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18
Correctness of Data Representations involving Heap Data Structures
 Science of Computer Programming
, 2003
"... While the semantics of local variables in programming languages is by now wellunderstood, the semantics of pointeraddressed heap variables is still an outstanding issue. In particular, the commonly assumed relational reasoning principles for data representations have not been validated in a se ..."
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Cited by 22 (7 self)
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While the semantics of local variables in programming languages is by now wellunderstood, the semantics of pointeraddressed heap variables is still an outstanding issue. In particular, the commonly assumed relational reasoning principles for data representations have not been validated in a semantic model of heap variables. In this paper, we de ne a parametricity semantics for a Pascallike language with pointers and heap variables which gives such reasoning principles. It is found that the correspondences between data representations are not simply relations between states, but more intricate correspondences that also need to keep track of visible locations whose pointers can be stored and leaked.
An Axiomatic Approach to Binary Logical Relations with Applications to Data Refinement
 Proc. TACS'97, Springer LNCS 1281
, 1997
"... We introduce an axiomatic approach to logical relations and data refinement. We consider a programming language and the monad on the category of small categories generated by it. We identify abstract data types for the language with sketches for the associated monad, and define an axiomatic notion o ..."
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Cited by 18 (1 self)
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We introduce an axiomatic approach to logical relations and data refinement. We consider a programming language and the monad on the category of small categories generated by it. We identify abstract data types for the language with sketches for the associated monad, and define an axiomatic notion of "relation" between models of such a sketch in a semantic category. We then prove three results: (i) such models lift to the whole language together with the sketch; (ii) any such relation satisfies a soundness condition, and (iii) such relations compose. We do this for both equality of data representations and for an ordered version. Finally, we compare our formulation of data refinement with that of Hoare. This work has been done with the support of the MITI Cooperative Architecture Project. This author also acknowledges the support of Kakenhi. y This author achnowledges the support of the MITI Cooperative Architecture Project. z This author acknowledges the support of EPSRC grant...
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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Cited by 10 (3 self)
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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.
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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Cited by 9 (6 self)
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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
The GirardReynolds isomorphism (second edition
 Theoretical Computer Science
, 2004
"... polymorphic lambda calculus, F2. Girard additionally proved a Representation Theorem: every function on natural numbers that can be proved total in secondorder intuitionistic predicate logic, P2, can be represented in F2. Reynolds additionally proved an Abstraction Theorem: every term in F2 satisfi ..."
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Cited by 6 (0 self)
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polymorphic lambda calculus, F2. Girard additionally proved a Representation Theorem: every function on natural numbers that can be proved total in secondorder intuitionistic predicate logic, P2, can be represented in F2. Reynolds additionally proved an Abstraction Theorem: every term in F2 satisfies a suitable notion of logical relation; and formulated a notion of parametricity satisfied by wellbehaved models. We observe that the essence of Girard’s result is a projection from P2 into F2, and that the essence of Reynolds’s result is an embedding of F2 into P2, and that the Reynolds embedding followed by the Girard projection is the identity. We show that the inductive naturals are exactly those values of type natural that satisfy Reynolds’s notion of parametricity, and as a consequence characterize situations in which the Girard projection followed by the Reynolds embedding is also the identity. An earlier version of this paper used a logic over untyped terms. This version uses a logic over typed term, similar to ones considered by Abadi and Plotkin and by Takeuti, which better clarifies the relationship between F2 and P2. This paper uses colour to enhance its presentation. If the link below is not blue, follow it for the colour version.
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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Cited by 5 (4 self)
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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
Relational Parametricity for Higher Kinds
"... Abstract. Reynolds ’ notion of relational parametricity has been extremely influential and well studied for polymorphic type theories such as System F. The extension of relational parametricity to higher kinded polymorphism, which allows quantification over type operators as well as types, has not b ..."
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Cited by 4 (0 self)
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Abstract. Reynolds ’ notion of relational parametricity has been extremely influential and well studied for polymorphic type theories such as System F. The extension of relational parametricity to higher kinded polymorphism, which allows quantification over type operators as well as types, has not been as well studied. In this paper we give a model of relational parametricity for System F ω and investigate some of its consequences. 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].
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
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.