| This paper discusses several properties related to K-triviality, including superlowness and the LR reducibility. In the second chapter, we answer a question of Andre Nies: is there a promptly simple set which is not superlow cuppable? In the third chapter (which is joint work with Keng Meng Ng), we obtain a stronger version of promptness which implies superlow cuppability, and further investigate properties of this promptness notion. In particular, we show that there are K-trivial sets which are strongly prompt, and that any benign cost function admits a strongly prompt c.e. set which obeys it. In the fourth chapter, we examine the LR reducibility, which can be seen as a partial relativization of K-triviality. We show that given any pair of low sets, there is a low c.e. set which is ≥LR both members of the pair. |
