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What Can be Efficiently Reduced to the K-Random Strings?

  • Eric Allender
  • Harry Buhrman
  • Michal Koucký
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2996)

Abstract

We investigate the question of whether one can characterize complexity classes (such as PSPACE or NEXP) in terms of efficient reducibility to the set of Kolmogorov-random strings R K . We show that this question cannot be posed without explicitly dealing with issues raised by the choice of universal machine in the definition of Kolmogorov complexity. Among other results, we show that although for every universal machine U, there are very complex sets that are \(\leq^{p}_{dtt}\)-reducible to R k ∪ , it is nonetheless true that P=REC \(\cap\bigcap\cup\{A:A\leq^{p}_{dtt} R_{k\cup}\}\). We also show for a broad class of reductions that the sets reducible to R K have small circuit complexity.

Keywords

Turing Machine Kolmogorov Complexity Random String Answer Sequence Universal Machine 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Eric Allender
    • 1
  • Harry Buhrman
    • 2
  • Michal Koucký
    • 3
  1. 1.Rutgers UniversityNew BrunswickUSA
  2. 2.CWI and University of AmsterdamAmsterdamNetherlands
  3. 3.McGill UniversityMontréalCanada

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