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Covering the Recursive Sets

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Evolving Computability (CiE 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9136))

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Abstract

We give solutions to two of the questions in a paper by Brendle, Brooke-Taylor, Ng and Nies. Our examples derive from a 2014 construction by Khan and Miller as well as new direct constructions using martingales.

At the same time, we introduce the concept of i.o. subuniformity and relate this concept to recursive measure theory. We prove that there are classes closed downwards under Turing reducibility that have recursive measure zero and that are not i.o. subuniform. This shows that there are examples of classes that cannot be covered with methods other than probabilistic ones. It is easily seen that every set of hyperimmune degree can cover the recursive sets. We prove that there are both examples of hyperimmune-free degree that can and that cannot compute such a cover.

This work was partially supported by a grant from the Simons Foundation (#315188 to Bjørn Kjos-Hanssen) and by a grant from the NUS (R146-000-181-112 to F. Stephan). A substantial part of the work was performed while the first and third authors were supported by the Institute for Mathematical Sciences of the National University of Singapore during the workshop on Algorithmic Randomness during 2–30 June 2014.

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References

  1. Brendle, J, Brooke-Taylor, A, Ng, K.M., Nies, A.: An analogy between cardinal characteristics and highness properties of oracles. Technical report on http://arxiv.org/abs/1404.2839 (2014)

  2. Demuth, O., Kučera, A.: Remarks on \(1\)-genericity, semigenericity and related concepts. Commentationes Math. Univ. Carol. 28(1), 85–94 (1987)

    MATH  Google Scholar 

  3. Downey, R.G., Hirschfeldt, D.R., Lempp, S., Solomon, R.: A \(\Delta ^0_2\) set with no infinite low subset in either it or its complement. J. Symbolic Logic 66(3), 1371–1381 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  4. Downey, R.G., Hirschfeldt, D.R.: Algorithmic Randomness and Complexity. Theory and Applications of Computability. Springer, New York (2010)

    Book  MATH  Google Scholar 

  5. Franklin, J.: Hyperimmune-free degrees and Schnorr triviality. J. Symbolic Logic 73, 999–1008 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  6. Franklin, J., Stephan, F.: Schnorr trivial sets and truth-table reducibility. J. Symbolic Logic 75, 501–521 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  7. Hirschfeldt, D.R., Terwijn, S.A.: Limit computability and constructive measure. In: Computational Prospects of Infinity II: Presented Talks. Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore, vol. 15, pp. 131–141. World Scientific Publishing Co., Pte. Ltd., Hackensack (2008)

    Google Scholar 

  8. Khan, M., Miller, J.S.: Forcing with bushy trees. Manuscript (2014)

    Google Scholar 

  9. Kjos-Hanssen, B., Merkle, W., Stephan, F.: Kolmogorov complexity and the recursion theorem. Trans. Am. Math. Soc. 363(10), 5465–5480 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  10. Kjos-Hanssen, B., Nies, A., Stephan, F.: Lowness for the class of Schnorr random reals. SIAM J. Comput. 35(3), 647–657 (2005)

    Article  MathSciNet  Google Scholar 

  11. Kurtz, S.A.: Randomness and genericity in the degrees of unsolvability. Thesis (Ph.D.) - University of Illinois at Urbana-Champaign, ProQuest LLC, Ann Arbor, MI (1981)

    Google Scholar 

  12. Li, M., Vitányi, P.: An Introduction to Kolmogorov Complexity and Its Applications, 3rd edn. Springer, Heidelberg (2008)

    Book  MATH  Google Scholar 

  13. Miller, W., Martin, D.A.: The degrees of hyperimmune sets. Zeitschrift für Math. Logik und Grundlagen der Math. 14, 159–166 (1968)

    Article  MATH  MathSciNet  Google Scholar 

  14. Nies, A.: Computability and Randomness. Oxford Science Publications, New York (2009)

    Book  MATH  Google Scholar 

  15. Odifreddi, P.G.: Classical Recursion Theory. Studies in Logic and the Foundations of Mathematics, vol. 125. North-Holland Publishing Co., Amsterdam (1989)

    MATH  Google Scholar 

  16. Odifreddi, P.G.: Classical Recursion Theory II. Studies in Logic and the Foundations of Mathematics, vol. 143. North-Holland Publishing Co., Amsterdam (1999)

    MATH  Google Scholar 

  17. Rupprecht, N.: Relativized Schnorr tests with universal behavior. Arch. Math. Logic 49(5), 555–570 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  18. Soare, R.I.: Recursively enumerable sets and degrees. In: Feferman, S., Lerman, M., Magidor, M., Scedrov, A. (eds.) Perspectives in Mathematical Logic. Springer, Berlin (1987)

    Google Scholar 

  19. Terwijn, S.A.: On the quantitative structure of \(\Delta _2^0\). In: Berger, U., Osswald, H., Schuster, P. (eds.) Reuniting the Antipodes–Constructive and Nonstandard Views of the Continuum (Venice. 1999). Synthese Library, pp. 271–283. Kluwer Academic Publishers, Dordrecht (2001)

    Chapter  Google Scholar 

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Acknowledgements

The authors would like to thank George Barmpalias and Michiel van Lambalgen for discussions about Sect. 3 and André Nies and Benoit Monin for correspondence and thorough checking of Theorem 10.

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Authors and Affiliations

  1. Department of Mathematics, University of Hawaii at Manoa, Honolulu, HI, 96822, USA

    Bjørn Kjos-Hanssen

  2. Department of Mathematics and Department of Computer Science, National University of Singapore, Singapore, 119076, Republic of Singapore

    Frank Stephan

  3. Department of Mathematics, Radboud University Nijmegen, P.O. Box 9010, 6500 GL, Nijmegen, The Netherlands

    Sebastiaan A. Terwijn

Authors
  1. Bjørn Kjos-Hanssen
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  2. Frank Stephan
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  3. Sebastiaan A. Terwijn
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Corresponding author

Correspondence to Frank Stephan .

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Editors and Affiliations

  1. Department of Computer Science, Swansea University, Swansea, United Kingdom

    Arnold Beckmann

  2. University of Bucharest, Bucharest, Romania

    Victor Mitrana

  3. Sofia University, Sofia, Bulgaria

    Mariya Soskova

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Kjos-Hanssen, B., Stephan, F., Terwijn, S.A. (2015). Covering the Recursive Sets. In: Beckmann, A., Mitrana, V., Soskova, M. (eds) Evolving Computability. CiE 2015. Lecture Notes in Computer Science(), vol 9136. Springer, Cham. https://doi.org/10.1007/978-3-319-20028-6_5

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  • DOI: https://doi.org/10.1007/978-3-319-20028-6_5

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  • Print ISBN: 978-3-319-20027-9

  • Online ISBN: 978-3-319-20028-6

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