Muchnik Degrees and Medvedev Degrees of Randomness Notions

The main theme of this paper is computational power when a machine is allowed to access random sets. The computability depends on the randomness notions and we compare them by Muchnik and Medvedev degrees. The central question is whether, given a random oracle, one can compute a more random set. The main result is that, for each Turing functional, there exists a Schnorr random set whose output is not computably random.