Depth as Randomness Deficiency

TL;DR

This paper explores the concept of depth in objects through the lens of Kolmogorov complexity, establishing a unified framework that relates logical and computational depth to randomness deficiency, and extends the notion to infinite sequences.

Contribution

It unifies various depth notions by linking them to randomness deficiency and introduces the concept of super deep sequences for infinite strings.

Findings

Established quantitative relations between logical and computational depth.

Unified different depth notions via randomness deficiency.

Introduced super deep sequences and related them to existing approaches.

Abstract

Depth of an object concerns a tradeoff between computation time and excess of program length over the shortest program length required to obtain the object. It gives an unconditional lower bound on the computation time from a given program in absence of auxiliary information. Variants known as logical depth and computational depth are expressed in Kolmogorov complexity theory. We derive quantitative relation between logical depth and computational depth and unify the different depth notions by relating them to A. Kolmogorov and L. Levin's fruitful notion of randomness deficiency. Subsequently, we revisit the computational depth of infinite strings, introducing the notion of super deep sequences and relate it with other approaches.