## Arithmetical proofs of strong normalization results for symmetric λ-calculi

### Cached

### Download Links

Citations: | 17 - 7 self |

### BibTeX

@MISC{David_arithmeticalproofs,

author = {René David and Karim Nour},

title = {Arithmetical proofs of strong normalization results for symmetric λ-calculi},

year = {}

}

### OpenURL

### Abstract

symmetric λµ-calculus

### Citations

321 |
λµ-calculus: an algorithmic interpretation of classic natural deduction
- Parigot
- 1992
(Show Context)
Citation Context ...hism relating proofs and programs can be extended to classical logic, various systems have been introduced: the λc-calculus (Krivine [12]), the λexn-calculus (de Groote [6]), the λµ-calculus (Parigot =-=[18]-=-), the λ Sym -calculus (Barbanera & Berardi [1]), the λ∆calculus (Rehof & Sorensen [24]), the λµ˜µ-calculus (Curien & Herbelin [3]), ... The first calculus which respects the intrinsic symmetry of cla... |

161 | The duality of computation - Curien, Herbelin |

161 | A New Constructive Logic: Classical Logic - Girard - 1991 |

75 |
Proofs of strong normalization for second order classical natural deduction
- Parigot
- 1997
(Show Context)
Citation Context ...) ∈ SN. Since (M N) = (x y)[x := M][y := N] where x, y are fresh variables, the result follows by applying theorem 19 twice and the induction hypothesis. □ 5 Why the usual candidates do not work ? In =-=[21]-=-, the proof of the strong normalization of the λµ-calculus is done by using the usual (i.e. defined without a fix-point operation) candidates of reducibility. This proof could be easily extended to th... |

66 |
A symmetric lambda calculus for classical program extraction, Information and computation
- Barbanera, Berardi
- 1996
(Show Context)
Citation Context ...d to classical logic, various systems have been introduced: the λc-calculus (Krivine [12]), the λexn-calculus (de Groote [6]), the λµ-calculus (Parigot [18]), the λ Sym -calculus (Barbanera & Berardi =-=[1]-=-), the λ∆calculus (Rehof & Sorensen [24]), the λµ˜µ-calculus (Curien & Herbelin [3]), ... The first calculus which respects the intrinsic symmetry of classical logic is λ Sym . It is somehow different... |

49 | An evaluation semantics for classical proofs - Murthy - 1991 |

49 | Classical proofs as programs - Parigot - 1993 |

37 | A simple calculus of exception handling
- Groote
- 1995
(Show Context)
Citation Context ...d that the Curry-Howard isomorphism relating proofs and programs can be extended to classical logic, various systems have been introduced: the λc-calculus (Krivine [12]), the λexn-calculus (de Groote =-=[6]-=-), the λµ-calculus (Parigot [18]), the λ Sym -calculus (Barbanera & Berardi [1]), the λ∆calculus (Rehof & Sorensen [24]), the λµ˜µ-calculus (Curien & Herbelin [3]), ... The first calculus which respec... |

26 |
Normalization without reducibility
- David
- 2001
(Show Context)
Citation Context ...hen so is (x M1 ... Mn). The proofs of strong normalization that are given here are extensions of the ones given by the first author for the simply typed λ-calculus. This proof can be found either in =-=[7]-=- (where it appears among many other things) or as a simple unpublished note on the web page of the first author (www.lama.univ-savoie.fr/~david ). The same proofs can be done for the λµ˜µ-calculus and... |

26 | A short proof of the strong normalization of the simply typed λµ-calculus - David, Nour - 2003 |

24 |
Confluence en λµ-calcul
- Py
- 1998
(Show Context)
Citation Context ...hese proofs are thus highly non arithmetical. We consider here the λµ-calculus with the rules β, µ and µ ′ . It was known that, for the un-typed calculus, the µ-reduction is strongly normalizing (see =-=[23]-=-) but the strong normalization of the µµ ′ -reduction for the un-typed calculus was an open problem raised long ago by Parigot. We give here a proof of this result. Studying this reduction by itself i... |

20 | A semantical proof of strong normalization theorem for full propositional classical natural deduction - Nour, Saber - 2005 |

19 |
The λ∆-calculus
- Rehof, Sorensen
(Show Context)
Citation Context ...ave been introduced: the λc-calculus (Krivine [12]), the λexn-calculus (de Groote [6]), the λµ-calculus (Parigot [18]), the λ Sym -calculus (Barbanera & Berardi [1]), the λ∆calculus (Rehof & Sorensen =-=[24]-=-), the λµ˜µ-calculus (Curien & Herbelin [3]), ... The first calculus which respects the intrinsic symmetry of classical logic is λ Sym . It is somehow different from the previous calculi since the mai... |

