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Schnorr trivial reals: a construction

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Abstract

A real is Martin-Löf (Schnorr) random if it does not belong to any effectively presented null \({\Sigma^0_1}\) (recursive) class of reals. Although these randomness notions are very closely related, the set of Turing degrees containing reals that are K-trivial has very different properties from the set of Turing degrees that are Schnorr trivial. Nies proved in (Adv Math 197(1):274–305, 2005) that all K-trivial reals are low. In this paper, we prove that if \({{\bf h'} \geq_T {\bf 0''}}\) , then h contains a Schnorr trivial real. Since this concept appears to separate computational complexity from computational strength, it suggests that Schnorr trivial reals should be considered in a structure other than the Turing degrees.

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Authors and Affiliations

  1. Department of Mathematics, National University of Singapore, 2, Science Drive 2, Singapore, 117543, Singapore

    Johanna N. Y. Franklin

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  1. Johanna N. Y. Franklin
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Correspondence to Johanna N. Y. Franklin.

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This material is based upon work supported under a National Science Foundation Graduate Research Fellowship and appears in the author’s Ph.D. thesis. A preliminary version of this paper appeared in Electronic Notes in Theoretical Computer Science

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Franklin, J.N.Y. Schnorr trivial reals: a construction. Arch. Math. Logic 46, 665–678 (2008). https://doi.org/10.1007/s00153-007-0061-3

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