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Topology, Domain Theory and Theoretical Computer Science
, 1997
"... In this paper, we survey the use of ordertheoretic topology in theoretical computer science, with an emphasis on applications of domain theory. Our focus is on the uses of ordertheoretic topology in programming language semantics, and on problems of potential interest to topologists that stem from ..."
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Cited by 10 (2 self)
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In this paper, we survey the use of ordertheoretic topology in theoretical computer science, with an emphasis on applications of domain theory. Our focus is on the uses of ordertheoretic topology in programming language semantics, and on problems of potential interest to topologists that stem from concerns that semantics generates. Keywords: Domain theory, Scott topology, power domains, untyped lambda calculus Subject Classification: 06B35,06F30,18B30,68N15,68Q55 1 Introduction Topology has proved to be an essential tool for certain aspects of theoretical computer science. Conversely, the problems that arise in the computational setting have provided new and interesting stimuli for topology. These problems also have increased the interaction between topology and related areas of mathematics such as order theory and topological algebra. In this paper, we outline some of these interactions between topology and theoretical computer science, focusing on those aspects that have been mo...
A fixed point theorem in a category of compact metric spaces
 Theoretical Computer Science
, 1995
"... Various results appear in the literature for deriving existence and uniqueness of fixed points for endofunctors on categories of complete metric spaces. All these results are proved for contracting functors which satisfy some further requirements, depending on the category in question. Following a n ..."
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Cited by 2 (2 self)
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Various results appear in the literature for deriving existence and uniqueness of fixed points for endofunctors on categories of complete metric spaces. All these results are proved for contracting functors which satisfy some further requirements, depending on the category in question. Following a new kind of approach, based on the notion of ηisometry, we show that the sole hypothesis of contractivity is enough for proving existence and uniqueness of fixed points for endofunctors on the category of compact metric spaces and embeddingprojection pairs. 1
Solutions of Generalized Recursive MetricSpace Equations
"... It is well known that one can use an adaptation of the inverselimit construction to solve recursive equations in the category of complete ultrametric spaces. We show that this construction generalizes to a large class of categories with metricspace structure on each set of morphisms: the exact nat ..."
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Cited by 1 (1 self)
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It is well known that one can use an adaptation of the inverselimit construction to solve recursive equations in the category of complete ultrametric spaces. We show that this construction generalizes to a large class of categories with metricspace structure on each set of morphisms: the exact nature of the objects is less important. In particular, the construction immediately applies to categories where the objects are ultrametric spaces with ‘extra structure’, and where the morphisms preserve this extra structure. The generalization is inspired by classical domaintheoretic work by Smyth and Plotkin. Our primary motivation for solving generalized recursive metricspace equations comes from recent and ongoing work on Kripkestyle models in which the sets of worlds must be recursively defined. For many of the categories we consider, there is a natural subcategory in which each set of morphisms is required to be a compact metric space. Our setting allows for a proof that such a subcategory always inherits solutions of recursive equations from the full category. As another application, we present a construction that relates solutions of generalized domain equations in the sense of Smyth and Plotkin to solutions of equations in our class of categories. 1
Elements of generalized . . .
, 1996
"... Generalized ultrametric spaces are a common generalization of preorders and ordinary ultrametric spaces, as was observed by Lawvere (1973). Guided by his enrichedcategorical view on (ultra)metric spaces, we generalize the standard notions of Cauchy sequence and limit in an (ultra)metric space, and ..."
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Generalized ultrametric spaces are a common generalization of preorders and ordinary ultrametric spaces, as was observed by Lawvere (1973). Guided by his enrichedcategorical view on (ultra)metric spaces, we generalize the standard notions of Cauchy sequence and limit in an (ultra)metric space, and of adjoint pair between preorders. This leads to a solution method for recursive domain equations that combines and extends the standard ordertheoretic (Smyth and Plotkin, 1982) and metric (America and Rutten, 1989) approaches.