Van Lambalgen's Theorem and High Degrees

We show that van Lambalgen’s Theorem fails with respect to recursive randomness and Schnorr randomness for some real in every high degree and provide a full characterization of the Turing degrees for which van Lambalgen’s Theorem can fail with respect to Kurtz randomness. However, we also show that there is a recursively random real that is not Martin-Löf random for which van Lambalgen’s Theorem holds with respect to recursive randomness.

[1]  Kenshi Miyabe,et al.  Truth‐table Schnorr randomness and truth‐table reducible randomness , 2011, Math. Log. Q..

[2]  A. Kucera Measure, Π10-classes and complete extensions of PA , 1985 .

[3]  Joseph S. Miller,et al.  Uniform almost everywhere domination , 2005, Journal of Symbolic Logic.

[4]  Liang Yu When van Lambalgen’s Theorem fails , 2006 .

[5]  Rodney G. Downey,et al.  Algorithmic Randomness and Complexity , 2010, Theory and Applications of Computability.

[6]  Stefan Kuhr,et al.  Department of Mathematics and Computer Science , 2002 .

[7]  A. Nies Computability and randomness , 2009 .

[8]  Max L. Warshauer,et al.  Lecture Notes in Mathematics , 2001 .

[9]  Johanna N. Y. Franklin,et al.  Relativizations of randomness and genericity notions , 2011 .

[10]  P. Odifreddi Classical recursion theory , 1989 .

[11]  R. Soare Recursively enumerable sets and degrees , 1987 .

[12]  Michiel van Lambalgen,et al.  The Axiomatization of Randomness , 1990, J. Symb. Log..

[13]  Wolfgang Merkle,et al.  On the construction of effectively random sets , 2004, J. Symb. Log..

[14]  Guohua Wu,et al.  Anti-Complex Sets and Reducibilities with Tiny Use , 2011, The Journal of Symbolic Logic.

[15]  Per Martin-Löf,et al.  The Definition of Random Sequences , 1966, Inf. Control..

[16]  André Nies,et al.  Randomness, relativization and Turing degrees , 2005, J. Symb. Log..

[17]  C. Schnorr Zufälligkeit und Wahrscheinlichkeit , 1971 .

[18]  S. Barry Cooper,et al.  Minimal degrees and the jump operator , 1973, Journal of Symbolic Logic.

[19]  Johanna N. Y. Franklin,et al.  Schnorr trivial sets and truth-table reducibility , 2010, The Journal of Symbolic Logic.

[20]  André Nies,et al.  Kolmogorov-Loveland randomness and stochasticity , 2005, Ann. Pure Appl. Log..