On the logical depth function

TL;DR

This paper explores the properties of logical depth in binary strings, demonstrating the relationship between two definitions, the existence of strings with large depth gaps, and bounds related to the Busy Beaver function.

Contribution

It provides a new proof relating two definitions of logical depth and analyzes the bounds of depth gaps, connecting them to the Busy Beaver function.

Findings

Two definitions of logical depth are closely related.

Existence of strings with uncomputable to computable depth transitions.

Depth gaps are bounded by the Busy Beaver function.

Abstract

For a finite binary string $x$ its logical depth $d$ for significance $b$ is the shortest running time of a program for $x$ of length $K(x)+b$. There is another definition of logical depth. We give a new proof that the two versions are close. There is an infinite sequence of strings of consecutive lengths such that for every string there is a $b$ such that incrementing $b$ by 1 makes the associated depths go from incomputable to computable. The maximal gap between depths resulting from incrementing appropriate $b$'s by 1 is incomputable. The size of this gap is upper bounded by the Busy Beaver function. Both the upper and the lower bound hold for the depth with significance 0. As a consequence, the minimal computation time of the associated shortest programs rises faster than any computable function but not so fast as the Busy Beaver function.