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Solving Recursive Domain Equations with Enriched Categories
, 1994
"... Both preorders and metric spaces have been used at various times as a foundation for the solution of recursive domain equations in the area of denotational semantics. In both cases the central theorem states that a `converging' sequence of `complete' domains/spaces with `continuous' ..."
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Cited by 27 (0 self)
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Both preorders and metric spaces have been used at various times as a foundation for the solution of recursive domain equations in the area of denotational semantics. In both cases the central theorem states that a `converging' sequence of `complete' domains/spaces with `continuous' retraction pairs between them has a limit in the category of complete domains/spaces with retraction pairs as morphisms. The preorder version was discovered first by Scott in 1969, and is referred to as Scott's inverse limit theorem. The metric version was mainly developed by de Bakker and Zucker and refined and generalized by America and Rutten. The theorem in both its versions provides the main tool for solving recursive domain equations. The proofs of the two versions of the theorem look astonishingly similar, but until now the preconditions for the preorder and the metric versions have seemed to be fundamentally different. In this thesis we establish a more general theory of domains based on the noti...
Quasimetric Properties of Complexity Spaces
 Topology Appl
, 1999
"... The complexity (quasimetric) space has been introduced as a part of the development of a topological foundation for the complexity analysis of algorithms ([12]). Applications of this theory to the complexity analysis of Divide & Conquer algorithms have been discussed in [12]. Here we obtain ..."
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Cited by 15 (5 self)
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The complexity (quasimetric) space has been introduced as a part of the development of a topological foundation for the complexity analysis of algorithms ([12]). Applications of this theory to the complexity analysis of Divide & Conquer algorithms have been discussed in [12]. Here we obtain several quasimetric properties of the complexity space. The main results obtained are the Smythcompleteness of the complexity space and the compactness of closed complexity spaces which possess a (complexity) lower bound. Finally, some implications of these results in connection to the above mentioned complexity analysis techniques are discussed and the total boundedness of complexity spaces with a lower bound is discussed in the light of Smyth's computational interpretation of this property ([14]). AMS (1991) Subject Classification: 54E15, 54E35, 54C30, 54C35. 1 Introduction The letters N , !, R and R + denote the set of positive integers, of nonnegative integers, of real numbers ...
A characterization of partial metrizability: Domains are quantifiable
 Theoretical Computer Science
, 2001
"... A characterization of partial metrizability is given which provides a partial solution to an open problem stated by Kunzi in the survey paper Nonsymmetric Topology ([Kun93], problem 7 ). The characterization yields a powerful tool which establishes a correspondence between partial metrics and ..."
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Cited by 14 (3 self)
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A characterization of partial metrizability is given which provides a partial solution to an open problem stated by Kunzi in the survey paper Nonsymmetric Topology ([Kun93], problem 7 ). The characterization yields a powerful tool which establishes a correspondence between partial metrics and special types of valuations, referred to as Qvaluations (cf. also [Sch00]). The notion of a Qvaluation essentially combines the wellknown notion of a valuation with a weaker version of the notion of a quasiunimorphism, i.e. an isomorphism in the context of quasiuniform spaces. As an application, we show that #continuous dcpo's are quantifiable in the sense of [O'N97], i.e. the Scott topology and partial order are induced by a partial metric. For #algebraic dcpo's the Lawson topology is induced by the associated metric. The partial metrization of general domains improves prior approaches in two ways:  The partial metric is guaranteed to capture the Scott topology as opposed to e.g. [Smy87],[BvBR95],[FS96] and [FK97], which in general yield a coarser topology.
The Correspondence Between Partial Metrics and Semivaluations
"... Partial metrics, or the equivalent weightable quasimetrics, have been introduced in [Mat94] as part of the study of the denotational semantics of data flow networks (cf. also [Mat95]). The interest in valuations in connection to Domain Theory derives from e.g. [JP89], [Jon89], [Eda94] and [Hec95 ..."
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Cited by 11 (5 self)
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Partial metrics, or the equivalent weightable quasimetrics, have been introduced in [Mat94] as part of the study of the denotational semantics of data flow networks (cf. also [Mat95]). The interest in valuations in connection to Domain Theory derives from e.g. [JP89], [Jon89], [Eda94] and [Hec95]. Connections between partial metrics and valuations have been discussed in the literature, e.g. [O'N97], [BS97] and [BSh98]. In each
Duality and QuasiNormability for Complexity Spaces
"... The complexity (quasimetric) space was introduced in [23] to study complexity analysis of programs. Recently, it was introduced in [22] the dual complexity (quasimetric) space, as a subspace of the function space [0, +#) . Several quasimetric properties of the complexity space were obtained via ..."
