Low upper bounds of ideals

We show that there is a low T-upper bound for the class of ^-trivial sets, namely those which are weak from the point of view of algorithmic randomness. This result is a special case of a more general characterization of ideals in A® T-degrees for which there is a low T-upper bound.

[1]  Klaus Ambos-Spies,et al.  Randomness in Computability Theory , 2000 .

[2]  D. Scott.  Algebras of sets binumerable in complete extensions of arithmetic , 1962 .

[3]  A. Nies.  Lowness properties and randomness , 2005 .

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

[5]  C. Yates,et al.  ON THE DEGREES OF INDEX SETS , 2010 .

[6]  C. E. M. Yates,et al.  On the degrees of index sets. II , 1966 .

[7]  Robert W. Robinson,et al.  Interpolation and Embedding in the Recursively Enumerable Degrees , 1971 .

[8]  André Nies,et al.  Reals which Compute Little , 2002 .

[9]  Emil L. Post,et al.  The Upper Semi-Lattice of Degrees of Recursive Unsolvability , 1954 .

[10]  Sebastiaan Terwijn,et al.  Lowness for The Class of Random Sets , 1999, J. Symb. Log..

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

[12]  Antonín Kučera.  On the use of Diagonally Nonrecursive Functions , 1989 .

[13]  André Nies,et al.  Calibrating Randomness , 2006, Bull. Symb. Log..

[14]  Marcus Hutter,et al.  Algorithmic Information Theory , 1977, IBM J. Res. Dev..

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

[16]  Antonín Kučera,et al.  On Relative Randomness , 1993, Ann. Pure Appl. Log..

[17]  P. Odifreddi.  The theory of functions and sets of natural numbers , 1989 .

[18]  André Nies,et al.  Trivial Reals , 2002, CCA.

[19]  Stephen G. Simpson,et al.  Degrees of Unsolvability: A Survey of Results , 1977 .

[20]  André Nies,et al.  Using random sets as oracles , 2007 .

[21]  Anil Nerode,et al.  Reducibility orderings: Theories, definability and automorphisms☆ , 1980 .

[22]  R. Soare,et al.  Π⁰₁ classes and degrees of theories , 1972 .

[23]  Manuel Lerman,et al.  Degrees of Unsolvability: Local and Global Theory , 1983 .

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

[25]  André Nies,et al.  Randomness and Computability: Open Questions , 2006, Bulletin of Symbolic Logic.

[26]  Paul M. B. Vitányi,et al.  An Introduction to Kolmogorov Complexity and Its Applications , 1993, Graduate Texts in Computer Science.

[27]  Shirley Dex,et al.  JR 旅客販売総合システム（マルス）における運用及び管理について , 1991 .