Space-Bounded Kolmogorov Extractors

  • Daniil Musatov
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7353)

Abstract

An extractor is a function that receives some randomness and either “improves” it or produces “new” randomness. There are statistical and algorithmical specifications of this notion. We study an algorithmical one called Kolmogorov extractors and modify it to resource-bounded version of Kolmogorov complexity. Following Zimand we prove the existence of such objects with certain parameters. The utilized technique is “naive” derandomization: we replace random constructions employed by Zimand by pseudo-random ones obtained by Nisan-Wigderson generator.

Keywords

Marked Cell Ordinal Number Kolmogorov Complexity Good Seed Balance Property 
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.
    Ajtai, M.: Approximate counting with uniform constant-depth circuits. In: Advances in Computational Complexity Theory, pp. 1–20. American Mathematical Society, Providence (1993)Google Scholar
  2. 2.
    Fortnow, L., Hitchcock, J., Pavan, A., Vinodchandran, N.V., Wang, F.: Extracting Kolmogorov complexity with applications to dimension zero-one laws. Information and Computation 209(4), 627–636 (2011); Preliminary version appeared in Proceedings of the 33rd International Colloquium on Automata, Languages, and Programming. LNCS, vol. 4051, pp. 335–345 (2006)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Hitchcock, J., Pavan, A., Vinodchandran, N.: Kolmogorov complexity in randomness extraction. Electronic Colloquium on Computational Complexity (ECCC) 16, 71 (2009)Google Scholar
  4. 4.
    Musatov, D.: Improving the Space-Bounded Version of Muchnik’s Conditional Complexity Theorem via “Naive” Derandomization. In: Kulikov, A., Vereshchagin, N. (eds.) CSR 2011. LNCS, vol. 6651, pp. 64–76. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  5. 5.
    Nisan, N., Wigderson, A.: Hardness vs. Randomness. Journal of Computer and System Sciences 49(2), 149–167 (1994)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Romashchenko, A.: Pseudo-random Graphs and Bit Probe Schemes with One-Sided Error. In: Kulikov, A., Vereshchagin, N. (eds.) CSR 2011. LNCS, vol. 6651, pp. 50–63. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  7. 7.
    Shaltiel, R.: An Introduction to Randomness Extractors. In: Aceto, L., Henzinger, M., Sgall, J. (eds.) ICALP 2011, Part II. LNCS, vol. 6756, pp. 21–41. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  8. 8.
    Viola, E.: Randomness buys depth for approximate counting. In: Proceedings of IEEE FOCS 2011, pp. 230–239 (2011)Google Scholar
  9. 9.
    Zimand, M.: Two Sources Are Better Than One for Increasing the Kolmogorov Complexity of Infinite Sequences. In: Hirsch, E.A., Razborov, A.A., Semenov, A., Slissenko, A. (eds.) CSR 2008. LNCS, vol. 5010, pp. 326–338. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  10. 10.
    Zimand, M.: Extracting the Kolmogorov complexity of strings and sequences from sources with limited independence. In: Proceedings 26th STACS, pp. 697–708 (2009)Google Scholar
  11. 11.
    Zimand, M.: Impossibility of Independence Amplification in Kolmogorov Complexity Theory. In: Hliněný, P., Kučera, A. (eds.) MFCS 2010. LNCS, vol. 6281, pp. 701–712. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  12. 12.
    Zimand, M.: Possibilities and impossibilities in Kolmogorov complexity extraction. Sigact News 41(4), 74–94 (2010)MathSciNetGoogle Scholar
  13. 13.
    Zimand, M.: Symmetry of information and bounds on nonuniform randomness extraction via Kolmogorov extractors. In: Proceedings of 26th IEEE Conference in Computational Complexity, pp. 148–156 (2011)Google Scholar
  14. 14.
    Zimand, M.: On the Optimal Compression of Sets in PSPACE. In: Owe, O., Steffen, M., Telle, J.A. (eds.) FCT 2011. LNCS, vol. 6914, pp. 65–77. Springer, Heidelberg (2011)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Daniil Musatov
    • 1
    • 2
  1. 1.Moscow Institute for Physics and TechnologyRussia
  2. 2.Branch for Theoretical and Applied ResearchYandex LLCRussia

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