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Conference on Computability in Europe

CiE 2018: Sailing Routes in the World of Computation pp 234–243Cite as

Weak Reduction Principle and Computable Metric Spaces

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10936))

Abstract

This paper is a part of the ongoing research on developing a foundation for studying arithmetical and descriptive complexity of partial computable functions in the framework of computable topology. We propose new principles for computable metric spaces. We start with a weak version of Reduction Principle (\(\mathrm {WRP}\)) and prove that the lattice of the effectively open subsets of any computable metric space meets \(\mathrm {WRP}\). We illustrate the role of \(\mathrm {WRP}\) within partial computability. Then we investigate the existence of a principal computable numbering for the class of partial computable functions from an effectively enumerable space to a computable metric space. We show that while in general such numbering does not exist in the important case of a perfect computable Polish space it does. The existence of a principal computable numbering gives an opportunity to make reasoning about complexity of well-known problems in computable analysis in terms of arithmetical and analytic complexity of their index sets.

The research has been partially supported by the DFG grants CAVER BE 1267/14-1 and WERA MU 1801/5-1, RFBR grant A-17-01-00247.

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Author information

Authors and Affiliations

  1. A.P. Ershov Institute of Informatics Systems, SbRAS, Novosibirsk, Russia

    Margarita Korovina

  2. Sobolev Institute of Mathematics, SbRAS, Novosibirsk, Russia

    Oleg Kudinov

Authors
  1. Margarita Korovina
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  2. Oleg Kudinov
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Corresponding author

Correspondence to Margarita Korovina .

Editor information

Editors and Affiliations

  1. Kiel University, Kiel, Germany

    Florin Manea

  2. Queens College, CUNY, Queens, New York, USA

    Russell G. Miller

  3. Kiel University, Kiel, Germany

    Dirk Nowotka

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Korovina, M., Kudinov, O. (2018). Weak Reduction Principle and Computable Metric Spaces. In: Manea, F., Miller, R., Nowotka, D. (eds) Sailing Routes in the World of Computation. CiE 2018. Lecture Notes in Computer Science(), vol 10936. Springer, Cham. https://doi.org/10.1007/978-3-319-94418-0_24

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  • DOI: https://doi.org/10.1007/978-3-319-94418-0_24

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-94417-3

  • Online ISBN: 978-3-319-94418-0

  • eBook Packages: Computer ScienceComputer Science (R0)

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