The Kolmogorov Birthday Paradox

TL;DR

This paper establishes a Kolmogorov complexity version of the birthday paradox, demonstrating that large random string sets contain pairs with low conditional complexity, unless they are exotic with high mutual information with the halting sequence.

Contribution

It introduces a novel Kolmogorov complexity-based birthday paradox and characterizes exotic sets with high mutual information with the halting sequence.

Findings

Large random subsets contain pairs with low conditional Kolmogorov complexity

Exotic sets have high mutual information with the halting sequence

Minimum conditional complexity within non-exotic sets is very low

Abstract

We prove a Kolmogorov complexity variant of the birthday paradox. Sufficiently sized random subsets of strings are guaranteed to have two members x and y with low K(x/y). To prove this, we first show that the minimum conditional Kolmogorov complexity between members of finite sets is very low if they are not exotic. Exotic sets have high mutual information with the halting sequence.