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Categorical and domain theoretic models of parametric polymorphism
, 2005
"... We present a domaintheoretic model of parametric polymorphism based on admissible per’s over a domaintheoretic model of the untyped lambda calculus. The model is shown to be a model of Abadi & Plotkin’s logic for parametricity, by the construction of an LAPLstructure as defined by the authors ..."
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We present a domaintheoretic model of parametric polymorphism based on admissible per’s over a domaintheoretic model of the untyped lambda calculus. The model is shown to be a model of Abadi & Plotkin’s logic for parametricity, by the construction of an LAPLstructure as defined by the authors in [7, 5]. This construction gives formal proof of solutions to a large class of recursive domain equations, which we explicate. As an example of a computation in the model, we explicitly describe the natural numbers object obtained using parametricity. The theory of admissible per’s can be considered a domain theory for (impredicative) polymorphism. By studying various categories of admissible and chain complete per’s and their relations, we discover a picture very similar to that of domain theory. 1
LINEAR ABADI & PLOTKIN LOGIC
, 2006
"... ABSTRACT. We present a formalization of a version of Abadi and Plotkin’s logic for parametricity for a polymorphic dual intuitionistic / linear type theory with fixed points, and show, following Plotkin’s suggestions, that it can be used to define a wide collection of types, including existential ty ..."
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ABSTRACT. We present a formalization of a version of Abadi and Plotkin’s logic for parametricity for a polymorphic dual intuitionistic / linear type theory with fixed points, and show, following Plotkin’s suggestions, that it can be used to define a wide collection of types, including existential types, inductive types, coinductive types and general recursive types. We show that the recursive types satisfy a universal property called dinaturality, and we develop reasoning principles for the constructed types. In the case of recursive types, the reasoning principle is a mixed induction / coinduction principle, with the curious property that coinduction holds for general relations, but induction only for a limited collection of “admissible ” relations. A similar property was observed in Pitts analysis of recursive types in domain theory [Pit95]. In a future paper we will develop a category theoretic notion of models of the logic presented here, and show how the results developed in the logic can be transferred to the models.
Copyright c © 2006, L. Birkedal
, 1600
"... is permitted for educational or research use on condition that this copyright notice is included in any copy. ..."
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is permitted for educational or research use on condition that this copyright notice is included in any copy.
Web www.itu.dk Parametric Completion for Models of Polymorphic Linear / Intuitionistic Lambda Calculus
, 1600
"... Reproduction of all or part of this work is permitted for educational or research use on condition that this copyright notice is included in any copy. ISSN 1600–6100 ..."
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Reproduction of all or part of this work is permitted for educational or research use on condition that this copyright notice is included in any copy. ISSN 1600–6100
Web www.itu.dk Parametric Domaintheoretic models of Linear Abadi & Plotkin Logic
, 1600
"... is permitted for educational or research use on condition that this copyright notice is included in any copy. ISSN 1600–6100 ISBN 8779490867 ..."
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is permitted for educational or research use on condition that this copyright notice is included in any copy. ISSN 1600–6100 ISBN 8779490867
MFPS XX1 Preliminary Version Parametric Domaintheoretic Models of Polymorphic Intuitionistic / Linear Lambda Calculus
"... We present a formalization of a version of Abadi and Plotkin’s logic for parametricity for a polymorphic dual intuitionistic / linear type theory with fixed points, and show, following Plotkin’s suggestions, that it can be used to define a wide collection of types, including solutions to recursive d ..."
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We present a formalization of a version of Abadi and Plotkin’s logic for parametricity for a polymorphic dual intuitionistic / linear type theory with fixed points, and show, following Plotkin’s suggestions, that it can be used to define a wide collection of types, including solutions to recursive domain equations. We further define a notion of parametric LAPLstructure and prove that it provides a sound and complete class of models for the logic, and conclude that such models have solutions for a wide class of recursive domain equations. Finally, we present a concrete parametric LAPLstructure based on suitable categories of partial equivalence relations over a universal model of the untyped lambda calculus. Key words: Parametric polymorphism, Categorical semantics, domain theory
MFPS XX1 Preliminary Version Synthetic domain theory and models of Linear
"... In a recent article [4] the first two authors and R.L. Petersen have defined a notion of parametric LAPLstructure. Such structures are parametric models of the equational theory PILLY, a polymorphic intuitionistic / linear type theory with fixed points, in which one can reason using parametricity a ..."
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In a recent article [4] the first two authors and R.L. Petersen have defined a notion of parametric LAPLstructure. Such structures are parametric models of the equational theory PILLY, a polymorphic intuitionistic / linear type theory with fixed points, in which one can reason using parametricity and, for example, solve a large class of domain equations [4,5]. Based on recent work by Simpson and Rosolini [22] we construct a family of parametric LAPLstructures using synthetic domain theory and use the results of loc. cit. and results about LAPLstructures to prove operational consequences of parametricity for a strict version of the Lily programming language. In particular we can show that one can solve domain equations in the strict version of Lily up to ground contextual equivalence. Key words: Synthetic domain theory, parametric polymorphism, categorical semantics, domain theory 1