On Characterizations of Randomized Computation Using Plain Kolmogorov Complexity

  • Shuichi Hirahara
  • Akitoshi Kawamura
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8635)

Abstract

Allender, Friedman, and Gasarch recently proved an upper bound of pspace for the class DTTRK of decidable languages that are polynomial-time truth-table reducible to the set of prefix-free Kolmogorov-random strings regardless of the universal machine used in the definition of Kolmogorov complexity. It is conjectured that DTTRK in fact lies closer to its lower bound BPP established earlier by Buhrman, Fortnow, Koucký, and Loff. It is also conjectured that we have similar bounds for the analogous class DTTRC defined by plain Kolmogorov randomness. In this paper, we provide further evidence for these conjectures. First, we show that the time-bounded analogue of DTTRC sits between BPP and pspace ∩ P/poly. Next, we show that the class DTTRC, α obtained from DTTRC by imposing a restriction on the reduction lies between BPP and pspace. Finally, we show that the class P/R\(^{=log}_{c}\) obtained by further restricting the reduction to ask queries of logarithmic length lies between BPP and \(\Sigma^{p}_{2} \cap\) P/poly.

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References

  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. In: Rovan, B., Sassone, V., Widmayer, P. (eds.) MFCS 2012. LNCS, vol. 7464, pp. 88–99. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  3. 3.
    Allender, E., Friedman, L., Gasarch, W.: Limits on the computational power of random strings. Information and Computation 222, 80–92 (2013)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Arora, S., Barak, B.: Computational Complexity: A Modern Approach, 1st edn. Cambridge University Press (2009)Google Scholar
  5. 5.
    Buhrman, H., Fortnow, L., Koucký, M., Loff, B.: Derandomizing from random strings. In: Proceedings of the 25th Annual Conference on Computational Complexity, CCC 2010, pp. 58–63 (2010)Google Scholar
  6. 6.
    Impagliazzo, R., Wigderson, A.: P = BPP if E requires exponential circuits: Derandomizing the xor lemma. In: Proceedings of the 29th Annual ACM Symposium on Theory of Computing, STOC 1997, pp. 220–229 (1997)Google Scholar
  7. 7.
    Li, M., Vitányi, P.: An Introduction to Kolmogorov Complexity and Its Applications, 3rd edn. Springer Publishing Company (2008)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Shuichi Hirahara
    • 1
  • Akitoshi Kawamura
    • 1
  1. 1.The University of TokyoJapan

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