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A Uniform Approach to Domain Theory in Realizability Models
 Mathematical Structures in Computer Science
, 1996
"... this paper we provide a uniform approach to modelling them in categories of modest sets. To do this, we identify appropriate structure for doing "domain theory" in such "realizability models". In Sections 2 and 3 we introduce PCAs and define the associated "realizability&quo ..."
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Cited by 21 (6 self)
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this paper we provide a uniform approach to modelling them in categories of modest sets. To do this, we identify appropriate structure for doing "domain theory" in such "realizability models". In Sections 2 and 3 we introduce PCAs and define the associated "realizability" categories of assemblies and modest sets. Next, in Section 4, we prepare for our development of domain theory with an analysis of nontermination. Previous approaches have used (relatively complicated) categorical formulations of partial maps for this purpose. Instead, motivated by the idea that A provides a primitive programming language, we consider a simple notion of "diverging" computation within A itself. This leads to a theory of divergences from which a notion of (computable) partial function is derived together with a lift monad classifying partial functions. The next task is to isolate a subcategory of modest sets with sufficient structure for supporting analogues of the usual domaintheoretic constructions. First, we expect to be able to interpret the standard constructions of total type theory in this category, so it should inherit cartesianclosure, coproducts and the natural numbers from modest sets. Second, it should interact well with the notion of partiality, so it should be closed under application of the lift functor. Third, it should allow the recursive definition of partial functions. This is achieved by obtaining a fixpoint object in the category, as defined in (Crole and Pitts 1992). Finally, although there is in principle no definitive list of requirements on such a category, one would like it to support more complicated constructions such as those required to interpret polymorphic and recursive types. The central part of the paper (Sections 5, 6, 7 and 9) is devoted to establish...
Complete Cuboidal Sets in Axiomatic Domain Theory (Extended Abstract)
 In Proceedings of 12th Annual Symposium on Logic in Computer Science
, 1997
"... ) Marcelo Fiore !mf@dcs.ed.ac.uk? Gordon Plotkin y !gdp@dcs.ed.ac.uk? John Power !ajp@dcs.ed.ac.uk? Department of Computer Science Laboratory for Foundations of Computer Science University of Edinburgh, The King's Buildings Edinburgh EH9 3JZ, Scotland Abstract We study the enrichme ..."
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Cited by 18 (4 self)
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) Marcelo Fiore !mf@dcs.ed.ac.uk? Gordon Plotkin y !gdp@dcs.ed.ac.uk? John Power !ajp@dcs.ed.ac.uk? Department of Computer Science Laboratory for Foundations of Computer Science University of Edinburgh, The King's Buildings Edinburgh EH9 3JZ, Scotland Abstract We study the enrichment of models of axiomatic domain theory. To this end, we introduce a new and broader notion of domain, viz. that of complete cuboidal set, that complies with the axiomatic requirements. We show that the category of complete cuboidal sets provides a general notion of enrichment for a wide class of axiomatic domaintheoretic structures. Introduction The aim of Axiomatic Domain Theory (ADT) is to provide a conceptual understanding of why domains are adequate as mathematical models of computation. (For a discussion see [12, x Axiomatic Domain Theory ].) The approach taken is to axiomatise the structure needed on a category so that its objects can be considered as domains, and its maps as continuous...
Enrichment and Representation Theorems for Categories of Domains and Continuous Functions
, 1996
"... This paper studies the notions of approximation and passage to the limit in an axiomatic setting. Our axiomatisation is subject to the following criteria: the axioms should be natural (so that they are available in as many contexts as possible) and nonordertheoretic (so that Research supported b ..."
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Cited by 8 (5 self)
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This paper studies the notions of approximation and passage to the limit in an axiomatic setting. Our axiomatisation is subject to the following criteria: the axioms should be natural (so that they are available in as many contexts as possible) and nonordertheoretic (so that Research supported by SERC grant RR30735 and EC project Programming Language Semantics and Program Logics grant SC1000 795 they explain the ordertheoretic structure). Our aim is 1. to provide a justification of Scott's original consideration of ordered structures, and 2. to deepen our understanding of the notion of passage to the limit