Depth, Highness and DNR degrees

TL;DR

This paper explores the properties of Bennett deep sequences in recursion theory, revealing their relationships with randomness, computational degrees, and depth notions, and establishing several limitations and characterizations.

Contribution

It introduces and analyzes new depth notions for sequences, connecting them with randomness and Turing degrees, and proves several key limitations and properties.

Findings

Martin-Loef random sets are not order-deepC

Every many-one degree contains a set not O(1)-deepC

O(1)-deepC and order-deepK sets have high or DNR degrees

Abstract

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