Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-19T13:29:59.827Z Has data issue: false hasContentIssue false
  • English
  • Français

Domain representations of spaces of compact subsets

Published online by Cambridge University Press:  25 March 2010

ULRICH BERGER ,
JENS BLANCK  and
PETTER KRISTIAN KØBER
ULRICH BERGER
Affiliation:
Department of Computer Science, Swansea University, Singleton Park, Swansea, SA2 8PP, Wales Email: [email protected]; [email protected]
JENS BLANCK
Affiliation:
Department of Computer Science, Swansea University, Singleton Park, Swansea, SA2 8PP, Wales Email: [email protected]; [email protected]
PETTER KRISTIAN KØBER
Affiliation:
Department of Mathematics, University of Oslo, Pb.1053 Blindern, N-0316 Oslo, Norway Email: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We present a method for constructing from a given domain representation of a space X with underlying domain D, a domain representation of a subspace of compact subsets of X where the underlying domain is the Plotkin powerdomain of D. We show that this operation is functorial over a category of domain representations with a natural choice of morphisms. We study the topological properties of the space of representable compact sets and isolate conditions under which all compact subsets of X are representable. Special attention is paid to admissible representations and representations of metric spaces.

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 20 , Special Issue 2: Domains , April 2010 , pp. 107 - 126
Copyright
Copyright © Cambridge University Press 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abramsky, S. and Jung, A. (1994) Domain Theory. In: Handbook of Logic in Computer Science 3, Oxford University Press 1–158.Google Scholar
Battenfeld, I. (2006) Computational Effects in Topological Domain Theory. Electronic Notes in Theoretical Computer Science 158 59–80.CrossRefGoogle Scholar
Blanck, J. (1997) Domain representability of metric spaces. Annals of Pure and Applied Logic 83 225–247.CrossRefGoogle Scholar
Blanck, J. (1999) Effective domain representations of ℋ(X), the space of compact subsets. Theoretical Computer Science 219 19–48.CrossRefGoogle Scholar
Blanck, J. (2000) Domain Representations of Topological Spaces. Theoretical Computer Science 247 229–255.Google Scholar
Blanck, J. (2008) Reducibility of Domain Representations and Cantor–Weihrauch Domain Representations. Mathematical Structures in Computer Science 18 1031–1056.CrossRefGoogle Scholar
Ershov, Y. L. (1977) Model C of partial continuous functionals. In: Gandy, R. and Hyland, M. (eds.) Logic Colloquium 1976, North-Holland455–467.Google Scholar
Escardó, M., Lawson, J. and Simpson, A. (2004) Comparing Cartesian Closed Categories of (Core) Compactly Generated Spaces. Topology and its applications 143 105–145.CrossRefGoogle Scholar
Escardó, M. (2008) Exhaustible Sets in Higher-Type Computation. Logical Methods in Computer Science 4 1–37.Google Scholar
Gierz, G., Hofmann, K. H., Keimel, K., Lawson, J. D., Mislove, M. W. and Scott, D. S. (2003) Continuous Lattices and Domains, Cambridge University Press.CrossRefGoogle Scholar
Hamrin, G. (2005) Effective Domains and Admissible Domain Representations, Ph.D. Thesis, Uppsala Dissertations in Mathematics 42, Uppsala University.Google Scholar
Kleene, S. C. (1959) Countable functionals. In: Heyting, A. (ed.) Constructivity in Mathematics, North-Holland81–100.Google Scholar
Kreisel, G. (1959) Interpretation of analysis by means of constructive functionals of finite types. In: Heyting, A. (ed.) Constructivity in Mathematics, North-Holland101–128.Google Scholar
Normann, D. (1980) Recursion on the countable functionals. Springer-Verlag Lecture Notes in Computer Science 811.Google Scholar
Normann, D. (2008) A rich hierarchy of functionals of finite types (manuscript).CrossRefGoogle Scholar
Plotkin, G. (1983) Pisa Notes (on Domains).Google Scholar
Schröder, M. (2003) Admissible Representations for Continuous Computations, Ph.D. Thesis, FernUniversität Hagen.Google Scholar
Scott, D. S. (1970) Outline of a mathematical theory of computation. In: 4th Annual Princeton Conference on Information Sciences and Systems 169–176.Google Scholar
Smyth, M. B. (1983) Power Domains and Predicate Transformers: A Topological View. Springer-Verlag Lecture Notes in Computer Science 154 662–675.CrossRefGoogle Scholar
Stoltenberg-Hansen, V., Lindström, I. and Griffor, E. R. (1994) Mathematical Theory of Domains, Cambridge University Press.CrossRefGoogle Scholar
Stoltenberg-Hansen, V. and Tucker, J. V. (1995) Effective algebra. In: Abramsky, S. et al. (eds.) Handbook of Logic in Computer Science 4 357–526.CrossRefGoogle Scholar
Stoltenberg-Hansen, V. and Tucker, J. (2008) Computability on topological spaces via domain representations. In: Cooper, S. B. et al. (eds.) New Computational Paradigms 153–194.CrossRefGoogle Scholar
Weihrauch, K. (2000) An Introduction to Computable Analysis, Springer-Verlag.CrossRefGoogle Scholar