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54
Domain Theory
 Handbook of Logic in Computer Science
, 1994
"... Least fixpoints as meanings of recursive definitions. ..."
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Cited by 518 (23 self)
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Least fixpoints as meanings of recursive definitions.
Properly injective spaces and function spaces
 TO APPEAR IN TOPOLOGY AND ITS APPLICATIONS
, 1997
"... Given an injective space D (a continuous lattice endowed with the Scott topology) and a subspace embedding j: X → Y, Dana Scott asked whether the higherorder function [X → D] → [Y → D] which takes a continuous map f: X → D to its greatest continuous extension ¯ f: Y → D along j is Scott continuous ..."
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Cited by 34 (12 self)
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Given an injective space D (a continuous lattice endowed with the Scott topology) and a subspace embedding j: X → Y, Dana Scott asked whether the higherorder function [X → D] → [Y → D] which takes a continuous map f: X → D to its greatest continuous extension ¯ f: Y → D along j is Scott continuous. In this case the extension map is a subspace embedding. We show that the extension map is Scott continuous iff D is the trivial onepoint space or j is a proper map in the sense of Hofmann and Lawson. In order to avoid the ambiguous expression “proper subspace embedding”, we refer to proper maps as finitary maps. We show that the finitary sober subspaces of the injective spaces are exactly the stably locally compact spaces. Moreover, the injective spaces over finitary embeddings are the algebras of the upper power space monad on the category of sober spaces. These coincide with the retracts of upper power spaces of sober spaces. In the full subcategory of locally compact sober spaces, these are known to be the continuous meetsemilattices. In the full subcategory of stably locally compact spaces these are again the continuous lattices. The above characterization of the injective spaces over finitary embeddings is an instance of a general result on injective objects in posetenriched categories with the structure of a KZmonad established in this paper, which we also apply to various full subcategories closed under the upper power space construction and to the upper and lower power locale monads. The above results also hold for the injective spaces over dense subspace embeddings (continuous Scott domains). Moreover, we show that every sober space has a smallest finitary dense sober subspace (its support). The support always contains the subspace of maximal points, and in the stably locally compact case (which includes densely injective spaces) it is the subspace of maximal points iff that subspace is compact.
Semantic Domains for Combining Probability and NonDeterminism
 ELECTRONIC NOTES IN THEORETICAL COMPUTER SCIENCE
, 2005
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Power Domain Constructions
 SCIENCE OF COMPUTER PROGRAMMING
, 1998
"... The variety of power domain constructions proposed in the literature is put into a general algebraic framework. Power constructions are considered algebras on a higher level: for every ground domain, there is a power domain whose algebraic structure is specified by means of axioms concerning the alg ..."
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Cited by 23 (9 self)
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The variety of power domain constructions proposed in the literature is put into a general algebraic framework. Power constructions are considered algebras on a higher level: for every ground domain, there is a power domain whose algebraic structure is specified by means of axioms concerning the algebraic properties of the basic operations empty set, union, singleton, and extension of functions. A host of derived operations is introduced and investigated algebraically. Every power construction is shown to be equipped with a characteristic semiring such that the resulting power domains become semiring modules. Power homomorphisms are introduced as a means to relate different power constructions. They also allow to define the notion of initial and final constructions for a fixed characteristic semiring. Such initial and final constructions are shown to exist for every semiring, and their basic properties are derived. Finally, the known power constructions are put into the general framewo...
Clausal Logic And Logic Programming In Algebraic Domains
 Information and Computation
, 2001
"... . We introduce a domaintheoretic foundation for disjunctive logic programming. This foundation is built on clausal logic, a representation of the Smyth powerdomain of any coherent algebraic dcpo. We establish the completeness of a resolution rule for inference in such a clausal logic; we introdu ..."
