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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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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