Asymmetry of the Kolmogorov complexity of online predicting odd and even bits

TL;DR

This paper demonstrates that the symmetry of information does not extend to online Kolmogorov complexity, revealing asymmetries between odd and even bit complexities and their relation to the overall complexity.

Contribution

It introduces the concept of online Kolmogorov complexity for odd and even bits and shows their asymmetry, contrasting with classical symmetry of information.

Findings

Existence of strings with high odd and even complexity close to total complexity

Flipping odd and even bits reduces combined complexity to the original Kolmogorov complexity

Symmetry of information does not hold in online Kolmogorov complexity

Abstract

Symmetry of information states that $C(x) + C(y|x) = C(x,y) + O(\log C(x))$. We show that a similar relation for online Kolmogorov complexity does not hold. Let the even (online Kolmogorov) complexity of an n-bitstring $x_1x_2... x_n$ be the length of a shortest program that computes $x_2$ on input $x_1$, computes $x_4$ on input $x_1x_2x_3$, etc; and similar for odd complexity. We show that for all n there exist an n-bit x such that both odd and even complexity are almost as large as the Kolmogorov complexity of the whole string. Moreover, flipping odd and even bits to obtain a sequence $x_2x_1x_4x_3\ldots$, decreases the sum of odd and even complexity to $C(x)$.