Compactness arguments with effectively closed sets for the study of relative randomness.

Compactness arguments with effectively closed sets for the study of relative randomness.

G. Barmpalias

Summary

We present a variety of compactness arguments with effectively closed sets which yield results about relative randomness, and in particular properties of the LR degrees. Recall that two sets A, B have the same LR degree if Martin-Löf randomness relative to A coincides with Martin-Löf randomness relative to B. It is remarkable that in some cases, these arguments currently seem to be the only way to prove certain facts about the LR degrees. Hence they seem to play a more important role than in the context of the Turing degrees, where they were originally applied by Jockusch and Soare in their study of effectively closed sets and degrees of theories.