Polynomial Depth, Highness and Lowness for E

TL;DR

This paper introduces a new notion of polylogarithmic depth based on time-bounded Kolmogorov complexity, exploring its properties and relationships with highness, lowness, and complexity classes like NP and E.

Contribution

It defines polylog depth in complexity theory, establishes its fundamental properties, and connects it with highness, lowness, and the structure of NP and E classes.

Findings

Polylog depth satisfies key logical depth properties.

NP contains polylog deep sets if it does not have p-measure zero.

High sets for E contain polylog deep sets; low(E,EXP) sets can also be polylog deep.

Abstract

We study the relations between the notions of highness, lowness and logical depth in the setting of complexity theory. We introduce a new notion of polylog depth based on time bounded Kolmogorov complexity. We show polylog depth satisfies all basic logical depth properties, namely sets in P are not polylog deep, sets with (time bounded)-Kolmogorov complexity greater than polylog are not polylog deep, and only polylog deep sets can polynomially Turing compute a polylog deep set. We prove that if NP does not have p-measure zero, then NP contains polylog deep sets. We show that every high set for E contains a polylog deep set in its polynomial Turing degree, and that there exist low(E,EXP) polylog deep sets. Keywords: algorithmic information theory; Kolmogorov complexity; Bennett logical depth.