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

Abstract

We present a variety of compactness arguments with Π01 classes that 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-Lof randomness relative to A coincides with Martin-Lof 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 Π01 classes and degrees of theories.