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52
Theorems for free!
 FUNCTIONAL PROGRAMMING LANGUAGES AND COMPUTER ARCHITECTURE
, 1989
"... From the type of a polymorphic function we can derive a theorem that it satisfies. Every function of the same type satisfies the same theorem. This provides a free source of useful theorems, courtesy of Reynolds' abstraction theorem for the polymorphic lambda calculus. ..."
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Cited by 380 (8 self)
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From the type of a polymorphic function we can derive a theorem that it satisfies. Every function of the same type satisfies the same theorem. This provides a free source of useful theorems, courtesy of Reynolds' abstraction theorem for the polymorphic lambda calculus.
Categorical Logic
 A CHAPTER IN THE FORTHCOMING VOLUME VI OF HANDBOOK OF LOGIC IN COMPUTER SCIENCE
, 1995
"... ..."
ECC, an Extended Calculus of Constructions
, 1989
"... We present a higherorder calculus ECC which can be seen as an extension of the calculus of constructions [CH88] by adding strong sum types and a fully cumulative type hierarchy. ECC turns out to be rather expressive so that mathematical theories can be abstractly described and abstract mathematics ..."
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Cited by 91 (4 self)
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We present a higherorder calculus ECC which can be seen as an extension of the calculus of constructions [CH88] by adding strong sum types and a fully cumulative type hierarchy. ECC turns out to be rather expressive so that mathematical theories can be abstractly described and abstract mathematics may be adequately formalized. It is shown that ECC is strongly normalizing and has other nice prooftheoretic properties. An !\GammaSet (realizability) model is described to show how the essential properties of the calculus can be captured settheoretically.
Domain theoretic models of polymorphism
 INF. COMPUT
, 1989
"... We give an illustration of a construction useful in producing and describing models of Girard and Reynolds' polymorphic λcalculus. The key unifying ideas are that of a Grothendieck fibration and the category of continuous sections associated with it, constructions used in indexed category theo ..."
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Cited by 36 (2 self)
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We give an illustration of a construction useful in producing and describing models of Girard and Reynolds' polymorphic λcalculus. The key unifying ideas are that of a Grothendieck fibration and the category of continuous sections associated with it, constructions used in indexed category theory; the universal types of the calculus are interpreted as the category of continuous sections of the fibration. As a major example a new model for the polymorphic λcalculus is presented. In it a type is interpreted as a Scott domain. In fact, understanding universal types of the polymorphic λcalculus as categories of continuous sections appears to be useful generally. For example, the technique also applies to the finitary projection model of Bruce and Longo, and a recent model of Girard. (Indeed the work here was inspired by Girard's and arose through trying to extend the construction of his model to Scott domains.) It is hoped that by pinpointing a key construction this paper will help towards a deeper understanding of models for the polymorphic λcalculus and the
The Discrete Objects in the Effective Topos
 Proc. London Math. Soc
, 1990
"... The original aim of this paper was to give a rather quick and undemanding proof that the effective topos contains two nontrivial small (i.e. internal) full subcategories which are closed under all small limits in the topos (and hence in particular are internally complete). The interest in such subc ..."
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Cited by 34 (7 self)
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The original aim of this paper was to give a rather quick and undemanding proof that the effective topos contains two nontrivial small (i.e. internal) full subcategories which are closed under all small limits in the topos (and hence in particular are internally complete). The interest in such subcategories arises from
Notes on Sconing and Relators
, 1993
"... This paper describes a semantics of typed lambda calculi based on relations. The main mathematical tool is a categorytheoretic method of sconing, also called glueing or Freyd covers. Its correspondence to logical relations is also examined. 1 Introduction Many modern programming languages feature ..."
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This paper describes a semantics of typed lambda calculi based on relations. The main mathematical tool is a categorytheoretic method of sconing, also called glueing or Freyd covers. Its correspondence to logical relations is also examined. 1 Introduction Many modern programming languages feature rather sophisticated typing mechanisms. In particular, languages such as ML include polymorphic data types, which allow considerable programming flexibility. Several notions of polymorphism were introduced into computer science by Strachey [Str67], among them the important notion of parametric polymorphism. Strachey's intuitive definition is that a polymorphic function is parametric if it has a uniformly given algorithm in all types, that is, if the function's behavior is independent of the type at which the function is instantiated. Reynolds [Rey83] proposed a mathematical definition of parametric polymorphic functions by means of invariance with respect to certain relations induced by typ...
Reflexive Graphs and Parametric Polymorphism
, 1993
"... this paper is to understand why that is a parametric categorical model. In [10] Ma and Reynolds propose a parametricity hypothesis for a functor between categorical models of polymorphism which essentially requires that there is an extension of (a certain form of) an identity relation functor which ..."
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Cited by 24 (1 self)
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this paper is to understand why that is a parametric categorical model. In [10] Ma and Reynolds propose a parametricity hypothesis for a functor between categorical models of polymorphism which essentially requires that there is an extension of (a certain form of) an identity relation functor which preserve the model structure. There is no mention in the paper of any case when the parametricity hypothesis is satified, nor if there is a canonical completion of a category to one which satisfies the hypothesis. We shall suggest how the construction of a PLcategory of relations on a given category presented in [10] can be viewed as a "parametric completion". We shall also follow the suggestion of Ma in [9] that subtyping is a kind of parametricity requirement and show how to fit subtyping in the same setup. The basic idea is to use reflexive graphs of categories as in [12]. We shall employ their construction to present a kind of parametric completion of a given category. We also give a different presentation of the RELconstruction in [10], and use it to discuss some examples. We show in particular that the RELconstruction acts (essentially) in the same way on a category and on its completion. Hence it follows that the identity functor on the completion satisfies the parametricity hypothesis. Discussions with Eugenio Moggi, Peter O'Hearn, Edmund Robinson, and Thomas Streicher were very useful. Paul Taylor's beutiful diagram macros were used for typesetting all the diagrams in the text. 1 Graphs of categories
Some properties of Fib as a fibred 2category
, 1997
"... We consider some basic properties of the 2category Fib of fibrations over arbitrary bases, exploiting the fact that it is fibred over Cat. We show a factorisation property for adjunctions in Fib, which has direct consequences for fibrations, e.g. a characterisation of limits and colimits for them. ..."
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Cited by 19 (1 self)
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We consider some basic properties of the 2category Fib of fibrations over arbitrary bases, exploiting the fact that it is fibred over Cat. We show a factorisation property for adjunctions in Fib, which has direct consequences for fibrations, e.g. a characterisation of limits and colimits for them. We also consider oplax colimits in Fib, with the construction of Kleisli objects as a particular example. All our constructions are based on an elementary characterisation of Fib as a fibration.