Archive for Mathematical Logic

, Volume 46, Issue 7–8, pp 665–678 | Cite as

Schnorr trivial reals: a construction

Article

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.

Keywords

Randomness Triviality Schnorr trivial 

Mathematical Subject Classification (2000)

03D15 

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Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Department of MathematicsNational University of SingaporeSingaporeSingapore

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