# On Resource-bounded versions of the van Lambalgen theorem

## TL;DR

This paper investigates how the classical van Lambalgen theorem, which relates the randomness of sequences, extends to resource-bounded settings, revealing both its validity and failure depending on the notion of time-bounded randomness.

## Contribution

It proves the classical van Lambalgen theorem using martingales and Kolmogorov complexity, and demonstrates the failure of the theorem in resource-bounded contexts with counterexamples.

## Key findings

- Classical van Lambalgen theorem holds with martingales and Kolmogorov complexity.
- The theorem fails in resource-bounded settings for certain notions of time-bounded randomness.
- Failure modes depend on the specific notion of resource-bounded randomness used.

## Abstract

The van Lambalgen theorem is a surprising result in algorithmic information theory concerning the symmetry of relative randomness. It establishes that for any pair of infinite sequences $A$ and $B$, $B$ is Martin-L\"of random and $A$ is Martin-L\"of random relative to $B$ if and only if the interleaved sequence $A \uplus B$ is Martin-L\"of random. This implies that $A$ is relative random to $B$ if and only if $B$ is random relative to $A$ \cite{vanLambalgen}, \cite{Nies09}, \cite{HirschfeldtBook}. This paper studies the validity of this phenomenon for different notions of time-bounded relative randomness. We prove the classical van Lambalgen theorem using martingales and Kolmogorov compressibility. We establish the failure of relative randomness in these settings, for both time-bounded martingales and time-bounded Kolmogorov complexity. We adapt our classical proofs when applicable to the time-bounded setting, and construct counterexamples when they fail. The mode of failure of the theorem may depend on the notion of time-bounded randomness.