatesands0 and hence be trivial. However, this is consistent with an idea underlying the geometry of interaction [9], and game semantics =-=[4, 1, 5]-=-, that cut elimination is a processsanalogous to computation. Non-determinism is both a standard property of computational processes and a key feature of many important algorithms and the physical sys... |

17 | Untersuchung über das logische Schliessen - Gentzen - 1934 |

15 |
On intuitionistic arithmetic and number theory
- Gödel
- 1965
(Show Context)
Citation Context ...ermining their behaviour, either systematically or by attaching additional information to proofs. This permits an interpretation in terms of a deterministic system such as double negation translation =-=[13]-=-, control operators [14] or `linear decoration' [7]. Although these interpretations have generated many key insights (ingredients of the approach described here) they seem inevitably to lose some proo... |

10 | Computational Content of Classical Logic
- Coquand
- 1996
(Show Context)
Citation Context ...that many non-deterministic algorithms may be extracted from proofs. A more difficult question is whether there are natural proofs which have computational content which is non-deterministic. Coquand =-=[5, 6]-=- has described examples of symmetric classical existence proofs from which two different witnesses can be extracted by different double negation translations, but much work remains to be done. The str... |

1 |
Parameter passing and non-determinism
- Hennessey, Ashcroft
- 1977
(Show Context)
Citation Context ...ich programs are run. But can a connection between non-deterministic computation and non-deterministic cut elimination be established ? Computationally, or corresponds to an erratic `choice operator' =-=[15]-=-, showing in principal that many non-deterministic algorithms may be extracted from proofs. A more difficult question is whether there are natural proofs which have computational content which is non-... |