J.Y. Girard, Une extension de l'interpr#tation de G#del # l'analyse, et son application # l'#limination des coupures et # la th#orie des types. Proceedings of the 2nd Scandinavian Logic Symposium, J.E. Fenstadt,ed.,p.63-92, North Holland, 1971.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....6.2 Comparison with the u:r: models. 27 6.3 Are all the conditions dening F webs necessary . 28 1 Introduction. Girard s System F [16] which is the fragment of calculus which can be controlled by second order propositional logic, dates back to [14], 15] It was rediscovered independently by Reynolds [28] in a Computer Science setting, where it is also called polymorphism. In this paper we assume that the reader is familiar with the syntax of F; as presented in [16] We present here a rather general family of concrete models for System F; ....

....dioeerent, in our setting. Webbed models vs realizability and PER models . It is worth to say a word about realizability and PER models, since these models are successfully used to study some programming aspects of F , and since Girard s rst model of F was Troelstra s realizability model HRO 2 ([14], 15] 34] However these models are rather orthogonal to ours since, as already mentioned, they realize few polymorphic maps while ours are polymax 6 . A connection with intersection type systems. We end with a brief connection with intersection type systems. Engeler s model is transparently ....

J.Y. Girard, Une extension de l'interpr#tation de G#del # l'analyse, et son application # l'#limination des coupures et # la th#orie des types. Proceedings of the 2nd Scandinavian Logic Symposium, J.E. Fenstadt,ed.,p.63-92, North Holland, 1971.

....which class of models we get. In particular, the model introduced in [5] having as polymorphic maps exactly all possible Scott continuous maps, is fij complete. 1 Introduction. In this paper we will progress in the study of non trivial models of the notion of polymorphic maps of System F ([15], 25] This study was started in Girard [15] Scott [29] and McCracken [19] Reynolds [26] Girard [16] and continued by Amadio Bruce Longo [2] Coquand Gunter Winskel [11] and Berardi [6] Our contribution is to prove that there exists a large class of non trivial fij complete models of F . ....

....the model introduced in [5] having as polymorphic maps exactly all possible Scott continuous maps, is fij complete. 1 Introduction. In this paper we will progress in the study of non trivial models of the notion of polymorphic maps of System F ( 15] 25] This study was started in Girard [15], Scott [29] and McCracken [19] Reynolds [26] Girard [16] and continued by Amadio Bruce Longo [2] Coquand Gunter Winskel [11] and Berardi [6] Our contribution is to prove that there exists a large class of non trivial fij complete models of F . Essentially, this class consists of models ....

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J.Y. Girard, Une extension de l'interpr#tation de G#del # l'analyse, et son application # l'#limination des coupures et # la th#orie des types. Proceedings of the 2nd Scandinavian Logic symposium, J.E. Fenstadt, ed.,p.63-92, North Holland, 1975.

....in this logic. In such systems one may thus represent only a subclass of the computable functions, and also not all algorithms for computing them. The rst point is not a drawback if the logic is powerful enough : then all the functions one will ever wish to compute are indeed representable (Girard [7], 8] cf. 15, p.161] or [6] and the second limitation, which is even present in the full calculus, can often be locally overcome [25] The expressive power of a typed calculus is thus limited by its logic, but the positive counterpart is that the computation can be controlled by the logic. ....

....theory. The simplest intuitionistic logic where the Curry Howard correspondence makes sense is Curry s simple type system [4] namely propositional logic over implication only. The most fruitful ones are the intuitionistic restrictions of Second order propositional logic (Girard s System F ([7], 8] 9] rediscovered by Reynolds [30] in a computer science context) and Second order predicate logic (Leivant [22] Krivine s System FA 2 [15, p.141] and extensions (Parigot [27] and Raoeali [29] 0.2.4 Programming with classical proofs. At the beginning of the 90 s it was realized by ....

J.Y. Girard, Une extension de l'interpr#tation de G#del # l'analyse, in Proceedings 2nd Scandinavian Logic Symposium, ed. J. Fenstadt, North Holland, p.63-92, 1971.

....written a #. The deduction rules are given in gure 2 where all propositions are supposed to be normal. An alternative presentation does not normalize the propositions after the quanti er rules but takes fi and j as axioms. This system is well known to be consistent and to enjoy cut elimination [Gir70, Gir72]. 3 HOL C Higher order logic can be expressed as a many sorted rst order theory whose sorts are all simple types. In such a presentation, when t is a term of type T U and u a term of type T we cannot write the application of the term t to the term u as (t u) but we need to introduce a ....

J.-Y. Girard. Une extension de l'interpr#tation de G#del # l'analyse et son application # l'#limination des coupures dans l'analyse et la th#orie des types. In J.E. Fenstad (Ed.), Second Scandinavian Logic Symposium. North-Holland, 1970.

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