Jean-Yves Girard. Une extension de l'interpr'etation de Godel `a l'analyse, et son application `a l"elimination des coupures dans l'analyse et la th'eorie des types. In J. E. Fenstad, editor, Proceedings of the Second Scanindavian Logic Symposium, Studies in Logic and the Foundations of Mathematics, pages 63--92. North-Holland, 1971.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... t) and (t t) t t) produces a result of type (t t) Once again, the identity (y:y) is an appropriate argument. The rank 2 systems we consider are subsystems of two widely studied type systems, System F and the system of intersection types. System F, introduced independently by Girard [7] and by Reynolds [28] predates ML and can type many more terms. A recent result of Wells [34] however, shows that typability in the system is undecidable, putting type inference out of reach. The system of intersection types, introduced independently by Coppo and Dezani [5] and by Sall e [29] ....

Jean-Yves Girard. Une extension de l'interpretation de Godel `a l'analyse, et son application `a l"elimination des coupures dans l'analyse et la th'eorie des types. In J.E. Fenstad, editor, 2 nd Scandinavian Logic Symp., pages 63--92. North-Holland Publishing Co., 1971.

.... t) and (t t) t t) produces a result of type (t t) Once again, the identity (y:y) is an appropriate argument. The rank 2 systems we consider are subsystems of two widely studied type systems, System F and the system of intersection types. System F, introduced independently by Girard [7] and by Reynolds [28] predates ML and can type many more terms. A recent result of Wells [34] however, shows that typability in the system is undecidable, putting type inference out of reach. The system of intersection types, introduced independently by Coppo and Dezani [5] and by Sall e [29] ....

Jean-Yves Girard. Une extension de l'interpretation de Godel `a l'analyse, et son application `a l"elimination des coupures dans l'analyse et la th'eorie des types. In J.E. Fenstad, editor, 2 nd Scandinavian Logic Symp., pages 63--92. North-Holland Publishing Co., 1971.

....x does not occur free in N . Indeed, the semantics of many programming languages including non functional languages can be given by translation to FPC. FPC s main deficiency as a metalanguage for denotational semantics is a lack of parametric polymorphism (as in the Girard Reynolds calculus [7, 27]) which precludes a good representation of abstract data types. 3 Category of Meanings In this section we construct a category suitable for interpreting FPC. Throughout the section, DCPO denotes the category of dcpos and partial continuous functions, where a dcpo is a directed complete poset ....

J.-Y. Girard. Une extension de l'interpr'etation de Godel `a l'analyse, et son application `a l"elimination des coupures dans l'analyse et la th'eorie des types. In J. E. Fenstad, editor, Proceedings of the Second Scandinavian Logic Symposium, volume 63 of Studies in Logic and the Foundations of Mathematics, pages 63--92. North-Holland, 1971.

....side, called the set of accessible subterms, by the use of computability preserving operations. Here, computability refers to Tait s computability predicate method for proving the termination of the simply typed calculus [46] which was later extended by Girard to the polymorphic calculus [22,24]. To explain our construction, we need to recall the basics of Tait s method. The starting observation is that it is not possible to prove the termination of fi reduction directly by induction on the structure of terms because of the application case: in the untyped calculus, the term (x:xx x:xx) ....

J.-Y. Girard. Une extension de l'interpr'etation de Godel `a l'analyse, et son application `a l"elimination des coupures dans l'analyse et la th'eorie des types. In J. E. Fenstad, editor, Proc. of the 2nd Scandinavian Logic Symp., volume 63 of Studies in Logic and the Foundations of Mathematics. North-Holland, 1971.

....a (difficult) result (the Genericity Theorem in section 3) In our views, this may contribute to set on more solid grounds impredicatively given properties. 2 System Fc and Impredicativity System F is known as Impredicative Type Theory, or Polymorphic Gammacalculus; it was introduced by Girard, [Gir71] (see [GLT89] for a more recent presentation and [AL91] for its categorical semantics) It consists of types and terms (well typed terms) As a key component of its expressive power, the type system of F allows explicit quantification on type variables (second order quantification) its ....

