Schnorr Trivial Reals: A construction

Authors:
Johanna N. Y. Franklin
Affiliations:
Group in Logic and the Methodology of Science, University of California, Berkeley, Berkeley, CA
Venue:
Electronic Notes in Theoretical Computer Science (ENTCS)
Year:
2007

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Abstract

A real is Martin-Lof (Schnorr) random if it does not belong to any effectively presented null @S"1^0 (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 [11] that all K-trivial reals are low. In this paper, we prove that if h is a high degree, then every degree a="Th 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.