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)-deep K , O(1)-deep C , order-deep K and order-deep C sequences. Our main results are that Martin-Löf 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 .

Limit-depth and DNR degrees

“…Bennett's original notion [1] considered O (1) terms instead of order functions. Several authors have considered different order functions (see [6] for a summary) and as seen in [7]…”

“…In [7] Moser and Stephan proved that every Bennett deep set is either high or DNR. Here we show limit-deep sets share a similar property, namely every limit-deep set has DNR wtt-degree.…”

“…It was shown in [7] that every Bennett deep set is high. We do not know if a direct analogue holds for limit-depth.…”

Polylog depth, highness and lowness for E

“…A sequence is Bennett deep [5] if every computable approximation of the Kolmogorov complexity of its initial segments satisfies that the difference between the approximation and the actual value of the Kolmogorov complexity of the initial segments dominates every constant function. This difference is called the depth magnitude of the sequence [16].…”

“…Moser and Stephan [16] studied the differences in computational power of sequences of different depth magnitudes; within the context of computability theory. They related logical depth to standard computability notions (e.g.…”