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A realizability model of impredicative hoare type theory
 In European Symposium on Programming (ESOP
, 2007
"... Abstract. We present a denotational model of impredicative Hoare Type Theory, a very expressive dependent type theory in which one can specify and reason about mutable abstract data types. The model ensures soundness of the extension of Hoare Type Theory with impredicative polymorphism; makes the co ..."
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Cited by 15 (9 self)
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Abstract. We present a denotational model of impredicative Hoare Type Theory, a very expressive dependent type theory in which one can specify and reason about mutable abstract data types. The model ensures soundness of the extension of Hoare Type Theory with impredicative polymorphism; makes the connections to separation logic clear, and provides a basis for investigation of further sound extensions of the theory, in particular equations between computations and types. 1
Synthetic domain theory and models of linear Abadi & Plotkin logic
, 2005
"... Plotkin suggested using a polymorphic dual intuitionistic / linear type theory (PILLY) as a metalanguage for parametric polymorphism and recursion. In recent work the first two authors and R.L. Petersen have defined a notion of parametric LAPLstructure, which are models of PILLY, in which one can r ..."
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Plotkin suggested using a polymorphic dual intuitionistic / linear type theory (PILLY) as a metalanguage for parametric polymorphism and recursion. In recent work the first two authors and R.L. Petersen have defined a notion of parametric LAPLstructure, which are models of PILLY, in which one can reason using parametricity and, for example, solve a large class of domain equations, as suggested by Plotkin. In this paper we show how an interpretation of a strict version of Bierman, Pitts and Russo’s language Lily into synthetic domain theory presented by Simpson and Rosolini gives rise to a parametric LAPLstructure. This adds to the evidence that the notion of LAPLstructure is a general notion suitable for treating many different parametric models, and it provides formal proofs of consequences of parametricity expected to hold for the interpretation. Finally, we show how these results in combination with Rosolini and Simpson’s computational adequacy result can be used to prove consequences of parametricity for Lily. In particular we show that one can solve domain equations in Lily up to ground contextual equivalence. 1
Interpreting polymorphic FPC into domain theoretic models of parametric polymorphism
 in: International Colloquium on Automata, Languages and Programming, Proceedings, Vol. 4052 of LNCS, SpringerVerlag
, 2006
"... Abstract. This paper shows how parametric PILLY (Polymorphic Intuitionistic / Linear Lambda calculus with a fixed point combinator Y) can be used as a metalanguage for domain theory, as originally suggested by Plotkin more than a decade ago. Using recent results about solutions to recursive domain e ..."
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Abstract. This paper shows how parametric PILLY (Polymorphic Intuitionistic / Linear Lambda calculus with a fixed point combinator Y) can be used as a metalanguage for domain theory, as originally suggested by Plotkin more than a decade ago. Using recent results about solutions to recursive domain equations in parametric models of PILLY, we show how to interpret FPC in these. Of particular interest is a model based on “admissible ” pers over a reflexive domain, the theory of which can be seen as a domain theory for (impredicative) polymorphism. We show how this model gives rise to a parametric and computationally adequate model of PolyFPC, an extension of FPC with impredicative polymorphism. This is the first model of a language with parametric polymorphism, recursive terms and recursive types in a nonlinear setting. 1
CATEGORYTHEORETIC MODELS OF LINEAR ABADI & PLOTKIN LOGIC
, 2008
"... This paper presents a sound and complete categorytheoretic notion of models for Linear Abadi & Plotkin Logic [Birkedal et al., 2006], a logic suitable for reasoning about parametricity in combination with recursion. A subclass of these called parametric LAPL structures can be seen as an axioma ..."
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This paper presents a sound and complete categorytheoretic notion of models for Linear Abadi & Plotkin Logic [Birkedal et al., 2006], a logic suitable for reasoning about parametricity in combination with recursion. A subclass of these called parametric LAPL structures can be seen as an axiomatization of domain theoretic models of parametric polymorphism, and we show how to solve general (nested) recursive domain equations in these. Parametric LAPL structures constitute a general notion of model of parametricity in a setting with recursion. In future papers we will demonstrate this by showing how many different models of parametricity and recursion give rise to parametric LAPL structures, including Simpson and Rosolini’s set theoretic models [Rosolini and Simpson, 2004], a syntactic model based on Lily [Pitts, 2000, Bierman et al., 2000] and a model based on admissible pers over a reflexive domain [Birkedal et al., 2007].
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
A Relationally Parametric Model of the Calculus of Constructions
"... In this paper, we give the first relationally parametric model of the (extensional) calculus of constructions. Our model remains as simple as traditional PER models of dependent types, but unlike them, our model additionally permits relating terms at different implementation types. Using this model, ..."
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In this paper, we give the first relationally parametric model of the (extensional) calculus of constructions. Our model remains as simple as traditional PER models of dependent types, but unlike them, our model additionally permits relating terms at different implementation types. Using this model, we can validate the soundness of quotient types, as well as derive strong equality axioms for Churchencoded data, such as the etalaw for strong dependent pair types. 1.
Internalizing Relational Parametricity in the Extensional Calculus of Constructions
"... Abstract—We give the first relationally parametric model of the extensional calculus of constructions. Our model remains as simple as traditional PER models of types, but unlike them, it types in different ways. Using our model, we can validate the soundness of quotient types, as well as derive stro ..."
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Abstract—We give the first relationally parametric model of the extensional calculus of constructions. Our model remains as simple as traditional PER models of types, but unlike them, it types in different ways. Using our model, we can validate the soundness of quotient types, as well as derive strong equality axioms for Churchencoded data, such as the usual induction principles for Church naturals and booleans, and the eta law for strong dependent pair types. Furthermore, we show that such equivalences, justified by relationally parametric reasoning, may soundly be internalized (i.e., added as equality axioms to our type theory). Thus, we demonstrate that it is possible to interpret equality in a dependentlytyped setting using parametricity. The key idea behind our approach is to interpret types as socalled quasiPERs (or zigzagcomplete relations), which enable us to model the symmetry and transitivity of equality while at the same time allowing for different representations of abstract types. 1
Free Theorems Involving . . .
, 2009
"... Free theorems are a charm, allowing the derivation of useful statements about programs from their (polymorphic) types alone. We show how to reap such theorems not only from polymorphism over ordinary types, but also from polymorphism over type constructors restricted by class constraints. Our prime ..."
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Free theorems are a charm, allowing the derivation of useful statements about programs from their (polymorphic) types alone. We show how to reap such theorems not only from polymorphism over ordinary types, but also from polymorphism over type constructors restricted by class constraints. Our prime application area is that of monads, which form the probably most popular type constructor class of Haskell. To demonstrate the broader scope, we also deal with a transparent way of introducing difference lists into a program, endowed with a neat and general correctness proof.