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Schnorr trivial reals: a construction. (English) Zbl 1142.03020
Summary: 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. A. Nies proved in [Adv. Math. 197, No. 1, 274–305 (2005; Zbl 1141.03017)] that all $$K$$-trivial reals are low. In this paper, we prove that if $${{\mathbf h'} \geq_T {\mathbf 0''}}$$, 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.

##### MSC:
 03D15 Complexity of computation (including implicit computational complexity) 68Q30 Algorithmic information theory (Kolmogorov complexity, etc.)
##### Keywords:
Randomness; Triviality; Schnorr trivial
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