Depth, Highness and DNR degrees

Abstract

We study Bennett deep sequences in the context of recursion theory; in particular we investigate the notions of \(O(1)\text {-deep}_K\), \(O(1)\text {-deep}_C\), order\(\text {-deep}_K\) and order\(\text {-deep}_C\) sequences. Our main results are that Martin-Lof random sets are not order\(\text {-deep}_C\), that every many-one degree contains a set which is not \(O(1)\text {-deep}_C\), that \(O(1)\text {-deep}_C\) sets and order\(\text {-deep}_K\) sets have high or DNR Turing degree and that no K-trival set is \(O(1)\text {-deep}_K\).