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Cited by 19 (5 self)
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. We introduce a domaintheoretic foundation for disjunctive logic programming. This foundation is built on clausal logic, a representation of the Smyth powerdomain of any coherent algebraic dcpo. We establish the completeness of a resolution rule for inference in such a clausal logic; we introduce a natural declarative semantics and a fixedpoint semantics for disjunctive logic programs, and prove their equivalence; finally, we apply our results to give both a syntax and semantics for default logic in any coherent algebraic dcpo. 1. Introduction Domain theory, as introduced by Scott in the 1970's, has many connections with logic. Such connections are usually made by extracting an appropriate language /syntax from a category of domains. To name a few examples, we have Abramsky's "domain theory in logical form" [Abr91], Scott's own representation of Scott domains as information systems [Sco82], extended to other domains by Zhang [Zha91], and Smyth's treatment of observable prope...
The regularlocallycompact coreflection of stably locally compact locale
 Journal of Pure and Applied Algebra
, 2001
"... The Scott continuous nuclei form a subframe of the frame of all nuclei. We refer to this subframe as the patch frame. We show that the patch construction exhibits (i) the category of regular locally compact locales and perfect maps as a coreflective subcategory of the category of stably locally comp ..."
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Cited by 18 (8 self)
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The Scott continuous nuclei form a subframe of the frame of all nuclei. We refer to this subframe as the patch frame. We show that the patch construction exhibits (i) the category of regular locally compact locales and perfect maps as a coreflective subcategory of the category of stably locally compact locales and perfect maps,
On the duality of compact vs. open
 Papers on General Topology and Applications: Eleventh Summer Conference at the University of Southern Maine, volume 806 of Annals of the New York Academy of Sciences
, 1996
"... It is a pleasant fact that Stoneduality may be described very smoothly when restricted to the category of compact spectral spaces: The Stoneduals of these spaces, arithmetic algebraic lattices, may be replaced by their sublattices of compact elements thus discarding infinitary operations. We presen ..."
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Cited by 17 (1 self)
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It is a pleasant fact that Stoneduality may be described very smoothly when restricted to the category of compact spectral spaces: The Stoneduals of these spaces, arithmetic algebraic lattices, may be replaced by their sublattices of compact elements thus discarding infinitary operations. We present a similar approach to describe the Stoneduals of coherent spaces, thus dropping the requirement of having a base of compactopens (or, alternatively, replacing algebraicity of the lattices by continuity). The construction via strong proximity lattices is resembling the classical case, just replacing the order by an order of approximation. Our development enlightens the fact that “open ” and “compact ” are dual concepts which merely happen to coincide in the classical case.
Stably Compact Spaces and Closed Relations
, 2001
"... Stably compact spaces are a natural generalization of compact Hausdorff spaces in the T 0 setting. They have been studied intensively by a number of researchers and from a variety of standpoints. In this paper we let the morphisms between stably compact spaces be certain \closed relations" and ..."
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Cited by 15 (3 self)
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Stably compact spaces are a natural generalization of compact Hausdorff spaces in the T 0 setting. They have been studied intensively by a number of researchers and from a variety of standpoints. In this paper we let the morphisms between stably compact spaces be certain \closed relations" and study the resulting categorical properties. Apart from extending ordinary continuous maps, these morphisms have a number of pleasing properties, the most prominent, perhaps, being that they correspond to preframe homomorphisms on the localic side. We exploit this Stonetype duality to establish that the category of stably compact spaces and closed relations has bilimits.
Power Domains and Second Order Predicates
 THEORETICAL COMPUTER SCIENCE
, 1993
"... Lower, upper, sandwich, mixed, and convex power domains are isomorphic to domains of second order predicates mapping predicates on the ground domain to logical values in a semiring. The various power domains differ in the nature of the underlying semiring logic and in logical constraints on the seco ..."
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Cited by 13 (7 self)
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Lower, upper, sandwich, mixed, and convex power domains are isomorphic to domains of second order predicates mapping predicates on the ground domain to logical values in a semiring. The various power domains differ in the nature of the underlying semiring logic and in logical constraints on the second order predicates.
Topological Games in Domain Theory
 Topology Appl
"... We prove that a metric space may be realized as the set of maximal elements in a continuous dcpo if and only if it is completely metrizable by showing more generally that the space of maximal elements in a domain is always complete in a sense rst introduced by Choquet. ..."
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Cited by 13 (0 self)
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We prove that a metric space may be realized as the set of maximal elements in a continuous dcpo if and only if it is completely metrizable by showing more generally that the space of maximal elements in a domain is always complete in a sense rst introduced by Choquet.