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Limits on the Computational Power of Random Strings

  • Eric Allender
  • Luke Friedman
  • William Gasarch
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6755)

Abstract

How powerful is the set of random strings? What can one say about a set A that is efficiently reducible to R, the set of Kolmogorov-random strings? We present the first upper bound on the class of computable sets in \(\mbox{\rm P}^R\) and \(\mbox{\rm NP}^R\).

The two most widely-studied notions of Kolmogorov complexity are the “plain” complexity C(x) and “prefix” complexity K(x); this gives two ways to define the set “R”: R C and R K . (Of course, each different choice of universal Turing machine U in the definition of C and K yields another variant \({R_{{C}_U}}\) or \({R_{{K}_U}}\).) Previous work on the power of “R” (for any of these variants [1,2,9]) has shown
  • \(\mbox{\rm BPP} \subseteq \{A : A \mbox{$\leq^{\rm p}_{\it tt}$} R\}\).

  • \(\mbox{\rm PSPACE} \subseteq \mbox{\rm P}^R\).

  • \(\mbox{\rm NEXP} \subseteq \mbox{\rm NP}^R\).

Since these inclusions hold irrespective of low-level details of how “R” is defined, we have e.g.: \(\mbox{\rm NEXP} \subseteq \mbox{$\Delta^0_1$} \cap \bigcap_U \mbox{\rm NP}^{{R_{{K}_U}}}\). (\(\mbox{$\Delta^0_1$}\) is the class of computable sets.)
Our main contribution is to present the first upper bounds on the complexity of sets that are efficiently reducible to \({R_{{K}_U}}\). We show:
  • \(\mbox{\rm BPP} \subseteq \mbox{$\Delta^0_1$} \cap \bigcap_U \{A : A \mbox{$\leq^{\rm p}_{\it tt}$} {R_{{K}_U}}\}\subseteq \mbox{\rm PSPACE}\).

  • \(\mbox{\rm NEXP} \subseteq \mbox{$\Delta^0_1$} \cap \bigcap_U \mbox{\rm NP}^{{R_{{K}_U}}}\subseteq \mbox{\rm EXPSPACE}\).

Hence, in particular, \(\mbox{\rm PSPACE}\) is sandwiched between the class of sets Turing- and truth-table-reducible to R.

As a side-product, we obtain new insight into the limits of techniques for derandomization from uniform hardness assumptions.

Keywords

Boolean Function Turing Machine Complexity Class Winning Strategy Kolmogorov Complexity 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Allender, E., Buhrman, H., Koucký, M., van Melkebeek, D., Ronneburger, D.: Power from random strings. SIAM Journal on Computing 35, 1467–1493 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Allender, E., Friedman, L., Gasarch, W.: Limits on the computational power of random strings. Technical Report TR10-139, Electronic Colloquium on Computational Complexity (2010)Google Scholar
  4. 4.
    Allender, E., Koucký, M., Ronneburger, D., Roy, S.: The pervasive reach of resource-bounded Kolmogorov complexity in computational complexity theory. Journal of Computer and System Sciences 77, 14–40 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Babai, L., Fortnow, L., Nisan, N., Wigderson, A.: BPP has subexponential time simulations unless EXPTIME has publishable proofs. Computational Complexity 3, 307–318 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Book, R.V.: On languages reducible to algorithmically random languages. SIAM Journal on Computing 23(6), 1275–1282 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Book, R.V., Lutz, J., Wagner, K.W.: An observation on probability versus randomness with applications to complexity classes. Mathematical Systems Theory 27(3), 201–209 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Book, R.V., Mayordomo, E.: On the robustness of ALMOST-r. RAIRO Informatique Théorique et Applications 30(2), 123–133 (1996)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Buhrman, H., Fortnow, L., Koucky, M., Loff, B.: Derandomizing from random strings. In: 25th IEEE Conference on Computational Complexity (CCC), pp. 58–63. IEEE Computer Society Press, Los Alamitos (2010)Google Scholar
  10. 10.
    Gutfreund, D., Vadhan, S.P.: Limitations of hardness vs. Randomness under uniform reductions. In: Goel, A., Jansen, K., Rolim, J.D.P., Rubinfeld, R. (eds.) APPROX and RANDOM 2008. LNCS, vol. 5171, pp. 469–482. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  11. 11.
    Hitchcock, J.M.: Lower bounds for reducibility to the kolmogorov random strings. In: Ferreira, F., Löwe, B., Mayordomo, E., Mendes Gomes, L. (eds.) CiE 2010. LNCS, vol. 6158, pp. 195–200. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  12. 12.
    Impagliazzo, R., Wigderson, A.: Randomness vs. time: de-randomization under a uniform assumption. J. Comput. Syst. Sci. 63(4), 672–688 (2001)CrossRefzbMATHGoogle Scholar
  13. 13.
    Kabanets, V., Cai, J.-Y.: Circuit minimization problem. In: Proc. ACM Symp. on Theory of Computing (STOC), pp. 73–79 (2000)Google Scholar
  14. 14.
    Li, M., Vitanyi, P.: Introduction to Kolmogorov Complexity and its Applications, 3rd edn. Springer, Heidelberg (2008)CrossRefzbMATHGoogle Scholar
  15. 15.
    Muchnik, A.A., Positselsky, S.: Kolmogorov entropy in the context of computability theory. Theoretical Computer Science 271, 15–35 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Reingold, O., Trevisan, L., Vadhan, S.P.: Notions of reducibility between cryptographic primitives. In: Naor, M. (ed.) TCC 2004. LNCS, vol. 2951, pp. 1–20. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  17. 17.
    Trevisan, L., Vadhan, S.P.: Pseudorandomness and average-case complexity via uniform reductions. Computational Complexity 16(4), 331–364 (2007)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Eric Allender
    • 1
  • Luke Friedman
    • 1
  • William Gasarch
    • 2
  1. 1.Department of Computer ScienceRutgers UniversityPiscatawayUSA
  2. 2.Dept. of Computer ScienceUniversity of MarylandCollege ParkUSA

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