Depth, Highness and DNR Degrees

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