On the polynomial depth of various sets of random strings

TL;DR

This paper introduces a new polynomial depth measure called monotone poly depth, based on polynomial monotone Kolmogorov complexity, and demonstrates its properties and the depth of certain random string sets.

Contribution

It defines monotone poly depth, proves its desirable properties, and shows that sets of Levin-random and Kolmogorov-random strings are monotone poly deep.

Findings

Monotone poly depth satisfies all key properties of a depth measure.

Both Levin-random and Kolmogorov-random string sets are proven to be monotone poly deep.

Abstract

This paper proposes new notions of polynomial depth (called monotone poly depth), based on a polynomial version of monotone Kolmogorov complexity. We show that monotone poly depth satisfies all desirable properties of depth notions i.e., both trivial and random sequences are not monotone poly deep, monotone poly depth satisfies the slow growth law i.e., no simple process can transform a non deep sequence into a deep one, and monotone poly deep sequences exist (unconditionally). We give two natural examples of deep sets, by showing that both the set of Levin-random strings and the set of Kolmogorov random strings are monotone poly deep.