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Parametric and TypeDependent Polymorphism
, 1995
"... Data Types, though, as Reynolds stresses, is not perfectly suited for higher type or higher order systems and, thus, he proposes a "relational" treatment of invariance: computations do not depend on types in the sense that they are "invariant" w.r.t. arbitrary relations on types ..."
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Cited by 10 (5 self)
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Data Types, though, as Reynolds stresses, is not perfectly suited for higher type or higher order systems and, thus, he proposes a "relational" treatment of invariance: computations do not depend on types in the sense that they are "invariant" w.r.t. arbitrary relations on types and between types. Reynolds's approach set the basis for most of the current work on parametricity, as we will review below (.3). Some twelve years earlier, Girard had given just a simple hint towards another understanding of the properties of "computing with types". In [Gir71], it is shown, as a side remark, that, given a type A, if one defines a term J A such that, for any type B, J A B reduces to 1, if A = B, and reduces to 0, if A ยน B, then F + J A does not normalize. In particular, then, J A is not definable in F. This remark on how terms may depend on types is inspired by a view of types which is quite different from Reynolds's. System F was born as the theory of proofs of second order intuitionis...
Typechecking is Undecidable When 'Type' is a Type
, 1989
"... A function has a dependent type when the type of its result depends upon the value of its argument. The type o all types is the type of every type, including itself. In a typed Acalculus, these two features synergize in a conceptually clean and uniform way to yield enormous expressive power at very ..."
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Cited by 3 (0 self)
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A function has a dependent type when the type of its result depends upon the value of its argument. The type o all types is the type of every type, including itself. In a typed Acalculus, these two features synergize in a conceptually clean and uniform way to yield enormous expressive power at very little apparent cost. By reconstructing and analyzing a paradox due to Girard, we argue that there is no effective typechecking algorithm for such a language.
Inductive, projective, and retractive types
, 1993
"... We give an analysis of classes of recursive types by presenting two extensions of the simplytyped lambda calculus. The first language only allows recursive types with builtin principles of wellfounded induction, while the second allows more general recursive types which permit nonterminating com ..."
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Cited by 1 (1 self)
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We give an analysis of classes of recursive types by presenting two extensions of the simplytyped lambda calculus. The first language only allows recursive types with builtin principles of wellfounded induction, while the second allows more general recursive types which permit nonterminating computations. We discuss the expressive power of the languages, examine the properties of reductionbased operational semantics for them, and give examples of their use in expressing iteration over large ordinals and in simulating both callbyname and callbyvalue versions of the untyped lambda calculus. The motivations for this work come from category theoretic models. 1