J.-Y. Girard. The system F of variable types: Fifteen years later. Theoretical Computer Science, 45:159--192, 1986.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....properties for dependent sums and products. It was clear that a conceptual analysis of totality was needed in order to at least motivate the seemingly ad hoc properties one had to asssume to make the inductive proofs work. Lill Kristiansen [7] considered an alternative situation. Girard [5] had used qualitative domains in order to give a semantics for system F introduced by himself [4] fifteen years earlier. In the Girard implementation each closed second order type is interpreted as a qualitative domain X with a set X of total objects. Each term of the type will be interpreted as a ....

Girard, J.-Y. The system F of variable types fifteen years later, Theor. Comp. Science 45 (1986) 159-192

....In order for types to be values, they need to belong to a type. We might try adding a single new type, say Type, for this purpose. But then it must belong to a type. We might try attaining closure by postulating. Type 2 Type. But this causes many troubles. It allows us to inhabit every type [21] and permits nonterminating functions in the function spaces [28] So following the traditions of predicative type theory [45, 34, 19, 20] we introduce a hierarchy of types. Following [35] we call them universes. They are denoted U 1 ; U 2 ; and we have that U i 2 U i 1 and if A 2 U i ....

J.-Y. Girard. The system F of variable types: Fifteen years later. Theoretical Computer Science, 45:159--192, 1986.

....B and that A b is non empty for all b 2 B. If E is a set, we note P fin (E) the set of its finite and non empty subsets. We write x fin E when x is a finite and non empty subset of E. 1. 2 About qualitative domains and stable functions We briefly recall the main definitions and results of [G1]. For the reader acquainted with domain theory, let us first say that a qualitative domain is a dI domain where all primes are atoms. We give here a direct definition, borrowed from [G1] Definition 1.1 A qualitative domain is a set E satisfying the following conditions 1 : ffl ; 2 E. ffl If ....

....E. 1.2 About qualitative domains and stable functions We briefly recall the main definitions and results of [G1] For the reader acquainted with domain theory, let us first say that a qualitative domain is a dI domain where all primes are atoms. We give here a direct definition, borrowed from [G1]. Definition 1.1 A qualitative domain is a set E satisfying the following conditions 1 : ffl ; 2 E. ffl If x 2 E and y x then y 2 E. ffl For any directed subset D of E, one has S D 2 E. If E is a qualitative domain, its web jEj is defined by jEj = fa j fag 2 Eg : This set is usually ....

[Article contains additional citation context not shown here]

J-Y. Girard. The system F of variable types fifteen years later. Theoretical Computer Science 45, 1986.

....are functions that are not sequential (typically the so called Berry function) and so, even if it has nice mathematical properties, the stable model is not very satisfactory with respect to the modelization of determinism. It should be noticed that stability has also been used by Girard (see [G1]) to model system F. He used a very simple kind of domains (qualitative domains) and he also observed that a subclass of these domains (coherence spaces) has very good properties with respect to stability. Hypercoherences: a strongly stable model of linear logic This work gave rise to the first ....

....C. If E is a set, we use P fin (E) to denote the set of its finite and non empty subsets. We write x fin E when x is a finite and non empty subset of E. The theory of hypercoherences is closely related to that of qualitative domains, so let us recall some basic definitions and results from [G1]. Definition 2 A qualitative domain is a pair (jQj ; Q) where jQj is a set (the web) and Q is a subset of P (jQj) satisfying the following conditions: ffl ; 2 Q and, if a 2 jQj, then fag 2 Q. ffl if x 2 Q and if y x then y 2 Q. ffl if D Q is directed with respect to inclusion, then S D 2 ....

[Article contains additional citation context not shown here]

J.-Y. Girard. The system F of variable types fifteen years later. Theor. Comp. Sci 45, 1986.

....incomparable. Stability captures sequentiality to some extend, e.g. POR is not a stable function, yet the stable model of PCF fails to be fully abstract. Stable functions also have applications in modelling concurrency [WN94] and in coherence space semantics of polymorphic programming languages [Gir86] and linear logic [Gir87] To model a typed programming language a cartesian closed category is needed. The classical examples are Scott domains and SFP domains for the continuous case and dI domains for the stable case. Whereas the cartesian closed categories with continuous maps have been ....

