We give two new characterizations of -triviality. We show that if for all such that is -random, is -random, then is -trivial. The other direction was proved by Stephan and Yu, giving us the first titular characterization of -triviality and answering a question of Yu. We also prove that if is -trivial, then for all such that is -random, . This answers a question of Merkle and Yu. The other direction is immediate, so we have the second characterization of -triviality.
The proof of the first characterization uses a new cupping result. We prove that if , then for every set there is a -random set such that is computable from .
"Two More Characterizations of K-Triviality." Notre Dame J. Formal Logic 59 (2) 189 - 195, 2018. https://doi.org/10.1215/00294527-2017-0021