....as soon as a sufficiently expressive system is given. This observation will be established in two steps. First, in this section, by a simple remark on the compatibility of an axiomatic extension of system F ; later on, by the Genericity Theorem. The first remark is inspired by a result in [Gir71]: in system F , there is no definable term that discriminates between types. That is, there is no term J oe such that J oe applied to type ae is 1 if oe = ae, and is 0 if oe 6= ae. In other 5 words, there is no term whose output values are all in the same type (the type of integers, or any other ....

J.-Y. Girard. Une extension de l'interpr'etation de Godel `a l'analyse, et son application `a l"elimination des coupures dans l'analyse et la th'eorie des types. Proceedings of the 2nd Scandinavian Logic Symposium, J.E. Fenstad, ed., pages 63--92, North-Holland, 1971. 13

....is bound to be tricky. Because it implies the consistency of Peano arithmetic, the proof must somehow go beyond the capabilities of that theory. W. Tait developed an elegant and flexible technique for proving normalization, using appropriate convertibility predicates (cf. Tait [1967] and Tait [1971]) 4.3.2. Definition. For each type oe, we define the set of reducible terms of type oe, denoted by Red oe : 1. If t is a term of type 0, then t is in Red 0 if and only if t is normalizing. 2. If t is a term of type oe , then t is in Red oe if and only if whenever s in Red oe , t(s) is in ....

....of T (BR) We have both a functional and a term model for T (BR) The former is given by the continuous functionals of Kleene and Kreisel indicated in section 4.1 and footnote 14, with the constants interpreted by recursively continuous functionals. The latter is established by work of Tait [1971] and independently Luckhardt [1973] which shows that T (BR) is normalizing and confluent. Both models can be formalized in PA 2 (CA) thus proving 6.5.4. Theorem. The provably total recursive functions of PA 2 (CA) are exactly the ones represented by bar recursive terms. 6.6. ....

[Article contains additional citation context not shown here]

Une extension de l'interpretation de Godel `a l'analyse, et son application `a l"elimination des coupures dans l'analyse et dans la th'eorie des types, in: Fenstad [1971], pp. 63--92.

....4, we define several variants of SRT that are better suited to practical use. We conclude with comparisons to related work. 2 Flow Analysis for Typed Languages We study a simple core language called i suitable for compiling languages such as ML. i extends the predicative subset of system F [7, 20] with recursive procedures. In i , as in the predicative subset of F , polymorphic functions cannot be applied to quantified types. 2.1 Language The expressions and types of i are defined as follows: e : f x e e e x oe :f (expressions) f : x oe :e ff:e (functions) b ....

Girard, J.-Y. Une extension de l'interpr'etation de Godel `a l'analyse, et son application `a l"elimination des coupures dans l'analyse et la th'eorie des types. In Proceedings of the 2nd Scandinavian Logic Symposium (1971), J. E. Fenstad, Ed., North--Holland, pp. 63--92.

....83] and some of the more recent object oriented database languages such as Gemstone [CM84] EXODUS [CDJS86] and Trellis Owl [OBS86] where, if it is at all possible to write polymorphic code, some dynamic type checking is required. Napier [MBCD89] attempts to combine parametric polymorphism [Rey74, Gir71] and persistence, but its polymorphism does not extend to operations on records and other database structures. The current practice in database programming is to use a query language embedded in a host language. In this arrangement, communication between programs in different languages is so ....

.... is a type representing the set of types that can be obtained by substituting its type variables with some types (such as int, bool or int int) This type can be understood as a representation of a polymorphic type of the form 10 8t:t t in the second order polymorphic lambda calculus [Rey74, Gir71] The most important property of the ML type system is that for any type consistent expression it infers a principal type. This is a type such that all its instances are types of the expression and conversely any type of the expression is its instance. This means that the type system infers a ....

J.-Y. Girard. Une extension de l'interpretation de godel `a l'analyse, et son application `a l"elimination des coupures dans l'analyse et th'eorie des types. In Second Scandinavian Logic Symposium. North-Holland, 1971.

