Generalized Ultrametric Spaces in Quantitative Domain Theory

Markus KrötzschMarkus Krötzsch

ISSN: 0304-3975
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Markus Krötzsch
Generalized Ultrametric Spaces in Quantitative Domain Theory
Theoretical Computer Science, 368(1--2):30--49, December 2006

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KurzfassungAbstract
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 via its formal-ball space. Furthermore, we can state new results on the topology of gums as well as two new fixed point theorems, which may be compared to the Prieß-Crampe and Ribenboim theorem, and the Banach fixed point theorem, respectively. Deeper insights into the nature of formal-ball 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 formal balls does not provide the hoped-for general connection to domain theory.
Bemerkung: Note: The PDF provided here is the TU Dresden Technical Report WV-04-02, which contains the same work but in a different format.
Forschungsgruppe:Research Group: Wissensbasierte SystemeKnowledge-Based Systems, WissensverarbeitungKnowledge Representation and Reasoning

@article{K2006,
  author    = {Markus Kr{\"{o}}tzsch},
  title     = {Generalized Ultrametric Spaces in Quantitative Domain Theory},
  journal   = {Theoretical Computer Science},
  volume    = {368},
  number    = {1--2},
  publisher = {Elsevier},
  year      = {2006},
  month     = {December},
  pages     = {30--49},
  doi       = {10.1016/j.tcs.2006.05.037}
}