J.-Y. Girard. The system F of variable types: Fifteen years later. Theoretical Computer Science, 45:159--192, 1986.

.... cardinality problems in finding solutions to domain equations such as D = D Gamma D] A much more recent development makes equally impressive use of continuity to give a finitary semantics for impredicative polymorphism, as in the second order lambda calculus (Girard s System F ) [Girard, 1986, Coquand et al. 1987, Coquand, 1989] This semantics makes essential use of the functorial aspects of Domain Theory. There have also been semantics for implicit polymorphism based on ideals [MacQueen et al. 1986] and partial equivalence relations [Abadi and Plotkin, 1990] over domains. We refer ....

....first step in an extensive development of Stable Domain Theory in which stable functions under the stable ordering take the place which continuous functions play in standard Domain Theory. Stable Domain theory was introduced by Berry [Berry, 1978, Berry, 1979] Some more recent references are [Girard, 1986, Coquand et al. 1987, Taylor, 1990, Ehrhard, 1993] Berry s motivation in introducing stable functions was actually to try to capture the notion of sequentially computable function at higher types. For the theory of sequential functions on concrete domains, we refer to [Kahn and Plotkin, 1993, ....

J.-Y. Girard. The system f of variable types: Fifteen years later. Theoretical Computer Science, 45:159--192, 1986.

....element a is called compact if a W X implies a x for some x 2 X for any directed X. A subset K ae D is called a basis if every element of D is a supremum of some elements from K [3] 2 It is not hard to show that this definition is equivalent to the usual definition of a qualitative domain [7]. that there are a countable number of values and a special symbol corresponding to incomplete information. The ordering of V is given by letting be less than any other value. The relations are finite sets of records, i.e. finite subsets of L V . However, not every finite subset of L ....

J.-Y. Girard. The system F of variable types : fifteen years later. Theoretical Computer Science 45:159--192, 1986.

.... circumvent cardinality problems in finding solutions to domain equations such as D = D 0 D] A much more recent development makes equally impressive use of continuity to give a finitary semantics for impredicative polymorphism, as in the second order lambda calculus (Girard s System F ) [Girard, 1986, Coquand et al. 1987, Coquand, 1989] This semantics makes essential use of the functorial aspects of Domain Theory. There have also been semantics for implicit polymorphism based on ideals [MacQueen et al. 1986] and partial equivalence relations [Abadi and Plotkin, 1990] over domains. We refer ....

....first step in an extensive development of Stable Domain Theory in which stable functions under the stable ordering take the place which continuous functions play in standard Domain Theory. Stable Domain theory was introduced by Berry [Berry, 1978, Berry, 1979] Some more recent references are [Girard, 1986, Coquand et al. 1987, Taylor, 1990, Ehrhard, 1993] Berry s motivation in introducing stable functions was actually to try to capture the notion of sequentially computable function at higher types. For the theory of sequential functions on concrete domains, we refer to [Kahn and Plotkin, 1993, ....

J.-Y. Girard. The system f of variable types: Fifteen years later. Theoretical Computer Science, 45:159--192, 1986.

....attack on impredicativity uses the technology originally developed for solving recursive domain equations. The earliest such model [McC79] interprets types as operators on a suitable universal domain. An alternative, that avoids universal domains, is the model construction given by Girard [Gir86] using ideas from stable domain theory; it was later adapted by Coquand, Gunter and Winskel [CGW89] to a more traditional cpo setting. The construction works by interpreting types as continuous functors of a certain kind, allowing a type to be constructed out of finite approximations. This ....

J.-Y. Girard. The system F of variable types: Fifteen years later. Theoretical Computer Science, 45:159--192, 1986.

....almost all of the categories of domains that have been proposed as a possible foundation for the semantics of programming languages have been (essentially equivalent to) dcpo s which satisfy the consistent completeness condition. This includes those categories which use stable continuous functions [Ber78, Gir86] as well as categories related to the Scott domains (such as the continuous lattices) 1 The one noteworthy exception is the category of bifinite domains which was introduced by Plotkin [Plo76] where it is called SFP) These will be discussed below. One might apply the following line of ....

