Reductions to the Set of Random Strings: The Resource-Bounded Case

  • Eric Allender
  • Harry Buhrman
  • Luke Friedman
  • Bruno Loff
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7464)


This paper is motivated by a conjecture [1,5] that BPP can be characterized in terms of polynomial-time nonadaptive reductions to the set of Kolmogorov-random strings. In this paper we show that an approach laid out in [5] to settle this conjecture cannot succeed without significant alteration, but that it does bear fruit if we consider time-bounded Kolmogorov complexity instead.

We show that if a set A is reducible in polynomial time to the set of time-t-bounded Kolmogorov-random strings (for all large enough time bounds t), then A is in P/poly, and that if in addition such a reduction exists for any universal Turing machine one uses in the definition of Kolmogorov complexity, then A is in PSPACE.


Turing Machine Random Oracle Winning Strategy Kolmogorov Complexity Random String 
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  1. 1.
    Allender, E.: Curiouser and Curiouser: The Link between Incompressibility and Complexity. In: Cooper, S.B., Dawar, A., Löwe, B. (eds.) CiE 2012. LNCS, vol. 7318, pp. 11–16. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  2. 2.
    Allender, E., Buhrman, H., Friedman, L., Loff, B.: Reductions to the set of random strings:the resource-bounded case. Technical Report TR12-054, ECCC (2012)Google Scholar
  3. 3.
    Allender, E., Buhrman, H., Koucký, M.: What can be efficiently reduced to the Kolmogorov-random strings? Annals of Pure and Applied Logic 138, 2–19 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Allender, E., Buhrman, H., Koucký, M., van Melkebeek, D., Ronneburger, D.: Power from random strings. SIAM Journal on Computing 35, 1467–1493 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Allender, E., Davie, G., Friedman, L., Hopkins, S.B., Tzameret, I.: Kolmogorov complexity, circuits, and the strength of formal theories of arithmetic. Technical Report TR12-028, ECCC (2012) (submitted for publication)Google Scholar
  6. 6.
    Allender, E., Friedman, L., Gasarch, W.: Limits on the computational power of random strings. Information and Computation (to appear, 2012); special issue on ICALP 2011, See also ECCC TR10-139Google Scholar
  7. 7.
    Balcázar, J.L., Días, J., Gabarró, J.: Structural Complexity I. Springer (1988)Google Scholar
  8. 8.
    Buhrman, H., Fortnow, L., Koucký, M., Loff, B.: Derandomizing from random strings. In: 25th IEEE Conference on Computational Complexity (CCC), pp. 58–63. IEEE (2010)Google Scholar
  9. 9.
    Buhrman, H., Fortnow, L., Newman, I., Vereshchagin, N.K.: Increasing Kolmogorov Complexity. In: Diekert, V., Durand, B. (eds.) STACS 2005. LNCS, vol. 3404, pp. 412–421. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  10. 10.
    Buhrman, H., Mayordomo, E.: An excursion to the Kolmogorov random strings. J. Comput. Syst. Sci. 54(3), 393–399 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Juedes, D.W., Lutz, J.H.: Modeling time-bounded prefix Kolmogorov complexity. Theory of Computing Systems 33(2), 111–123 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Li, M., Vitanyi, P.: Introduction to Kolmogorov Complexity and its Applications, 3rd edn. Springer (2008)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Eric Allender
    • 1
  • Harry Buhrman
    • 2
  • Luke Friedman
    • 1
  • Bruno Loff
    • 3
  1. 1.Department of Computer ScienceRutgers UniversityPiscatawayUSA
  2. 2.CWI and University of AmsterdamThe Netherlands
  3. 3.CWIThe Netherlands

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