Adjusted Kolmogorov Complexity of Binary Words with Empirical Entropy Normalization

TL;DR

This paper introduces an entropy-normalized Kolmogorov complexity measure for binary words that accounts for symbol distribution, providing a more intrinsic measure of complexity and linking it to randomness and structure in data.

Contribution

It proposes a novel complexity measure dividing Kolmogorov complexity by empirical entropy, enabling better analysis of randomness and structure in binary sequences.

Findings

Adjusted complexity grows linearly for Martin L"of random sequences

Normalized complexity converges to one for random sequences under certain measures

Framework connects Kolmogorov complexity, empirical entropy, and randomness

Abstract

Kolmogorov complexity of a finite binary word reflects both algorithmic structure and the empirical distribution of symbols appearing in the word. Words with symbol frequencies far from one half have smaller combinatorial richness and therefore appear less complex under the standard definition. In this paper an entropy-normalized complexity measure is introduced that divides the Kolmogorov complexity of a word by the empirical entropy of its observed distribution of zeros and ones. This adjustment isolates intrinsic descriptive complexity from the purely combinatorial effect of symbol imbalance. For Martin L\"of random sequences under constructive exchangeable measures, the adjusted complexity grows linearly and converges to one. A pathological construction shows that regularity of the underlying measure is essential. The proposed framework connects Kolmogorov complexity, empirical entropy, and randomness in a natural manner and suggests applications in randomness testing and in the analysis of structured binary data.