Depth, Highness and DNR degrees - Moser, Philippe; Stephan, Frank (Discrete Mathematics and Theoretical Computer Science, 2017)
We study Bennett deep sequences in the context of recursion theory; in particular we investigate the notions of O(1)-[deep.sub.K], O(1)-[deep.sub.C], order-[deep.sub.K] and order-[deep.sub.C] sequences. Our main results are that Martin-Lof random sets are not order-[deep.sub.C], that every many-one degree contains a set which is not O(1)-[deep.sub.C], that O(1)-[deep.sub.C] sets and order-[deep.sub.K] sets have high or DNR Turing degree and that no K-trival set is O(1)-[deep.sub.K]. Keywords: Bennett logical depth, Kolmogorov complexity, algorithmic randomness theory, computability and randomness.