On Resource-Bounded Versions of the van Lambalgen Theorem

  • Diptarka Chakraborty
  • Satyadev Nandakumar
  • Himanshu Shukla
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10185)


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.


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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Diptarka Chakraborty
    • 1
  • Satyadev Nandakumar
    • 2
  • Himanshu Shukla
    • 2
  1. 1.Computer Science Institute of Charles UniversityPraha 1Czech Republic
  2. 2.Department of Computer Science and EngineeringIndian Institute of Technology KanpurKanpurIndia

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