Additivity of on-line decision complexity is violated by a linear term in the length of a binary string

TL;DR

The paper demonstrates that the additivity property of on-line decision complexity does not hold when a linear term in string length is involved, revealing limitations in the complexity measure.

Contribution

It provides a counterexample showing the violation of additivity in on-line decision complexity with a linear term in string length.

Findings

Existence of infinitely many binary strings violating additivity

On-line decision complexity can exceed Kolmogorov complexity by a linear term

Challenges assumptions about the additive nature of decision complexity

Abstract

We show that there are infinitely many binary strings z, such that the sum of the on-line decision complexity of predicting the even bits of z given the previous uneven bits, and the decision complexity of predicting the uneven bits given the previous event bits, exceeds the Kolmogorov complexity of z by a linear term in the length of z.