Demuth randomness and computational complexity

Demuth tests generalize Martin-Lof tests (Gm)m∈N in that one can exchange the m-th component a computably bounded number of times. A set Z⊆N fails a Demuth test if Z is in infinitely many final versions of the Gm. If we only allow Demuth tests such that Gm⊇Gm+1 for each m, we have weak Demuth randomness. We show that a weakly Demuth random set can be high and Δ20, yet not superhigh. Next, any c.e. set Turing below a Demuth random set is strongly jump-traceable. We also prove a basis theorem for non-empty Π10 classes P. It extends the Jockusch–Soare basis theorem that some member of P is computably dominated. We use the result to show that some weakly 2-random set does not compute a 2-fixed point free function.

[1]  André Nies,et al.  Benign cost functions and lowness properties , 2011, The Journal of Symbolic Logic.

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

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

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

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

[6]  Stephen G. Simpson,et al.  Mass problems and almost everywhere domination , 2007, Math. Log. Q..

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

[8]  Antonín Kucera,et al.  An Alternative, Priority-Free, Solution to Post's Problem , 1996, MFCS.

[9]  G. E. Hardy,et al.  A Generalization of a Fixed Point Theorem of Reich , 1973, Canadian Mathematical Bulletin.

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

[11]  O. Demuth,et al.  Remarks on 1-genericity, semigenericity and related concepts , 1987 .

[12]  André Nies,et al.  Counting the Changes of Random D02 Sets , 2010, CiE.

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

[14]  André Nies,et al.  Lowness Properties and Approximations of the Jump , 2006, Electron. Notes Theor. Comput. Sci..

[15]  ROBERT IRVING SOARE,et al.  DEGREES OF MEMBERS OF Π\ CLASSES , 2012 .

[16]  Robert I. Soare,et al.  Degrees of members of Π10 classes , 1972 .

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

[18]  Noam Greenberg,et al.  Characterizing the strongly jump-traceable sets via randomness , 2011, 1109.6749.

[19]  Ludomir Newelski,et al.  Logic Colloquium 2005 (Lecture Notes in Logic) , 2007 .

[20]  E. Hille,et al.  On some generalizations of a theorem of A. Markoff , 1937 .

[21]  André Nies,et al.  Randomness notions and partial relativization , 2012 .

[22]  Péter Gács,et al.  Every Sequence Is Reducible to a Random One , 1986, Inf. Control..

[23]  Noam Greenberg,et al.  A random set which only computes strongly jump-traceable c.e. sets , 2011, The Journal of Symbolic Logic.

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

[25]  Donald A. Martin,et al.  The Degrees of Hyperimmune Sets , 1968 .