On Logical Depth and the Running Time of Shortest Programs

TL;DR

This paper explores the properties of logical depth in strings, showing that the maximal gap in logical depth caused by small changes in parameters can grow faster than any computable function, bounded by the Busy Beaver function.

Contribution

It establishes new theorems relating different definitions of logical depth and demonstrates the uncomputability and extreme growth of logical depth gaps in strings.

Findings

Logical depth can grow faster than any computable function.

Incrementing parameter b by 1 can cause uncomputable increases in logical depth.

The maximal gap in logical depth growth is bounded by the Busy Beaver function.

Abstract

The logical depth with significance $b$ of a finite binary string $x$ is the shortest running time of a binary program for $x$ that can be compressed by at most $b$ bits. There is another definition of logical depth. We give two theorems about the quantitative relation between these versions: the first theorem concerns a variation of a known fact with a new proof, the second theorem and its proof are new. We select the above version of logical depth and show the following. There is an infinite sequence of strings of increasing length such that for each $j$ there is a $b$ such that the logical depth of the $j$th string as a function of $j$ is incomputable (it rises faster than any computable function) but with $b$ replaced by $b+1$ the resuling function is computable. Hence the maximal gap between the logical depths resulting from incrementing appropriate $b$'s by 1 rises faster than any computable function. All functions mentioned are upper bounded by the Busy Beaver function. Since for every string its logical depth is nonincreasing in $b$, the minimal computation time of the shortest programs for the sequence of strings as a function of $j$ rises faster than any computable function but not so fast as the Busy Beaver function.