Monthly Archives: May 2011

Truth-table Schnorr randomness and truth-table reducible randomness

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

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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}
}

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