....polymorphism. The type systems of most wide spread (typed) functional programming languages (Haskell, Hope, Miranda 1 , SML) are based on Milner s polymorphic types [11] for functional programs. This polymorphism is a proper subsystem of the type system of second order typed calculus of Girard [3] and Reynolds [13] An important difference is that ML types can be inferred from the program, while type inference for second order typed calculus is not decidable 2 . The idea of polymorphism is to avoid writing the same function several times for types that are different but have the same ....

J. Y. Girard. Une extension de l'interpr'etation de godel a l'analyse, et son application `a l"elimination des coupures dans l'analyse et la th'eorie des types. In J. E. Fenstad, editor, Proceedings of the Second Scandinavian Logic Symposium. North Holland, 1971.

....also derivable in F. Proof The proof corresponds exactly to Kolmogorov s double negation embedding for second order propositional logic. Corollary 13 If all terms in F are weakly normalisable then they are strongly normalisable. Proof Parallel to the proof of Corollary 11 It is first shown by Girard (1971) that all terms in F are weakly normalisable. Though it is proven later that all terms in F are strongly normalisable by Prawitz (1971) the formulation of 6 reducibility candidates gets complicated accordingly. This comparison can also be done with a sharp weak normalisation proof for the theory ....

Girard, J.-Y., (1971) Une extension de L'interpretation de Godel `a l'analyse, et son application `a l"elimination des coupures dans l'analyse et la th'eorie des types, Proceedings of the 2nd Scandinavian logic symposium, editor J.E. Fenstad, North-Holland Publishing Company, Amsterdam.

....is best captured by universal quantification. For example, the polymorphic identity x : x does not care what the type of its argument is. Its generic type is therefore 8ff : ff ff. The contemporary study of parametric polymorphism is based on Girard Reynold s polymorphic second order calculus [Gir71, Gir72, Rey74, GLT89]. The inheritance (or vertical) polymorphism is based on the idea of hierarchical data organization where each new descendant type possesses all the properties of its ancestor types. Therefore, a function applicable to an ancestor type should successfully work on all its offsprings, i.e. ....

J.-Y. Girard. Une extension de l'interpretation fonctionnelle de Godel `a l'analyse et son application `a l"elimination des coupures dans l'analyse et la th'eorie des types. In 2nd Scandinavian Logic Symposium, pages 63--92. North-Holland Publishing Company, 1971.

....side, called the set of accessible subterms, by the use of computability preserving operations. Here, computability refers to Tait s computability predicate method for proving the termination of the simply typed calculus [43] which was later extended by Girard to the polymorphic calculus [22, 24]. To explain our construction, we need to recall the basics of Tait s method. The basic observation is that it is not possible to prove the termination of fi reduction directly by induction on the structure of terms because of the application case: in the untyped calculus, the term (x:xx x:xx) ....

J.-Y. Girard. Une extension de l'interpr'etation de Godel `a l'analyse, et son application `a l"elimination des coupures dans l'analyse et la th'eorie des types. In J. E. Fenstad, editor, Proceedings of the Second Scandinavian Logic Symposium, volume 63 of Studies in Logic and the Foundations of Mathematics. North-Holland, 1971.

....languages with products is easy. 6] gives a complete axiomatization of the equations in the simply typed calculus with coproducts. Extending our results to this calculus is likely to be challenging. Likewise, we believe that results along our lines for type systems with second order polymorphism [8, 14] would be interesting and challenging as well. Acknowledgements Riecke was partially supported by NSF grant number CCR 8912778 and CCR 90 57570, NRL grant number N00014 91 J 2022, NOSC grant number 19 920123 31, and ONR Grant N00014 88 K 0634 during a stay at the University of ....

J.-Y. Girard. Une extension de l'interpr'etation de Godel `a l'analyse, et son application `a l"elimination des coupures dans l'analyse et la th'eorie des types. In J. E. Fenstad, editor, Proceedings of the Second Scandinavian Logic Symposium, volume 63 of Studies in Logic and the Foundations of Mathematics, pages 63--92. North-Holland, 1971.

