## A-translation and Looping Combinators in Pure Type Systems (1994)

Venue: | Journal of Functional Programming |

Citations: | 9 - 1 self |

### BibTeX

@ARTICLE{Coquand94a-translationand,

author = {Thierry Coquand and Hugo Herbelin},

title = {A-translation and Looping Combinators in Pure Type Systems},

journal = {Journal of Functional Programming},

year = {1994},

volume = {4},

pages = {77--88}

}

### OpenURL

### Abstract

We present here a generalization of A-translation to a class of Pure Type Systems. We apply this translation to give a direct proof of the existence of a looping combinator in a large class of inconsistent type systems, class which includes type systems with a type of all types. This is the first non-automated solution to this problem.

### Citations

240 |
Interprétation Fonctionnelle et Élimination des Coupures de L’arithmétique D’order Supérieur
- Girard
- 1972
(Show Context)
Citation Context ...eralizes an A-translation of Paulin-Mohring [11] for the Calculus of Constructions with data types distinguished from propositions, and is inspired by a classical/intuitionistic translation of Girard =-=[5]-=- for higher order -calculi. Remark 2: if we assume Church-Rosser property for the Pure Type System we are considering, lemma 5 holds also for fij-conversion and therefore proposition 1 and theorem 1 s... |

74 |
Classicaly and intuitionistically provable recursive functions
- Friedman
- 1978
(Show Context)
Citation Context ...ystems, class which includes type systems with a type of all types. This is the first non-automated solution to this problem. Introduction The term A-translation first appeared in a paper of Friedman =-=[3]-=-. It denotes there a technical tool used in a proof of closure under Markov's rule of several intuitionistic systems. Combined with Godel's translation from classical arithmetic into intuitionistic ar... |

62 |
Extracting Constructive Content from Classical Proofs
- Murthy
- 1990
(Show Context)
Citation Context ...ormulas. Leivant [8] is a good reference about A-translation. Recently, connections between A-translation and Continuation Passing Style have been investigated. See for instance Murthy's Ph. D. thesis=-=[10]-=-. We are going to generalise A-translation to a large class of Pure Type Systems, introduced recently by Barendregt [1, 4]. This generalisation is motivated by the following problem: to extract constr... |

61 |
Introduction to generalized type systems
- Barendregt
- 1991
(Show Context)
Citation Context ...on Passing Style have been investigated. See for instance Murthy's Ph. D. thesis[10]. We are going to generalise A-translation to a large class of Pure Type Systems, introduced recently by Barendregt =-=[1, 4]. This gen-=-eralisation is motivated by the following problem: to extract constructive informations from paradoxes in inconsistent type systems. More specifically, let us define a "looping combinator " ... |

19 |
A modular proof of strong normalisation for the calculus of constructions
- Geuvers, Nederhof
- 1991
(Show Context)
Citation Context ...on Passing Style have been investigated. See for instance Murthy's Ph. D. thesis[10]. We are going to generalise A-translation to a large class of Pure Type Systems, introduced recently by Barendregt =-=[1, 4]. This gen-=-eralisation is motivated by the following problem: to extract constructive informations from paradoxes in inconsistent type systems. More specifically, let us define a "looping combinator " ... |

14 | The computational behaviour of Girard’s paradox - Howe |

13 |
Syntactic translations and provably recursive functions
- Leivant
- 1985
(Show Context)
Citation Context ... Godel's translation from classical arithmetic into intuitionistic arithmetic, this was used to give a new proof of the intuitionistic provability of classically provable \Sigma 0 1 formulas. Leivant =-=[8]-=- is a good reference about A-translation. Recently, connections between A-translation and Continuation Passing Style have been investigated. See for instance Murthy's Ph. D. thesis[10]. We are going t... |

11 |
M.B.Reinhold: 'Type' is not a type
- Meyer
- 1986
(Show Context)
Citation Context ...n a context \Gamma; x : Y and X is not a proof in \Gamma then p[x := X] is long A-applicative in \Gamma. Proof By induction on the structure of p: 4 Looping combinators The idea of Meyer and Reinhold =-=[9] to obtain-=- a recursion combinator in the inconsistent system Type : Type was to exploit the non normalisability of the proof of the inconsistency by inserting some "f " in it in order to obtain a term... |

4 |
Extraction de programmes dans le calcul des constructions
- Paulin
- 1989
(Show Context)
Citation Context ...sion is done by lemma 7. Proposition 1 treats the case of judgements \Gamma ` X : Y with X not a proof. Remark 1: is a Kolmogorov-like A-translation. It generalizes an A-translation of Paulin-Mohring =-=[11]-=- for the Calculus of Constructions with data types distinguished from propositions, and is inspired by a classical/intuitionistic translation of Girard [5] for higher order -calculi. Remark 2: if we a... |

1 |
A New Paradox in Type Theory. To appear
- Coquand
- 1991
(Show Context)
Citation Context ... the only sort: T ype the only axiom: T ype : Type and the only rule: (Type; Type; Type) Both systems U and U \Gamma are nondependent logical Pure Type System. They are both inconsistent, as shown in =-=[2, 5]-=-. Hence, by theorem 2, they contain a looping combinator of sort P rop. It is clear that a looping combinator for one of this system translates directly in a looping combinator of sort Type for T ype ... |

1 |
On the principle of excluded middle. 1925. in From Frege to Godel : a source book in mathematical logic
- Kolmogorov
- 1967
(Show Context)
Citation Context ...plication of Atranslation for Pure Type Systems. The basic idea motivating this use can be traced back to the earliest known translation from classical logic to intuitionistic logic due to Kolmogorov =-=[7]. This tra-=-nslation was actually a translation of classical logic into minimal logic: the rule "ab falso quodlibet" is never used, and the absurd proposition ? in Kolmogorov's paper can thus be replace... |