Random noise increases Kolmogorov complexity and Hausdorff dimension

TL;DR

This paper investigates how randomly flipping bits in a binary string affects its Kolmogorov complexity and Hausdorff dimension, providing bounds and a self-contained proof using combinatorial and probabilistic methods.

Contribution

It establishes bounds on the increase in complexity caused by random bit flips and offers a self-contained, simplified proof of the probabilistic technique involved.

Findings

Random bit flips increase Kolmogorov complexity linearly with high probability.

The increase in complexity depends on the original string's complexity.

Exact lower and upper bounds for the complexity increase are provided.

Abstract

Consider a binary string $x$ of length $n$ whose Kolmogorov complexity is $\alpha n$ for some $\alpha<1$. We want to increase the complexity of $x$ by changing a small fraction of bits in $x$. This is always possible: Buhrman, Fortnow, Newman and Vereshchagin (2005) showed that the increase can be at least $\delta n$ for large $n$ (where $\delta$ is some positive number that depends on $\alpha$ and the allowed fraction of changed bits). We consider a related question: what happens with the complexity of $x$ when we randomly change a small fraction of the bits (changing each bit independently with some probability $\tau$)? It turns out that a linear increase in complexity happens with high probability, but this increase is smaller than in the case of arbitrary change. We note that the amount of the increase depends on $x$ (strings of the same complexity could behave differently), and give an exact lower and upper bounds for this increase (with $o(n)$ precision). The proof uses the combinatorial and probabilistic technique that goes back to Ahlswede, G\'acs and K\"orner (1976). For the reader's convenience (and also because we need a slightly stronger statement) we provide a simplified exposition of this technique, so the paper is self-contained.