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Domain Theory
 Handbook of Logic in Computer Science
, 1994
"... Least fixpoints as meanings of recursive definitions. ..."
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Least fixpoints as meanings of recursive definitions.
βηcomplete models for System F
, 2000
"... We show that Friedman's proof of the existence of nontrivial βηcomplete models of λ→ can be extended to system F. We isolate a set of conditions which are sufficient to ensure βηcompleteness for a model of F (and αcompleteness at the level of types), and we dis ..."
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Cited by 3 (2 self)
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We show that Friedman's proof of the existence of nontrivial βηcomplete models of &lambda;&rarr; can be extended to system F. We isolate a set of conditions which are sufficient to ensure βηcompleteness for a model of F (and &alpha;completeness at the level of types), and we discuss which class of models we get. In particular, the model introduced in [5], having as polymorphic maps exactly all possible Scott continuous maps, is &beta;&eta;complete and is hence the first known complete nonsyntactic model of F. In order to have a suitable framework where to express the conditions and develop the proof, we also introduce the very natural notion of "polymax models" of System F.
Building continuous webbed models for System F
, 2000
"... We present here a large family of concrete models for Girard and Reynolds polymorphism (System F ), in a non categorical setting. The family generalizes the construction of the model of Barbanera and Berardi [2], hence it contains complete models for F [5] and we conjecture that it contains models w ..."
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We present here a large family of concrete models for Girard and Reynolds polymorphism (System F ), in a non categorical setting. The family generalizes the construction of the model of Barbanera and Berardi [2], hence it contains complete models for F [5] and we conjecture that it contains models which are complete for F . It also contains simpler models, the simplest of them, E 2 ; being a second order variant of the EngelerPlotkin model E . All the models here belong to the continuous semantics and have underlying prime algebraic domains, all have the maximum number of polymorphic maps. The class contains models which can be viewed as two intertwined compatible webbed models of untyped calculus (in the sense of [8]), but it is much larger than this. Finally many of its models might be read as two intertwined strict intersection type systems. Contents 1
A Full Continuous Model of Polymorphism
"... Abstract. We introduce a model of the secondorder lambda calculus. Such a model is a Scott domain whose elements are themselves Scott domains, and in it polymorphic maps are interpreted by generic continous maps. Keywords: Secondorder lambda calculus, model, Scott domain, nonparametric. 1 ..."
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Abstract. We introduce a model of the secondorder lambda calculus. Such a model is a Scott domain whose elements are themselves Scott domains, and in it polymorphic maps are interpreted by generic continous maps. Keywords: Secondorder lambda calculus, model, Scott domain, nonparametric. 1
This text is based on the chapter Domain Theory in the Handbook for Logic in
"... E. Maibaum, published by Clarendon Press, Oxford in 1994. While the numbering of all theorems and definitions has been kept the same, we have included comments and corrections which we have received over the years. For ease of reading, small typographical errors have simply been corrected. Where we ..."
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E. Maibaum, published by Clarendon Press, Oxford in 1994. While the numbering of all theorems and definitions has been kept the same, we have included comments and corrections which we have received over the years. For ease of reading, small typographical errors have simply been corrected. Where we felt the original text gave a misleading impression, we have included additional explanations, clearly marked as such. If you wish to refer to this text, then please cite the published original version where possible, or otherwise this online version which we try to keep available from the page
Domain Theory  Corrected and expanded version
"... bases were introduced in [Smy77] where they are called "Rstructures". Examples of abstract bases are concrete bases of continuous domains, of course, where the relation is the restriction of the order of approximation. Axiom (INT) is satisfied because of Lemma 2.2.15 and because we have ..."
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bases were introduced in [Smy77] where they are called "Rstructures". Examples of abstract bases are concrete bases of continuous domains, of course, where the relation is the restriction of the order of approximation. Axiom (INT) is satisfied because of Lemma 2.2.15 and because we have required bases in domains to have directed sets of approximants for each element.