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Bounded Reducibility for Computable Numberings

  • Nikolay Bazhenov
  • Manat MustafaEmail author
  • Sergei Ospichev
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11558)

Abstract

The theory of numberings gives a fruitful approach to studying uniform computations for various families of mathematical objects. The algorithmic complexity of numberings is usually classified via the reducibility \(\le \) between numberings. This reducibility gives rise to an upper semilattice of degrees, which is often called the Rogers semilattice. For a computable family S of c.e. sets, its Rogers semilattice R(S) contains the \(\le \)-degrees of computable numberings of S. Khutoretskii proved that R(S) is always either one-element, or infinite. Selivanov proved that an infinite R(S) cannot be a lattice.

We introduce a bounded version of reducibility between numberings, denoted by \(\le _{bm}\). We show that Rogers semilattices \(R_{bm}(S)\), induced by \(\le _{bm}\), exhibit a striking difference from the classical case. We prove that the results of Khutoretskii and Selivanov cannot be extended to our setting: For any natural number \(n\ge 2\), there is a finite family S of c.e. sets such that its semilattice \(R_{bm}(S)\) has precisely \(2^n-1\) elements. Furthermore, there is a computable family T of c.e. sets such that \(R_{bm}(T)\) is an infinite lattice.

Notes

Acknowledgements

Part of the research contained in this paper was carried out while the first and the last authors were visiting the Department of Mathematics of Nazarbayev University, Astana. The authors wish to thank Nazarbayev University for its hospitality. The authors also thank the anonymous reviewers for their helpful suggestions.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Sobolev Institute of MathematicsNovosibirskRussia
  2. 2.Novosibirsk State UniversityNovosibirskRussia
  3. 3.Department of Mathematics, School of Science and TechnologyNazarbayev UniversityAstanaKazakhstan

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