....polymorphism. The type systems of most wide spread (typed) functional programming languages (Haskell, Hope, Miranda 1 , SML) are based on Milner s polymorphic types [10] for functional programs. This polymorphism is a proper subsystem of the type system of second order typed calculus of Girard [3] and Reynolds [12] An important difference is that ML types can be inferred from the program, while type inference for second order typed calculus is not decidable 2 . The idea of polymorphism is to avoid writing the same function several times for types that are different but have the same ....

J. Y. Girard. Une extension de l'interpr'etation de godel a l'analyse, et son application `a l"elimination des coupures dans l'analyse et la th'eorie des types. In J. E. Fenstad, editor, Proceedings of the Second Scandinavian Logic Symposium. North Holland, 1971.

....form a simply typed calculus, it is a routine matter to show that equality of well formed constructors (and, consequently, types) in ML is decidable. It is then easy to show that type checking in ML is decidable. This is a well known property of the polymorphic lambda calculus F (c.f. Gir71, Gir72, Rey74, BMM89] which may be seen as an impredicative extension of the ML calculus. Lemma 2.2 There is a straightforward one pass algorithm which decides, for an arbitrary well formed theory T and formation judgement F , whether or not ML [T ] F . The main technical ....

J.-Y. Girard. Une extension de l'interpretation de Godel `a l'analyse, et son application `a l"elimination des coupures dans l'analyse et la th'eorie des types. In J.E. Fenstad, editor, 2nd Scandinavian Logic Symposium, pages 63--92. NorthHolland, 1971.

....behaves uniformly on any record containing an l field. An important achievement in type theory of programming languages is the development of polymorphic type systems where generic behavior of a program such as above is cleanly represented by a polymorphic type. In Girard Reynolds type system [7, 31], the function id is given the following type id : 8t:t t representing the polymorphic behavior of id. This form of polymorphism is embodied in the type system of ML [19] Its practical usefulness has been widely recognized and an ML style 3 A preliminary version of this article was ....

J.-Y. Girard. Une extension de l'interpretation de godel `a l'analyse, et son application `a l"elimination des coupures dans l'analyse et th'eorie des types. In Second Scandinavian Logic Symposium. North-Holland, 1971.

....fixed type to the elements of the group. This problem is usually solved by using typical ambiguity i.e. more or less type variables. However, in some situations, type theory is better suited than set theory. For instance, proofs as objects interpretations are always developed in typed languages [8, 34, 21, 22, 12]. Also when one is interested in expressing computations as rewrite rules on the expressions of the language, type theories seem better suited than set theories [34, 21, 22, 12] At last automated theorem proving methods have been developed for type theory [3, 28, 29] and very rarely for set ....

....better suited than set theory. For instance, proofs as objects interpretations are always developed in typed languages [8, 34, 21, 22, 12] Also when one is interested in expressing computations as rewrite rules on the expressions of the language, type theories seem better suited than set theories [34, 21, 22, 12]. At last automated theorem proving methods have been developed for type theory [3, 28, 29] and very rarely for set theory. Several reasons explain these success of type theory. 1) Type theory provides an explicit notation for objects ( calculus) while set theory merely provides axioms expressing ....

[Article contains additional citation context not shown here]

J.Y. Girard, Une extension de l'interpr'etation de Godel `a l'analyse et son application `a l"elimination des coupures dans l'analyse et la th'eorie des types, Proceedings of the 2 nd Scandinavian Logic Symposium, Fenstad (ed.), North-Holland, Amsterdam (1970).

....pay for the convenience of type inference. For example, it is often enough to be able to pass particular monomorphic instances of polymorphic values to and from functions, rather than the polymorphic values themselves. On the other hand, comparisons with explicitly typed languages, like System F [4, 29], that do support first class polymorphism, reveal significant differences in expressiveness. We will see a number of practical examples in later sections that use first class polymorphism in essential ways, but cannot be coded in a standard Hindley Milner type system. 1.1 This paper In this ....

