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Types, Abstraction, and Parametric Polymorphism, Part 2
, 1991
"... The concept of relations over sets is generalized to relations over an arbitrary category, and used to investigate the abstraction (or logicalrelations) theorem, the identity extension lemma, and parametric polymorphism, for Cartesianclosedcategory models of the simply typed lambda calculus and P ..."
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Cited by 53 (1 self)
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The concept of relations over sets is generalized to relations over an arbitrary category, and used to investigate the abstraction (or logicalrelations) theorem, the identity extension lemma, and parametric polymorphism, for Cartesianclosedcategory models of the simply typed lambda calculus and PLcategory models of the polymorphic typed lambda calculus. Treatments of Kripke relations and of complete relations on domains are included.
On functors expressible in the polymorphic typed lambda calculus
 Logical Foundations of Functional Programming
, 1990
"... This is a preprint of a paper that has been submitted to Information and Computation. ..."
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This is a preprint of a paper that has been submitted to Information and Computation.
An Introduction to Polymorphic Lambda Calculus
 Logical Foundations of Functional Programming
, 1994
"... Introduction to the Polymorphic Lambda Calculus John C. Reynolds Carnegie Mellon University December 23, 1994 The polymorphic (or secondorder) typed lambda calculus was invented by JeanYves Girard in 1971 [11, 10], and independently reinvented by myself in 1974 [24]. It is extraordinary that ..."
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Introduction to the Polymorphic Lambda Calculus John C. Reynolds Carnegie Mellon University December 23, 1994 The polymorphic (or secondorder) typed lambda calculus was invented by JeanYves Girard in 1971 [11, 10], and independently reinvented by myself in 1974 [24]. It is extraordinary that essentially the same programming language was formulated independently by the two of us, especially since we were led to the language by entirely different motivations. In my own case, I was seeking to extend conventional typed programming languages to permit the definition of "polymorphic" procedures that could accept arguments of a variety of types. I started with the ordinary typed lambda calculus and added the ability to pass types as parameters (an idea that was "in the air" at the time, e.g. [4]). For example, as in the ordinary typed lambda calculus one can write f int!int : x int : f(f (x)) to denote the "doubling" function for the type int, which accepts a function from integers
A Full Continuous Model of Polymorphism
"... Abstract. We introduce a model of the secondorder lambda calculus. Such a model is a Scott domain whose elements are themselves Scott domains, and in it polymorphic maps are interpreted by generic continous maps. Keywords: Secondorder lambda calculus, model, Scott domain, nonparametric. 1 ..."
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Abstract. We introduce a model of the secondorder lambda calculus. Such a model is a Scott domain whose elements are themselves Scott domains, and in it polymorphic maps are interpreted by generic continous maps. Keywords: Secondorder lambda calculus, model, Scott domain, nonparametric. 1
A New Model Construction for the Polymorphic Lambda Calculus
"... Various models for the GirardReynolds secondorder lambda calculus have been presented in the literature. Except the term model they are either realizability or domain models. In this paper a further model construction is introduced. Types are interpreted as inverse limits of #cochains of finite s ..."
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Various models for the GirardReynolds secondorder lambda calculus have been presented in the literature. Except the term model they are either realizability or domain models. In this paper a further model construction is introduced. Types are interpreted as inverse limits of #cochains of finite sets. The corresponding morphisms are sequences of maps acting locally on the finte sets in the omegacochains. The model can easily be turned into an effectively given one. Moreover, it can be arranged in such a way that the universal type (ForAll t.t) representing absurdity in the higherorder logic defined by the type structure is interpreted by the empty set, which means that it is also a model of this logic.
This is a preprint of a paper that has been submitted to Information and Computation. On Functors Expressible in the Polymorphic Typed Lambda Calculus
, 1991
"... Given a model of the polymorphic typed lambda calculus based upon a Cartesian closed category K, there will be functors from K to K whose action on objects can be expressed by type expressions and whose action on morphisms can be expressed by ordinary expressions. We show that if T is such a functor ..."
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Given a model of the polymorphic typed lambda calculus based upon a Cartesian closed category K, there will be functors from K to K whose action on objects can be expressed by type expressions and whose action on morphisms can be expressed by ordinary expressions. We show that if T is such a functor then there is a weak initial Talgebra and if, in addition, K possesses equalizers of all subsets of its morphism sets, then there is an initial Talgebra. These results are used to establish the impossibility of certain models, including those in which types denote sets and S → S ′ denotes the set of all functions from S to S ′.