## STORAGE OPERATORS and ∀-POSITIVE TYPES in TTR TYPE SYSTEM

### BibTeX

@MISC{Nour_storageoperators,

author = {Karim Nour},

title = {STORAGE OPERATORS and ∀-POSITIVE TYPES in TTR TYPE SYSTEM},

year = {}

}

### OpenURL

### Abstract

In 1990, J.L. Krivine introduced the notion of storage operator to simulate ”call by value ” in the ”call by name” strategy. J.L. Krivine has shown that, using Gődel translation of classical into intuitionitic logic, we can find a simple type for the storage operators in AF 2 type system. This paper studies the ∀-positive types (the universal second order quantifier appears positively in these types), and the Gődel transformations (a generalization of classical Gődel translation) of T T R type system. We generalize, by using syntaxical methods, the J.L. Krivine’s Theorem about these types and for these transformations. We give a proof of this result in the case of the type of recursive integers.

### Citations

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(Show Context)
Citation Context ...Cx1...xnA < t1, .., tn >)*= µCx1...xnA*< t1, .., tn >. We give the proof of this result in the case of the type of recurcive integers. 2 Basic notions of pure λ-calculus Our notation is standard (see =-=[1]-=- and [5]). We denote by Λ the set of terms of pure λ-calculus, also called λ-terms. Let t, u, u1, ..., un ∈ Λ, the application of t to u is denoted by (t)u. In the same way we write [8]). 4 J.L. Krivi... |

123 |
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(Show Context)
Citation Context ...A < t1, .., tn >)*= µCx1...xnA*< t1, .., tn >. We give the proof of this result in the case of the type of recurcive integers. 2 Basic notions of pure λ-calculus Our notation is standard (see [1] and =-=[5]-=-). We denote by Λ the set of terms of pure λ-calculus, also called λ-terms. Let t, u, u1, ..., un ∈ Λ, the application of t to u is denoted by (t)u. In the same way we write [8]). 4 J.L. Krivine and t... |

55 | Programming with proofs - Krivine, Parigot - 1990 |

35 | Typing and computation properties of lambda expressions - Leivant - 1986 |

34 |
Opérateurs de mise en mémoire et traductions de Gödel, Archiv for Mathematical Logic (30
- Krivine
- 1990
(Show Context)
Citation Context ...e typing of t ′. We suggest the following proposition : N*[x] = ∀X{∀y[(X(y) → O) → (X(sy) → O)] → [(X(0) → O) → (X(x) → O)]}. It is easy to chech that ⊢AF 2 T1, T2 : ∀x{N*[x] → [(N[x] → O) → O]} (see =-=[6]-=- and [12]). For each formula F of AF 2, we indicate by F * the formula obtained by putting ¬ in front of each atomic formulas of F (F * is called the Gődel translation of F ). J.L. Krivine has shown t... |

28 |
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(Show Context)
Citation Context ... < x1, ..., xm >→ E] where C is not free in E and G, and Y is the Turing’s fixed point. The rule (Y ) expresses also the fact that µCx1...xmD < t1, ..., tm > is a least fixed point. Theorem 3.2 ([12],=-=[18]-=-). 1) Conservation Theorem If Γ ⊢T T R t : A, and t →β t ′ , then Γ ⊢T T R t ′ : A. 2) Strong normalization If Γ ⊢T T R t : A without using the rule (Y ), then t is strongly normalizable. 3) Weak norm... |

19 |
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- 1993
(Show Context)
Citation Context ... variable f never comes in head position during the reduction, and we may then replace f by any λ-term. • The computation time of the head reduction (T )θnF ≻ (F )n depends only on θn. We showed (see =-=[12]-=-) that it is not possible to get the normal form of θn. We then change the definition : A closed λ-term T is called storage operator for N if and only if for every n ∈ IN, there is a closed λ-term τn ... |

19 |
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- 1989
(Show Context)
Citation Context ...efine the recursive integer n by induction : 0 = λfλxx and n + 1 = λfλx(f)n. Let N be the set of recursive integers. We have N = {t / t is a closed normal λ-term / ⊢T T R t : N r [sn(0)], n ≥ 0} (see =-=[19]-=-). Let s = λnλfλx(f)n. It is easy to check that s is a λ-term for successor, and ⊢T T R s : ∀y(N r [y] → N r [sy]). Define T1 = (Y )H where H = λxλy((y)λz(G)(x)z)δ, G = λxλy(x)λz(y)(s)z, and δ = λf(f)... |

17 | Opérateurs de mise en mémoire et types ∀-positifs - Nour - 1996 |

15 |
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(Show Context)
Citation Context ...ms 3.2 and 3.3. ✷ Remark We cannot if the reverse of 2)-Theorem 3.2 is true, but the λ-term t = λx(λy((x)(y)λxx)(y)λxλyx)λx(x)x (which is strongly normalizable, and untypable in AF 2 type system (see =-=[3]-=-)) is typable in T T R type system. Indeed, if we take B = µC(∀XX → C), we check easily that ⊢T T R⋄ t : [B → (B → B)] → B. 4 Properties of T T R type system 4.1 Permutations Lemmas Lemma 4.1 1) The t... |

13 |
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(Show Context)
Citation Context ... standard (see [1] and [5]). We denote by Λ the set of terms of pure λ-calculus, also called λ-terms. Let t, u, u1, ..., un ∈ Λ, the application of t to u is denoted by (t)u. In the same way we write =-=[8]-=-). 4 J.L. Krivine and the author proved independely the same result for AF 2 type system (see [7] and [12]). 5 This types were studied by some authors (in particular R. Labib-Sami), and have remarkabl... |

7 | Preuve syntaxique d'un theoreme de J.L. Krivine sur les operateurs de mise en memoire - Nour |

7 | Storage operators and directed lambda-calculus - David, Nour - 1995 |

6 |
Mise en mémoire (preuve générale
- Krivine
- 1993
(Show Context)
Citation Context ...rms. Let t, u, u1, ..., un ∈ Λ, the application of t to u is denoted by (t)u. In the same way we write [8]). 4 J.L. Krivine and the author proved independely the same result for AF 2 type system (see =-=[7]-=- and [12]). 5 This types were studied by some authors (in particular R. Labib-Sami), and have remarkable properties (see 4(t)u1...un instead of (...((t)u1)...)un. The β-reduction (resp. β-equivalence... |

6 | Strong storage operators and data types - Nour - 1995 |

4 | Lambda calcul, évaluation paresseuse et mise en mémoire - Krivine - 1991 |

3 |
The Inf function in system F
- DAVID
- 1994
(Show Context)
Citation Context ...N = ∀X{[X → X] → [X → X]}) in F type system that computes the minimum of two Church integers in time O(min.Log(min)). The notion of storage operators plays an important tool in this constraction (see =-=[2]-=-). 3The T T R type system is an extension of AF 2 based on recursive definitions of types, which is intented to solve the basic problems of efficiency mentioned before. In T T R we have a logical ope... |

3 | Polomorphic type - MITCHELL - 1988 |

3 | Opérateurs propre de mise en mémoire - Nour - 1993 |