J. Y. Girard. The system F of variable types: fifteen years later. Theoretical Computer Science, 45:159--192, 1986.

.... Theta 2 y. Thus, p A 1 (z) x; p A 2 (z) y and z x; y. Therefore, x y exists, and by Proposition 19, D A 1 Theta A 2 . Theorem 21 is proved. 2 A simpler result can be stated for qualitative domains. Recall that a domain D is called qualitative iff #x is a Boolean lattice for every x 2 D [10]. Since all congruences are permutable in a Boolean lattice, we have: Corollary 23 General decompositions of a qualitative domain are given by pairs of congruences ( Theta 1 ; Theta 2 ) such that Theta x 1 and Theta x 2 are complementary elements in Con(x) for every x 2 D max . 6 ....

J.-Y. Girard. The system F of variable types: Fifteen years later. Theoretical Computer Science, 45:159--192, 1986.

....and unicity of the representation of data have no equivalent in other systems. Moreover, the class of data types we consider is very large with some unusual ones. 1 Introduction Since the work of Curry, a lot of type systems have been created (e.g. De Bruijn s Automath [4] Girard s system F [5]; Martin Lof s type theory [10] Coquand Huet s Calculus of construction [3] etc) One of their purposes is program extraction via the CurryHoward isomorphism [6] which establishes a correspondence between programs and proofs of specifications. One of these systems is AF 2 (second order ....

J-Y. Girard. The system F of variable types: fifteen years later. Theoretical Computer Science, 45:159--192, 1986.

....calculus with recursive types. As we mentioned above, this translation interprets inheritances in Fun as coercion functions already definable in the extended polymorphic lambda calculus. Then, we can use existing techniques for modeling polymorphism and recursion (such as those described in [ABL86, Gir86, CGW87, CGW89]) to interpret the extended polymorphic lambda calculus, thus providing models for the original language with inheritance. This method achieves simultaneous modeling of parametric polymorphism, recursive types, and inheritance. In the process, the paradigm inheritance as definable coercion ....

.... call TARGET, is an extension with record and recursive types of the Girard Reynolds polymorphic lambda calculus (see [CGW87] for the semantics of TARGET) Therefore, SOURCE extends with inheritance and bounded generics TARGET, which is at its turn an extension of what Girard calls System F in [Gir86]. Our translation takes derivations of inheritance and typing judgements in SOURCE into derivations of typing judgements in TARGET. We translate the inheritance judgements of SOURCE into definable terms of TARGET which can be thought of as canonical explicit coercions. Bounded generics translate ....

[Article contains additional citation context not shown here]

J. Y. Girard. The system F of variable types: fifteen years later. Theoretical Computer Science, 45:159--192, 1986.

....such that the previous diagram commutes for call by name compilation. If the reader is not familiar with second order stuff and system AF 2 , he could read this paper forgetting all about first order to use only its propositional part (This restriction of system AF 2 gives the Curry s system F [3]) But doing this, the reader will lose the argument about the correctness of programs in Section 6, because the type characterizes the function in system AF 2 but not in system F. 2 The deduction system. We use classical second order formulas. First we define first order terms from a language ....

J.-Y. Girard. The system F of variable types: fifteen years later. Theoretical Computer Science, 45:159--192, 1986.

....in which types are coded up as particular kinds of retracts. The latter are models for stronger calculi with a type of types and so are not tailored directly to the requirements of polymorphic calculus and do not in themselves suggest a general definition of model for the calculus. In his paper [Gir86], Girard produced an interesting new model in which types of the polymorphic calculus are represented as certain kinds of objects called qualitative domains, work which was extended in [CGW87] The category of domains used in [Gir86] and [CGW87] is not the usual one taken in denotational ....

....a general definition of model for the calculus. In his paper [Gir86] Girard produced an interesting new model in which types of the polymorphic calculus are represented as certain kinds of objects called qualitative domains, work which was extended in [CGW87] The category of domains used in [Gir86] and [CGW87] is not the usual one taken in denotational semantics in particular the morphisms are functions which are stable in the sense of Berry and not just Scott continuous. The work left open the question of whether or not a model similar to Girard s could be found in the more traditional ....

[Article contains additional citation context not shown here]

J. Y. Girard. The system F of variable types: fifteen years later. Theoretical Computer Science, 45:159--192, 1986.

No context found.

J.-Y. Girard. The system F of variable types: Fifteen years later. Theoretical Computer Science, 45:159--192, 1986.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC