**News**

Mar 19, 2011. Added citation information

Mar 7, 2011. Published online

May 18, 2010. The paper “Truth-table Schnorr randomness and truth-table reducible randomness” is accepted by Mathematical Logic Quarterly on May 18, 2010.

**Title**

Truth-table Schnorr randomness and truth-table reducible randomness

**Type**

Fullpaper

**Journal**

Mathematical Logic Quarterly 57(3):323-338, 2011

DOI 10.1002/malq.200910128

Journal Page

**Download**

preprint

Japanese summary

**Abstract**

Schnorr randomness and computably randomness are natural concepts of random sequences. However van Lambalgen’s Theorem fails for both randomnesses. In this paper we define truth-table Schnorr randomness (defined by Franklin and Stephan too only by martingales) and truth-table reducible randomness, for which we prove that van Lambalgen’s Theorem holds. We also show that the classes of truth-table Schnorr random reals relative to a high set contain reals Turing equivalent to the high set. It follows that each high Schnorr random real is half of a real for which van Lambalgen’s Theorem fails. Moreover we establish the coincidence between triviality and lowness notions for truth-table Schnorr randomness.

**Cited by**

@article{franklin2009van,

title={{van Lambalgen’s Theorem and high degrees}},

author={Franklin, J.N.Y. and Stephan, F.},

year={2009},

publisher={Submitted}

}

@misc{bienvenucharacterizing,

title={Characterizing lowness for Demuth randomness},

author={Bienvenu, L. and Downey, R. and Greenberg, N. and Nies, A. and Turetsky, D.},

publisher={Submitted}

}