Computability on subsets of metric spaces

  • Authors:

  • Vasco Brattka;Gero Presser

  • Affiliations:

  • Theoretische Informatik I, FernUniversität Hagen, D-58084 Hagen, Germany;Lehrstuhl Informatik I, Universität Dortmund, D-44221 Dortmund, Germany

  • Venue:
  • Theoretical Computer Science - Topology in computer science
  • Year:
  • 2003

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Abstract

The notions "recursively enumerable" and "recursive" are the basic notions of effectivity in classical recursion theory. In computable analysis, these notions are generalized to closed subsets of Euclidean space using their metric distance functions. We study a further generalization of these concepts to subsets of computable metric spaces. It appears that different characterizations, which coincide in case of Euclidean space, lead to different notions in the general case. However, under certain additional conditions, such as completeness and effective local compactness, the situation is similar to the Euclidean case. We present all results in the framework of "Type-2 Theory of Effectivity" which allows to express effectivity properties in a very uniform way: instead of comparing properties of single subsets, we compare corresponding representations of the hyperspace of closed subsets. Such representations do not only induce a concept of computability for single subsets, but they even yield a concept of computability for operations on hyperspaces, such as union, intersection, etc. We complete our investigation by studying the special situation of compact subsets.