....Stack xs ( tail head null push : a Stack a Stack a push x (Stack self push pop top empty) Stack (push x self) push pop top empty top : Stack a a top (Stack self push pop top empty) top self testExpr : Int] testExpr = map (top . push 1) makeListStack [1,2,3] makeListStack [4,5]] Figure 5: A simple encoding of stack packages tion, hidden by an existential quantifier. Note that our prototype implementation adopts the convention that variables beginning with an x are existentially quantified, avoiding the need for an explicit quantifier 2 . Our final example, in Figure ....

[Article contains additional citation context not shown here]

J.-Y. Girard. Une extension de l'interpr'etation de Godel `a l'analyse et son application `a l"elimination des coupures dans l'analyse et la th'eorie de types. In Fenstad, editor, Proceedings of the Scandanavian logic symposium. North Holland, 1971.

....Intuitively, each element of an existential type 9t:oe consists of a type and an element of [ t]oe . Using products to combine s , first , and rest , an implementation of stream would have type 9t: t Theta(t nat) Theta(t t) for example. Existential types were part of Girard s System F [Gir71, Gir72] but not linked to abstract data types) When we write abstract data type implementations apart from the declarations that use them, it is necessary to include some type information which might at first appear redundant. We will write an implementation in the form ht = M : oei , where ....

J.-Y. Girard. Une extension de l'interpretation de Godel `a l'analyse, et son application `a l"elimination des coupures dans l'analyse et la th'eorie des types. In J.E. Fenstad, editor, 2nd Scandinavian Logic Symposium, pages 63--92. North-Holland, Amsterdam, 1971.

....to LOOM providing support for virtual types. In section 5 we explain the semantics of this extension as a generalization of that given for MyType in LOOM. Finally we conclude. We presume the reader is familiar with the representation of type parameters in the polymorphic lambda calculus [Gir71, Rey74] and with existential types [MP88] Mitchell [Mit90] and [Mit96] are good references for these topics. 2 A brief introduction to LOOM and MyType In this section we provide a very brief overview of our object oriented language, LOOM. The reader is referred to [BFP97] for details. LOOM is ....

J.-Y. Girard. Une extension de l'interpretation de Godel `a l'analyse, et son application `a l"elimination des coupures dans l'analyse et la th'eorie des types. In J.E. Fenstad, editor, 2nd Scandinavian Logic Symposium, pages 63--92. North-Holland, 1971.

....each element of an existential type 9t:oe consists of a type and an element of [ t]oe . Using products to combine s , first , and rest , an implementation of stream would have type 9t: t Theta(t nat ) Theta(t t) for example. Existential types were part of Girard s System F [Gir71, Gir72] but not linked to abstract data types) When we write abstract data type implementations apart from the declarations that use them, it is necessary to include some type information which might at first appear redundant. We will write an implementation in the form ht = M : oei , where ....

J.-Y. Girard. Une extension de l'interpretation de Godel `a l'analyse, et son application `a l"elimination des coupures dans l'analyse et la th'eorie des types. In J.E. Fenstad, editor, 2nd Scandinavian Logic Symposium, pages 63--92. North-Holland, Amsterdam, 1971.

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Jean-Yves Girard. Une extension de l'interpr'etation de Godel `a l'analyse, et son application `a l"elimination des coupures dans l'analyse et la th'eorie des types. In J. E. Fenstad, editor, Proceedings of the Second Scanindavian Logic Symposium, Studies in Logic and the Foundations of Mathematics, pages 63--92. North-Holland, 1971.

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J.-Y. Girard. Une extension de l'interpretation de Godel `a l'analyse, et son application `a l"elimination des coupures dans l'analyse et la th'eorie des types. In J.E. Fenstad, editor, 2nd Scandinavian Logic Symposium, pages 63--92. NorthHolland, 1971.

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J.-Y. Girard. Une extension de l'interpr'etation de Godel `a l'analyse, et son application `a l"elimination des coupures dans l'analyse et la th'eorie des types. In Proceedings of the 2nd Scandinavian Logic Symposium. North-Holland, 1970. 205

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J.-Y. Girard. Une extension de l'interpr'etation de Godel `a l'analyse, et son application `a l"elimination des coupures dans l'analyse et la t'eorie des types. In Proceedings of the 2nd Scandinavian Logic Symposium. North-Holland, 1970.

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