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Truth‐table Schnorr randomness and truth‐table reducible randomness
Author(s) -
Miyabe Kenshi
Publication year - 2011
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/malq.200910128
Subject(s) - randomness , mathematics , truth table , table (database) , discrete mathematics , computer science , algorithm , statistics , data mining
Abstract Schnorr randomness and computable 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 in 6 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. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim