Feasible Depth

TL;DR

This paper proposes two new complexity-theoretic versions of Bennett's logical depth, demonstrating that deep sequences are hard to generate from shallow ones and showing the depth of certain languages within complexity classes.

Contribution

It introduces finite-state and polynomial-time formulations of logical depth, establishing their properties and depth characteristics of specific languages in complexity theory.

Findings

Trivial and random sequences are shallow in both formulations.

A slow growth law prevents easy creation of deep sequences from shallow ones.

Certain languages in E are proven to be polynomial-time deep.

Abstract

This paper introduces two complexity-theoretic formulations of Bennett's logical depth: finite-state depth and polynomial-time depth. It is shown that for both formulations, trivial and random infinite sequences are shallow, and a slow growth law holds, implying that deep sequences cannot be created easily from shallow sequences. Furthermore, the E analogue of the halting language is shown to be polynomial-time deep, by proving a more general result: every language to which a nonnegligible subset of E can be reduced in uniform exponential time is polynomial-time deep.