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24
Generalized Metrics and Uniquely Determined Logic Programs
 Theoretical Computer Science
"... The introduction of negation into logic programming brings the benefit of enhanced syntax and expressibility, but creates some semantical problems. Specifically, certain operators which are monotonic in the absence of negation become nonmonotonic when it is introduced, with the result that standard ..."
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The introduction of negation into logic programming brings the benefit of enhanced syntax and expressibility, but creates some semantical problems. Specifically, certain operators which are monotonic in the absence of negation become nonmonotonic when it is introduced, with the result that standard approaches to denotational semantics then become inapplicable. In this paper, we show how generalized metric spaces can be used to obtain fixedpoint semantics for several classes of programs relative to the supported model semantics, and investigate relationships between the underlying spaces we employ. Our methods allow the analysis of classes of programs which include the acyclic, locally hierarchical, and acceptable programs, amongst others, and draw on fixedpoint theorems which apply to generalized ultrametric spaces and to partial metric spaces.
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.
Generalized ultrametric spaces in quantitative domain theory
 Theoretical Computer Science
"... Domains and metric spaces are two central tools for the study of denotational semantics in computer science, but are otherwise very different in many fundamental aspects. A construction that tries to establish links between both paradigms is the space of formal balls, a continuous poset which can be ..."
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Cited by 11 (0 self)
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Domains and metric spaces are two central tools for the study of denotational semantics in computer science, but are otherwise very different in many fundamental aspects. A construction that tries to establish links between both paradigms is the space of formal balls, a continuous poset which can be defined for every metric space and that reflects many of its properties. On the other hand, in order to obtain a broader framework for applications and possible connections to domain theory, generalized ultrametric spaces (gums) have been introduced. In this paper, we employ the space of formal balls as a tool for studying these more general metrics by using concepts and results from domain theory. It turns out that many properties of the metric can be characterized by conditions on its formalball space. Furthermore, we can state new results on the topology of gums as well as two modified fixed point theorems, which may be compared to the PrießCrampe and Ribenboim theorem and the Banach fixed point theorem, respectively. Deeper insights into the nature of formalball spaces are gained by applying methods from category theory. Our results suggest that, while being a useful tool for the study of gums, the space of
Domain Theoretic Models of Topological Spaces
 Proceedings of Comprox III, ENTCS
, 1998
"... A model of a space X is simply a continuous dcpo D and a homeomorphism OE : X ! max D, where max D is given its inherited Scott topology. We show that a space has a coherent model iff it has a Scott domain model and investigate the topological structure of spaces which have G ffi models. ..."
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Cited by 10 (4 self)
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A model of a space X is simply a continuous dcpo D and a homeomorphism OE : X ! max D, where max D is given its inherited Scott topology. We show that a space has a coherent model iff it has a Scott domain model and investigate the topological structure of spaces which have G ffi models.
Nonclassical Techniques for Models of Computation
 Topology Proceedings
, 1999
"... After surveying recent work and new techniques in domain theoretic models of spaces, we introduce a new topological concept called recurrence, and consider some of its applications to the model problem. ..."
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Cited by 9 (4 self)
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After surveying recent work and new techniques in domain theoretic models of spaces, we introduce a new topological concept called recurrence, and consider some of its applications to the model problem.
Metric semantics for reactive probabilistic processes
, 1997
"... In this thesis we present three mathematical frameworks for the modelling of reactive probabilistic communicating processes. We first introduce generalised labelled transition systems as a model of such processes and introduce an equivalence, coarser than probabilistic bisimulation, over these syst ..."
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Cited by 6 (1 self)
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In this thesis we present three mathematical frameworks for the modelling of reactive probabilistic communicating processes. We first introduce generalised labelled transition systems as a model of such processes and introduce an equivalence, coarser than probabilistic bisimulation, over these systems. Two processes are identified with respect to this equivalence if, for all experiments, the probabilities of the respective processes passing a given experiment are equal. We next consider a probabilistic process calculus including external choice, internal choice, actionguarded probabilistic choice, synchronous parallel and recursion. We give operational semantics for this calculus be means of our generalised labelled transition systems and show that our equivalence is a congruence for this language. Following the methodology introduced by de Bakker & Zucker, we then give denotational semantics to the calculus by means of a complete metric space of probabilistic processes. The derived metric, although not an ultrametric, satisfies the intuitive property that the distance between two processes tends to 0 if a measure of the dif
Extension of Valuations on Locally Compact Sober Spaces.
, 2000
"... We show that every locally finite continuous valuation defined on the lattice of open sets of a regular or locally compact sober space extends uniquely to a Borel measure. ..."
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Cited by 6 (0 self)
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We show that every locally finite continuous valuation defined on the lattice of open sets of a regular or locally compact sober space extends uniquely to a Borel measure.
A "Converse" of the Banach Contraction Mapping Theorem
, 2001
"... this paper that such an ultrametric space underlying these processes can always be found. Thus, our main result, which is stated precisely in Theorem 2, is in a sense a converse of the Banach contraction mapping theorem, and permits that theorem to be "applied" in some circumstances where ..."
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Cited by 4 (1 self)
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this paper that such an ultrametric space underlying these processes can always be found. Thus, our main result, which is stated precisely in Theorem 2, is in a sense a converse of the Banach contraction mapping theorem, and permits that theorem to be "applied" in some circumstances where no metric rendering the operators in question to be contractions was readily to hand in advance