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Cited by 9 (2 self)
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The complexity (quasimetric) space was introduced in [23] to study complexity analysis of programs. Recently, it was introduced in [22] the dual complexity (quasimetric) space, as a subspace of the function space [0, +#) . Several quasimetric properties of the complexity space were obtained via the analysis of its dual. We here show that the structure of a quasinormed semilinear space provides a suitable setting to carry out an analysis of the dual complexity space. We show that if (E, is a biBanach space (i.e. a quasinormed space whose induced quasimetric is bicomplete), then the function space (B # E ) is biBanach, where # n=0 2 n (#f(n)###f(n)#) < +#}, and # n=0 2 n .
Complexity Spaces: Lifting Directedness
 Topology Proceedings 22 Summer
, 1999
"... The theory of complexity spaces has been introduced in [Sch95] as part of the development of a topological foundation for Complexity Analysis. ..."
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The theory of complexity spaces has been introduced in [Sch95] as part of the development of a topological foundation for Complexity Analysis.
Extendible spaces
 Appl. General Topology
, 2002
"... The domain theoretic notion of lifting allows one to extend a partial order in a trivial way by a minimum. In the context of Quantitative Domain Theory (e.g. [BvBR98] and [FK93]) partial orders are represented as quasimetric spaces. For such spaces, the notion of the extension by an extremal elemen ..."
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The domain theoretic notion of lifting allows one to extend a partial order in a trivial way by a minimum. In the context of Quantitative Domain Theory (e.g. [BvBR98] and [FK93]) partial orders are represented as quasimetric spaces. For such spaces, the notion of the extension by an extremal element turns out to be non trivial. To some extent motivated by these considerations, we characterize the directed quasimetric spaces extendible by an extremum. The class is shown to include the Scompletable directed quasimetric spaces. As an application of this result, we show that for the case of the invariant quasimetric (semi)lattices, weightedness can be characterized by order convexity combined with the extension property. 1
Similarity, Topology, and Uniformity Dedicated to Dieter Spreen on the occasion of his 60th birthday
"... We generalize various notions of generalized metrics even further to one general concept comprising them all. For convenience, we turn around the ordering in the target domain of the generalized metrics so that we speak of similarity instead of distance. Starting from an extremely general situation ..."
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We generalize various notions of generalized metrics even further to one general concept comprising them all. For convenience, we turn around the ordering in the target domain of the generalized metrics so that we speak of similarity instead of distance. Starting from an extremely general situation without axioms, we examine which axioms or additional properties are needed to obtain useful results. For instance, we shall see that commutativity and associativity of the generalized version of addition occurring in the triangle inequality is not really needed, nor do we require a generalized version of subtraction. Each similarity space comes with its own domain of possible similarity values. Therefore, we consider nonexpanding functions modulo some rescaling between different domains of similarity values. We show that nonexpanding functions with locally varying rescaling functions correspond to topologically continuous functions, while nonexpanding functions with a globally fixed rescaling generalize uniformly continuous functions.
Similarity, Topology, and Uniformity
, 2007
"... We generalize various notions of generalized metrics even further to one general concept comprising them all. For convenience, we turn around the ordering in the target domain of the generalized metrics so that we speak of similarity instead of distance. Starting from an extremely general situation ..."
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We generalize various notions of generalized metrics even further to one general concept comprising them all. For convenience, we turn around the ordering in the target domain of the generalized metrics so that we speak of similarity instead of distance. Starting from an extremely general situation without axioms, we examine which axioms or additional properties are needed to obtain useful results. For instance, we shall see that commutativity and associativity of the generalized version of addition occurring in the triangle inequation is not really needed, nor do we require a generalized version of subtraction. Each similarity space comes with its own domain of possible similarity values. Therefore, we consider nonexpanding functions modulo some rescaling between different domains of similarity values. We show that nonexpanding functions with locally varying rescaling functions correspond to topologically continuous functions, while nonexpanding functions with a globally fixed rescaling generalize uniformly continuous functions.