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Notes on Sconing and Relators
, 1993
"... This paper describes a semantics of typed lambda calculi based on relations. The main mathematical tool is a categorytheoretic method of sconing, also called glueing or Freyd covers. Its correspondence to logical relations is also examined. 1 Introduction Many modern programming languages feature ..."
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This paper describes a semantics of typed lambda calculi based on relations. The main mathematical tool is a categorytheoretic method of sconing, also called glueing or Freyd covers. Its correspondence to logical relations is also examined. 1 Introduction Many modern programming languages feature rather sophisticated typing mechanisms. In particular, languages such as ML include polymorphic data types, which allow considerable programming flexibility. Several notions of polymorphism were introduced into computer science by Strachey [Str67], among them the important notion of parametric polymorphism. Strachey's intuitive definition is that a polymorphic function is parametric if it has a uniformly given algorithm in all types, that is, if the function's behavior is independent of the type at which the function is instantiated. Reynolds [Rey83] proposed a mathematical definition of parametric polymorphic functions by means of invariance with respect to certain relations induced by typ...
Reflexive Graphs and Parametric Polymorphism
, 1993
"... this paper is to understand why that is a parametric categorical model. In [10] Ma and Reynolds propose a parametricity hypothesis for a functor between categorical models of polymorphism which essentially requires that there is an extension of (a certain form of) an identity relation functor which ..."
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this paper is to understand why that is a parametric categorical model. In [10] Ma and Reynolds propose a parametricity hypothesis for a functor between categorical models of polymorphism which essentially requires that there is an extension of (a certain form of) an identity relation functor which preserve the model structure. There is no mention in the paper of any case when the parametricity hypothesis is satified, nor if there is a canonical completion of a category to one which satisfies the hypothesis. We shall suggest how the construction of a PLcategory of relations on a given category presented in [10] can be viewed as a "parametric completion". We shall also follow the suggestion of Ma in [9] that subtyping is a kind of parametricity requirement and show how to fit subtyping in the same setup. The basic idea is to use reflexive graphs of categories as in [12]. We shall employ their construction to present a kind of parametric completion of a given category. We also give a different presentation of the RELconstruction in [10], and use it to discuss some examples. We show in particular that the RELconstruction acts (essentially) in the same way on a category and on its completion. Hence it follows that the identity functor on the completion satisfies the parametricity hypothesis. Discussions with Eugenio Moggi, Peter O'Hearn, Edmund Robinson, and Thomas Streicher were very useful. Paul Taylor's beutiful diagram macros were used for typesetting all the diagrams in the text. 1 Graphs of categories
Parametric and TypeDependent Polymorphism
, 1995
"... Data Types, though, as Reynolds stresses, is not perfectly suited for higher type or higher order systems and, thus, he proposes a "relational" treatment of invariance: computations do not depend on types in the sense that they are "invariant" w.r.t. arbitrary relations on types and between types. R ..."
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Cited by 10 (5 self)
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Data Types, though, as Reynolds stresses, is not perfectly suited for higher type or higher order systems and, thus, he proposes a "relational" treatment of invariance: computations do not depend on types in the sense that they are "invariant" w.r.t. arbitrary relations on types and between types. Reynolds's approach set the basis for most of the current work on parametricity, as we will review below (.3). Some twelve years earlier, Girard had given just a simple hint towards another understanding of the properties of "computing with types". In [Gir71], it is shown, as a side remark, that, given a type A, if one defines a term J A such that, for any type B, J A B reduces to 1, if A = B, and reduces to 0, if A ยน B, then F + J A does not normalize. In particular, then, J A is not definable in F. This remark on how terms may depend on types is inspired by a view of types which is quite different from Reynolds's. System F was born as the theory of proofs of second order intuitionis...
Using synthetic domain theory to prove operational properties of a polymorphic programming language based on strictness
 Manuscript
"... We present a simple and workable axiomatization of domain theory within intuitionistic set theory, in which predomains are (special) sets, and domains are algebras for a simple equational theory. We use the axioms to construct a relationally parametric settheoretic model for a compact but powerful ..."
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We present a simple and workable axiomatization of domain theory within intuitionistic set theory, in which predomains are (special) sets, and domains are algebras for a simple equational theory. We use the axioms to construct a relationally parametric settheoretic model for a compact but powerful polymorphic programming language, given by a novel extension of intuitionistic linear type theory based on strictness. By applying the model, we establish the fundamental operational properties of the language. 1.
Parametricity as Isomorphism
 Theoretical Computer Science
, 1993
"... . We investigate a simple form of parametricity, based on adding "abstract" copies of preexisting types. Connections are made with the ReynoldsMa theory of parametricity by logical relations, with the theory of parametricity via dinaturality, and with the categorical notion of equivalence. Introdu ..."
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. We investigate a simple form of parametricity, based on adding "abstract" copies of preexisting types. Connections are made with the ReynoldsMa theory of parametricity by logical relations, with the theory of parametricity via dinaturality, and with the categorical notion of equivalence. Introduction In his fundamental paper on the notion of parametricity in connection with type theories [Rey83], John Reynolds links the notion of parametricity firmly to the notion of data abstraction. This, unlike Strachey's earlier characterization via algorithm reuse, is a needdriven analysis. We need things to be parametric because otherwise our data abstractions will no longer be abstract. In his subsequent paper with Ma [MR91], two further points are made. One is that the problems reside more at the level of parametrized types than at the level of the quantified polymorphic types, and the other is that the notion of parametricity is not absolute, but relative. The MaReynolds work produces ...