There is a deep 1-generic set

TL;DR

This paper demonstrates the existence of a deep 1-generic set, contrasting with the known shallow nature of weakly 2-generic sets, by analyzing Kolmogorov complexity properties.

Contribution

It proves that deep 1-generic sets exist, expanding understanding of the complexity hierarchy in computability theory.

Findings

Existence of deep 1-generic sets established

Weakly 2-generic sets are shallow, not deep

Deepness characterized via Kolmogorov complexity differences

Abstract

An infinite binary sequence is Bennett deep if, for any computable time bound, the difference between the time-bounded prefix-free Kolmogorov complexity and the prefix-free Kolmogorov complexity of its initial segments is eventually unbounded. It is known that weakly 2-generic sets are shallow, i.e. not deep. In this paper, we show that there is a deep 1-generic set.