17 |
Finding computational content in classical proofs
- Constable, Murthy
- 1991
(Show Context)
Citation Context ...i.e. there is M ′ ≺ M and σ ′ such that M ′ [σ ′ ] ∈ SN, and (M ′ [σ ′ ] P ) ̸∈ SN. We prove something more general. (1) Let U ∈ U and ρ ∈ Σm. Assume U[ρ] ∈ SN and U[ρ][δ =r P ] ̸∈ SN. Then, C holds. =-=(2)-=- Let V ∈ V and τ ∈ Σn. Assume V [τ] ∈ SN and V [τ][δ =r P ] ̸∈ SN. Then, C holds. The conclusion C follows from (1) with M and σ. The properties (1) and (2) are proved by a simultaneous induction on η... |

16 |
Classical logic, storage operators and 2nd order lambda-calculus
- Krivine
- 1994
(Show Context)
Citation Context ...oduction Since it has been understood that the Curry-Howard isomorphism relating proofs and programs can be extended to classical logic, various systems have been introduced: the λc-calculus (Krivine =-=[12]-=-), the λexn-calculus (de Groote [6]), the λµ-calculus (Parigot [18]), the λ Sym -calculus (Barbanera & Berardi [1]), the λ∆calculus (Rehof & Sorensen [24]), the λµ˜µ-calculus (Curien & Herbelin [3]), ... |

13 | La valeur d’un entier classique en λµ-calcul, Archive for Mathematical Logic (36 - Nour - 1997 |

11 |
Why the usual candidates of reducibility do not work for the symmetric λµ-calculus
- David, Nour
- 2005
(Show Context)
Citation Context ... ∈ SN but (λxM µαN) ̸∈ SN. – M ′ [β =r µαN] ∈ SN but (µβM ′ µαN) ̸∈ SN. This comes from the fact that (M0 M0) and (M1 M1) are in SN but (M1 M0) and (M0 M1) are not in SN. More details can be found in =-=[10]-=-. The third property is true and its proof is essentially the same as the one of the strong normalization of µµ ′ . This comes from the fact that, since (x M1...Mn) never reduces to a λ, there is no “... |

9 | A CPS-translation of the lambda-mu-calculus - Groote - 1994 |

9 |
Free Deduction: An Analysis of ”Computations
- Parigot
- 1990
(Show Context)
Citation Context ...een λµ˜µ. The logical part is the (classical) sequent calculus instead of natural deduction. Natural deduction is not, intrinsically, symmetric but Parigot has introduced the so called Free deduction =-=[17]-=- which is completely symmetric. The λµ-calculus comes from there. To get a confluent calculus he had, in his terminology, to fix the inputs on the left. To keep the symmetry, it is enough to keep the ... |

8 | A non-deterministic classical logic (the λµ ++ -calculus - Nour - 2002 |

5 | Strong Normalization of Second Order Symmetric Lambda-mu Calculus
- Yamagata
(Show Context)
Citation Context ...candidates of reducibility but, unlike the usual construction (for example for Girard’s system F ), the definition of the interpretation of a type needs a rather complex fix-point operation. Yamagata =-=[25]-=- has used the same technic to prove the strong normalization of the symmetric λµ-calculus where the types are those of system F and Parigot, again using the same ideas, has extended Barbanera & Berard... |

3 |
Arithmetical proofs of the strong normalization of the λµ˜µcalculus
- David, Nour
- 2004
(Show Context)
Citation Context ...d calculus (again using candidates of reducibility and a fix point operator) can also be found there. Due to the lack of space, we do not give our proofs of these results here but they will appear in =-=[11]-=-. The paper is organized as follows. In section 2 we give the syntax of the terms and the reduction rules. An arithmetical proof of strong normalization is given in section 3 for the µµ ′ -reduction o... |

3 |
Substitutions explicites, logique et normalisation
- Polonovsky
(Show Context)
Citation Context ...nction is impossible since a term on the right of an application can go on the left of an application after some reductions. The proof of the strong normalization of the µ˜µ-reduction can be found in =-=[22]-=-. The proof is done (by using candidates of reducibility and a fix point operator) for a typed calculus but, in fact, since the type system is such that every term is typable, the result is valid for ... |