On Kolmogorov Structure Functions

TL;DR

This paper explores the properties of Kolmogorov Structure Functions within Algorithmic Statistics, highlighting limitations in modeling physical strings and discussing implications for the Minimum Description Length Principle.

Contribution

It demonstrates that strings with significant information about the halting sequence are purely mathematical and not found in nature, questioning the applicability of Algorithmic Statistics to real-world data.

Findings

Strings with low mutual information have flat structure functions

Strings with non-negligible information about the halting sequence are mathematical constructs

Two-part codes face limitations as shown in the MDL principle

Abstract

All strings with low mutual information with the halting sequence will have flat Kolmogorov Structure Functions, in the context of Algorithmic Statistics. Assuming the Independence Postulate, strings with non-negligible information with the halting sequence are purely mathematical constructions, and cannot be found in nature. Thus Algorithmic Statistics does not study strings in the physical world. This leads to the general thesis that two part codes require limitations as shown in the Minimum Description Length Principle. We also discuss issues with set-restricted Kolmogorov Structure Functions.