On Resource-Bounded Versions of the van Lambalgen Theorem
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-Löf random and A is Martin-Löf random relative to B if and only if the interleaved sequence \(A \uplus B\) is Martin-Löf random. This implies that A is relative random to B if and only if B is random relative to A [1, 2, 3]. 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.
The authors would like to acknowledge Jack Lutz, Manjul Gupta and Michal Koucký for helpful discussions.
- 3.Downey, R., Hirschfeldt, D.: Algorithmic randomness and complexity. Book Draft (2006)Google Scholar
- 9.Lutz, J.H.: Resource-bounded measure. In: Proceedings of the 13th IEEE Conference on Computational Complexity, pp. 236–248. IEEE Computer Society Press, New York (1998)Google Scholar