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Abstraction and Invariance for Algebraically Indexed Types
"... Reynolds ’ relational parametricity provides a powerful way to reason about programs in terms of invariance under changes of data representation. A dazzling array of applications of Reynolds ’ theory exists, exploiting invariance to yield “free theorems”, noninhabitation results, and encodings of al ..."
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Reynolds ’ relational parametricity provides a powerful way to reason about programs in terms of invariance under changes of data representation. A dazzling array of applications of Reynolds ’ theory exists, exploiting invariance to yield “free theorems”, noninhabitation results, and encodings of algebraic datatypes. Outside computer science, invariance is a common theme running through many areas of mathematics and physics. For example, the area of a triangle is unaltered by rotation or flipping. If we scale a triangle, then we scale its area, maintaining an invariant relationship between the two. The transformations under which properties are invariant are often organised into groups, with the algebraic structure reflecting the composability and invertibility of transformations. In this paper, we investigate programming languages whose types are indexed by algebraic structures such as groups of geometric transformations. Other examples include types indexed by principals–for information flow security–and types indexed by distances–for analysis of analytic uniform continuity properties. Following Reynolds, we prove a general Abstraction Theorem that covers all these instances. Consequences of our Abstraction Theorem include free theorems expressing invariance properties of programs, type isomorphisms based on invariance properties, and nondefinability results indicating when certain algebraically indexed types are uninhabited or only inhabited by trivial programs. We have fully formalised our framework and most examples in Coq.
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 Church-encoded 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 dependently-typed setting using parametricity. The key idea behind our approach is to interpret types as so-called quasi-PERs (or zigzag-complete 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
General Terms
"... SML.NET is a compiler for Standard ML that targets the Common Language Runtime and is integrated into the Visual Studio development environment. It supports easy interoperability with other.NET languages via a number of language extensions, which go considerably beyond those of our earlier compiler, ..."
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SML.NET is a compiler for Standard ML that targets the Common Language Runtime and is integrated into the Visual Studio development environment. It supports easy interoperability with other.NET languages via a number of language extensions, which go considerably beyond those of our earlier compiler, MLj. This paper describes the new language extensions and the features of the Visual Studio plugin, including syntax highlighting, Intellisense, continuous type inference and debugger support. We discuss our experiences using SML.NET to write SML programs that interoperate with other.NET languages, libraries and frameworks. Examples include the Visual Studio plugin itself (written in SML.NET, using.NET’s COM interop features to integrate in a C++ application) and writing ASP.NET and Pocket PC applications in SML.
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 Church-encoded data, such as the eta-law for strong dependent pair types. 1.
