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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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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 9 (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.
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 6 (1 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 .
Static SpaceTimes Naturally Lead to QuasiPseudometrics
"... The standard 4dimensional Minkowski spacetime of special relativity is based on the 3dimensional Euclidean metric. In 1967, H. Busemann showed that similar static spacetime models can be based on an arbitrary metric space. In this paper, we search for the broadest possible generalization of a me ..."
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The standard 4dimensional Minkowski spacetime of special relativity is based on the 3dimensional Euclidean metric. In 1967, H. Busemann showed that similar static spacetime models can be based on an arbitrary metric space. In this paper, we search for the broadest possible generalization of a metric under which a construction of a static spacetime leads to a physically reasonable spacetime model. It turns out that this broadest possible generalization is related to the known notion of a quasipseudometric. 1 Computational Motivations Status of this section. In this introductory section, we present the main motivation for this paper—to enhance computational modelling of spacetime. The area of computational modelling of spacetime is, by definition, very interdisciplinary, it includes mathematicians and computer scientists interested in physical applications and physicists interested in computational aspects of their research. To help readers with different backgrounds better understand our motivations, we decided to describe these motivations in a special section. To some readers, these motivations are well known; other readers may be interested only in our mathematical results and are thus not interested in reading 0Keywords: spacetime, quasipseudometric, causality, antitriangle inequality, relativity, kinematic metric.
Quantitative Concept Analysis
"... Abstract. Formal Concept Analysis (FCA) begins from a context, given as a binary relation between some objects and some attributes, and derives a lattice of concepts, where each concept is given as a set of objects and a set of attributes, such that the first set consists of all objects that satisfy ..."
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Abstract. Formal Concept Analysis (FCA) begins from a context, given as a binary relation between some objects and some attributes, and derives a lattice of concepts, where each concept is given as a set of objects and a set of attributes, such that the first set consists of all objects that satisfy all attributes in the second, and vice versa. Many applications, though, provide contexts with quantitative information, telling not just whether an object satisfies an attribute, but also quantifying this satisfaction. Contexts in this form arise as rating matrices in recommender systems, as occurrence matrices in text analysis, as pixel intensity matrices in digital image processing, etc. Such applications have attracted a lot of attention, and several numeric extensions of FCA have been proposed. We propose the framework of proximity sets (proxets), which subsume partially ordered sets (posets) as well as metric spaces. One feature of this approach is that it extracts from quantified contexts quantified concepts, and thus allows full use of the available information. Another feature is that the categorical approach allows analyzing any universal properties that the classical FCA and the new versions may have, and thus provides structural guidance for aligning and combining the approaches.
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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Cited by 1 (1 self)
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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
Bicompletions of Distance Matrices To Samson Abramsky on the occasion of his 60th birthday
"... Abstract. In the practice of information extraction, the input data are usually arranged into pattern matrices, and analyzed by the methods of linear algebra and statistics, such as principal component analysis. In some applications, the tacit assumptions of these methods lead to wrong results. The ..."
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Abstract. In the practice of information extraction, the input data are usually arranged into pattern matrices, and analyzed by the methods of linear algebra and statistics, such as principal component analysis. In some applications, the tacit assumptions of these methods lead to wrong results. The usual reason is that the matrix composition of linear algebra presents information as flowing in waves, whereas it sometimes flows in particles, which seek the shortest paths. This waveparticle duality in computation and information processing has been originally observed by Abramsky. In this paper we pursue a particle view of information, formalized in distance spaces, which generalize metric spaces, but are slightly less general than Lawvere’s generalized metric spaces. In this framework, the task of extracting the ’principal components ’ from a given matrix of data boils down to a bicompletion, in the sense of enriched category theory. We describe the bicompletion construction for distance matrices. The practical goal that motivates this research is to develop a method to estimate the hardness of attack constructions in security. 1
Special RelativityType SpaceTimes Naturally Lead to QuasiPseudometrics
"... The standard 4dimensional Minkowski spacetime of special relativity is based on the 3dimensional Euclidean metric. In 1967, H. Busemann showed that similar spacetime models can be based on an arbitrary metric space. In this paper, we search for the broadest possible generalization of a metric un ..."
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The standard 4dimensional Minkowski spacetime of special relativity is based on the 3dimensional Euclidean metric. In 1967, H. Busemann showed that similar spacetime models can be based on an arbitrary metric space. In this paper, we search for the broadest possible generalization of a metric under which a construction similar to Minkowski spacetime leads to a physically reasonable spacetime model. It turns out that this broadest possible generalization is related to the known notion of a quasipseudometric. 1 Computational Motivations Status of this section. In this introductory section, we present the main motivation for this paper—to enhance computational modelling of spacetime. The area of computational modelling of spacetime is, by definition, very interdisciplinary, it includes mathematicians and computer scientists interested in physical applications and physicists interested in computational aspects of their research. To help readers with different backgrounds better understand our motivations, we decided to describe these motivations in a special section. To some readers, these motivations are well known; other readers may be interested only in our mathematical results and are thus not interested in reading 0Keywords: spacetime, quasipseudometric, causality, antitriangle inequality, relativity, kinematic metric.