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A General Storage Theorem for Integers in CallByName
"... The notion of storage operator introduced in [5, 6] appears to be an important tool in the study of data types in second order #calculus. These operators are #terms which simulate callbyvalue in the callbyname strategy, and they can be used in order to modelize assignment instructions. The mai ..."
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The notion of storage operator introduced in [5, 6] appears to be an important tool in the study of data types in second order #calculus. These operators are #terms which simulate callbyvalue in the callbyname strategy, and they can be used in order to modelize assignment instructions. The main result about storage operators is that there is a very simple second order type for them, using Godel's "notnot translation" of classical into intuitionistic logic. We give here a new and simpler proof of a strengthened version of this theorem, which contains all previous results in intuitionistic and in classical logic ([6, 7]), and gives rise to new "storage theorems". Moreover, this result has a simple and intuitive meaning, in terms of realizability.
La valeur d’un entier classique en λµcalcul
 Archive for Mathematical Logic
, 1997
"... de mathématiques, équipe de logique, ..."
Résultats de complétude pour des classes de types du système AF2
 Theoretical Informatics and Applications
, 1998
"... Abstract. J.L. Krivine introduced the AF2 type system in order to obtain programs (λterms) which calculate functions, by writing demonstrations of their totalities. We present in this paper two results of completness for some types of AF2 and for many notions of reductions. These results generalize ..."
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Abstract. J.L. Krivine introduced the AF2 type system in order to obtain programs (λterms) which calculate functions, by writing demonstrations of their totalities. We present in this paper two results of completness for some types of AF2 and for many notions of reductions. These results generalize a theorem of R. LabibSami established in the system F of J.Y. Girard. Résumé. J.L. Krivine a introduit le système de typage AF2 pour obtenir des programmes (λtermes) calculant des fonctions en écrivant des démonstrations de leur totalité. Nous présentons dans ce papier des résultats de complétude pour certains types de AF2 et pour plusieurs notions de réductions. Ces résultats généralisent un théorème de R. LabibSami établi dans le système F de J.Y. Girard.
Mixed Logic and Storage Operators
 ARCHIVE FOR MATHEMATICAL LOGIC
, 2000
"... In 1990 JL. Krivine introduced the notion of storage operators. They are λterms which simulate callbyvalue in the callbyname strategy and they can be used in order to modelize assignment instructions. JL. Krivine has shown that there is a very simple second order type in AF 2 type system for ..."
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Cited by 7 (2 self)
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In 1990 JL. Krivine introduced the notion of storage operators. They are λterms which simulate callbyvalue in the callbyname strategy and they can be used in order to modelize assignment instructions. JL. Krivine has shown that there is a very simple second order type in AF 2 type system for storage operators using Gődel translation of classical to intuitionistic logic. In order to modelize the control operators, JL. Krivine has extended the system AF 2 to the classical logic. In his system the property of the unicity of integers representation is lost, but he has shown that storage operators typable in the system AF 2 can be used to find the values of classical integers. In this paper, we present a new classical type system based on a logical system called mixed logic. We prove that in this system we can characterize, by types, the storage operators and the control operators. We present also a similar result in the M. Parigot’s λµcalculus.
Storage operators and directed Lambda Calculus
 J. Symb. Logic
, 1995
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Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at
A general type for storage operator
 Mathematical Logic Quaterly
, 1995
"... In 1990, J.L. Krivine introduced the notion of storage operator to simulate, in λcalculus, the ”call by value ” in a context of a ”call by name”. J.L. Krivine has shown that, using Gődel translation from classical into intuitionistic logic, we can find a simple type for storage operators in AF 2 ty ..."
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Cited by 5 (4 self)
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In 1990, J.L. Krivine introduced the notion of storage operator to simulate, in λcalculus, the ”call by value ” in a context of a ”call by name”. J.L. Krivine has shown that, using Gődel translation from classical into intuitionistic logic, we can find a simple type for storage operators in AF 2 type system,. In this present paper, we give a general type for storage operators in a slight extension of AF 2. We give at the end (without proof) a generalization of this result to other types. 1
Rescorla, Quarterly
 Journal of Experimental Psychology
, 2003
"... Abstract. We show that Shoenfield’s functional interpretation of Peano arithmetic can be factorized as a negative translation due to J. L. Krivine followed by Gödel’s Dialectica interpretation. Mathematics Subject Classification: 03F03, 03F10. ..."
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Abstract. We show that Shoenfield’s functional interpretation of Peano arithmetic can be factorized as a negative translation due to J. L. Krivine followed by Gödel’s Dialectica interpretation. Mathematics Subject Classification: 03F03, 03F10.
A Semantical Storage Operator Theorem For All Types
, 1997
"... Storage operators are terms which simulate callbyvalue in callbyname for a given set of terms. Krivine's storage operator theorem shows that any term of type :D ! :D , where D is the Godel translation of D, is a storage operator for the terms of type D when D is a datatype or a formula ..."
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Storage operators are terms which simulate callbyvalue in callbyname for a given set of terms. Krivine's storage operator theorem shows that any term of type :D ! :D , where D is the Godel translation of D, is a storage operator for the terms of type D when D is a datatype or a formula with only positive second order quantifiers. We prove that a new semantical version of Krivine's theorem is valid for every types. This also gives a simpler proof of Krivine's theorem using the properties of datatypes. Key words: calculus. Types. AF 2 type system. Storage operators. Godel translations. 1 Introduction. The notion of storage operator was introduced by Krivine in [3]. A storage operator for a set of terms D is a term T simulating callbyvalue in headreduction for all elements in D: for t in D, (T ' t) headreduces to (' t 0 ) where t 0 fi t only depends on the normal form of t (the actual definition is slightly more complex). The storage operator theorem is valid for a typ...
About Classical Logic and Imperative Programming
 Annals of mathematics and Articial Intelligence
, 1996
"... Introduction In this lecture, we shall consider a very well known typed #calculus system, which is the second order #calculus (also called "system F") of Girard [2], rediscovered by Reynolds [16] in a computer science frame. We shall extend it in two ways: . Types will be formulas of second orde ..."
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Introduction In this lecture, we shall consider a very well known typed #calculus system, which is the second order #calculus (also called "system F") of Girard [2], rediscovered by Reynolds [16] in a computer science frame. We shall extend it in two ways: . Types will be formulas of second order predicate calculus, and not only, as in system F, second order propositional calculus [5, 6]. In a certain sense, this is a harmless extension, since the #terms which are typable are the same. This kind of extension has already been considered by D. Leivant [11]. . A much more serious extension is the following: the underlying logic will be classical logic, and not only, as in system F, intuitionistic logic. Extraction of programs from classical proofs has been considered, since two or three years by several people (C. Murthy [12], J.Y. Girardapproach has the following features: 1. We