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Computability and Complexity pp 633–643Cite as

Effective Bi-immunity and Randomness

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10010))

Abstract

We study the relationship between randomness and effective bi-immunity. Greenberg and Miller have shown that for any oracle X, there are arbitrarily slow-growing \(\mathrm {DNR}\) functions relative to X that compute no Martin-Löf random set. We show that the same holds when Martin-Löf randomness is replaced with effective bi-immunity. It follows that there are sequences of effective Hausdorff dimension 1 that compute no effectively bi-immune set.

We also establish an important difference between the two properties. The class Low(MLR, EBI) of oracles relative to which every Martin-Löf random is effectively bi-immune contains the jump-traceable sets, and is therefore of cardinality continuum.

This work was partially supported by a grant from the Simons Foundation (#315188 to Bjørn Kjos-Hanssen). This material is based upon work supported by the National Science Foundation under Grant No. 1545707.

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Notes

  1. 1.

    A simple set is an r.e. set whose complement is immune.

  2. 2.

    We use the recursion theorem here.

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Acknowledgements

The authors would like to thank Uri Andrews, Daniel Turetsky, Linda Westrick, Rohit Nagpal, and Ashutosh Kumar for helpful discussions.

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Authors and Affiliations

  1. Department of Mathematics, University of Hawai‘i at Mānoa, Honolulu, Hawaii, 96822, USA

    Achilles A. Beros, Mushfeq Khan & Bjørn Kjos-Hanssen

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  1. Achilles A. Beros
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  2. Mushfeq Khan
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  3. Bjørn Kjos-Hanssen
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Corresponding author

Correspondence to Bjørn Kjos-Hanssen .

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Editors and Affiliations

  1. Victoria University of Wellington, Wellington, New Zealand

    Adam Day

  2. University of Bergen, Bergen, Norway

    Michael Fellows

  3. Victoria University of Wellington, Wellington, New Zealand

    Noam Greenberg

  4. University of Auckland, Auckland, New Zealand

    Bakhadyr Khoussainov

  5. Massey University, Auckland, New Zealand

    Alexander Melnikov

  6. University of Bergen, Bergen, Norway

    Frances Rosamond

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Beros, A.A., Khan, M., Kjos-Hanssen, B. (2017). Effective Bi-immunity and Randomness. In: Day, A., Fellows, M., Greenberg, N., Khoussainov, B., Melnikov, A., Rosamond, F. (eds) Computability and Complexity. Lecture Notes in Computer Science(), vol 10010. Springer, Cham. https://doi.org/10.1007/978-3-319-50062-1_38

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  • DOI: https://doi.org/10.1007/978-3-319-50062-1_38

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-50061-4

  • Online ISBN: 978-3-319-50062-1

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