Shannon Information and Kolmogorov Complexity

TL;DR

This paper systematically compares Shannon information theory and Kolmogorov complexity, highlighting their similarities, differences, and new relations, providing a comprehensive overview of their foundational concepts and connections.

Contribution

It offers the first comprehensive systematic comparison of Shannon information and Kolmogorov complexity, including new relations such as rate distortion theory versus Kolmogorov's structure function.

Findings

Comparison of Shannon entropy and Kolmogorov complexity

Relation of mutual information in both theories

Introduction of new relations like rate distortion versus structure function

Abstract

We compare the elementary theories of Shannon information and Kolmogorov complexity, the extent to which they have a common purpose, and where they are fundamentally different. We discuss and relate the basic notions of both theories: Shannon entropy versus Kolmogorov complexity, the relation of both to universal coding, Shannon mutual information versus Kolmogorov (`algorithmic') mutual information, probabilistic sufficient statistic versus algorithmic sufficient statistic (related to lossy compression in the Shannon theory versus meaningful information in the Kolmogorov theory), and rate distortion theory versus Kolmogorov's structure function. Part of the material has appeared in print before, scattered through various publications, but this is the first comprehensive systematic comparison. The last mentioned relations are new.