Advertisement

Dimension Characterizations of Complexity Classes

  • Xiaoyang Gu
  • Jack H. Lutz
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4162)

Abstract

We use derandomization to show that sequences of positive pspace-dimension – in fact, even positive Δ\(^{\rm p}_{\rm k}\)-dimension for suitable k – have, for many purposes, the full power of random oracles. For example, we show that, if S is any binary sequence whose Δ\(^{\rm p}_{\rm 3}\)-dimension is positive, then BPP ⊆ P S and, moreover, every BPP promise problem is P S -separable. We prove analogous results at higher levels of the polynomial-time hierarchy.

The dimension-almost-class of a complexity class \(\mathcal{C}\), denoted by dimalmost-\(\mathcal{C}\), is the class consisting of all problems A such that \(A \in \mathcal{C}^S\) for all but a Hausdorff dimension 0 set of oracles S. Our results yield several characterizations of complexity classes, such as BPP = dimalmost-P and AM = dimalmost-NP, that refine previously known results on almost-classes. They also yield results, such as Promise-BPP = almost-P-Sep = dimalmost-P-Sep, in which even the almost-class appears to be a new characterization.

Keywords

Complexity Class Binary Sequence SIAM Journal Random Oracle 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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Allender, E.: When worlds collide: Derandomization, lower bounds, and kolmogorov complexity. In: Hariharan, R., Mukund, M., Vinay, V. (eds.) FSTTCS 2001. LNCS, vol. 2245, pp. 1–15. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  2. 2.
    Allender, E., Buhrman, H., Koucký, M., van Melkebeek, D., Ronneburger, D.: Power from random strings. In: Proceedings of the 43rd Annual IEEE Symposium on Foundations of Computer Science, pp. 669–678 (2002); SIAM Journal on Computing (to appear)Google Scholar
  3. 3.
    Allender, E., Strauss, M.: Measure on small complexity classes with applications for BPP. In: Proceedings of the 35th Symposium on Foundations of Computer Science, pp. 807–818 (1994)Google Scholar
  4. 4.
    Balcázar, J.L., Díaz, J., Gabarró, J.: Structural Complexity I, 2nd edn. Springer, Berlin (1995)Google Scholar
  5. 5.
    Bennett, C.H., Gill, J.: Relative to a random oracle A, PA ≠ NPA ≠ co − NPA with probability 1. SIAM Journal on Computing 10, 96–113 (1981)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Buhrman, H., Fortnow, L.: One-sided versus two-sided randomness. In: Proceedings of the sixteenth Symposium on Theoretical Aspects of Computer Science, pp. 100–109 (1999)Google Scholar
  7. 7.
    Chomsky, N., Miller, G.A.: Finite state languages. Information and Control 1(2), 91–112 (1958)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Falconer, K.: Fractal Geometry: Mathematical Foundations and Applications, 2nd edn. Wiley, Chichester (2003)MATHCrossRefGoogle Scholar
  9. 9.
    Fortnow, L.: Comparing notions of full derandomization. In: Proceedings of the 16th IEEE Conference on Computational Complexity, pp. 28–34 (2001)Google Scholar
  10. 10.
    Grollman, J., Selman, A.: Complexity measures for public-key cryptosystems. SIAM J. Comput. 11, 309–335 (1988)CrossRefGoogle Scholar
  11. 11.
    Hausdorff, F.: Dimension und äusseres Mass. Mathematische Annalen 79, 157–179 (1919)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Hitchcock, J.M.: Effective fractal dimension: foundations and applications. PhD thesis, Iowa State University (2003)Google Scholar
  13. 13.
    Hitchcock, J.M.: Hausdorff dimension and oracle constructions. Theoretical Computer Science 355(3), 382–388 (2006)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Hitchcock, J.M., Lutz, J.H., Mayordomo, E.: The fractal geometry of complexity classes. SIGACT News 36(3), 24–38 (2005)CrossRefGoogle Scholar
  15. 15.
    Hitchcock, J.M., Vinodchandran, N.V.: Dimension, entropy rates, and compression. Journal of Computer and System Sciences 72(4), 760–782 (2006)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Impagliazzo, R., Wigderson, A.: P = BPP if E requires exponential circuits: Derandomizing the XOR lemma. In: Proceedings of the 29th Symposium on Theory of Computing, pp. 220–229 (1997)Google Scholar
  17. 17.
    Kuich, W.: On the entropy of context-free languages. Information and Control 16(2), 173–200 (1970)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Lutz, J.H.: A pseudorandom oracle characterization of BPP. SIAM Journal on Computing 22(5), 1075–1086 (1993)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Lutz, J.H.: Dimension in complexity classes. SIAM Journal on Computing 32, 1236–1259 (2003)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Lutz, J.H.: Effective fractal dimensions. Mathematical Logic Quarterly 51, 62–72 (2005)MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Martin-Löf, P.: The definition of random sequences. Information and Control 9, 602–619 (1966)CrossRefMathSciNetGoogle Scholar
  22. 22.
    Mayordomo, E.: Effective Hausdorff dimension. In: Proceedings of Foundations of the Formal Sciences III, pp. 171–186. Kluwer Academic Publishers, Dordrecht (2004)Google Scholar
  23. 23.
    Moser, P.: Relative to P promise-BPP equals APP. Technical Report TR01-68, Electronic Colloquium on Computational Complexity (2001)Google Scholar
  24. 24.
    Moser, P.: Random nondeterministic real functions and Arthur Merlin games. Technical Report TR02-006, ECCC (2002)Google Scholar
  25. 25.
    Nisan, N., Wigderson, A.: Hardness vs randomness. Journal of Computer and System Sciences 49, 149–167 (1994)MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Staiger, L.: Kolmogorov complexity and Hausdorff dimension. Information and Computation 103, 159–194 (1993)MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Staiger, L.: A tight upper bound on Kolmogorov complexity and uniformly optimal prediction. Theory of Computing Systems 31, 215–229 (1998)MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Wilson, C.B.: Relativized circuit complexity. Journal of Computer and System Sciences 31, 169–181 (1985)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Xiaoyang Gu
    • 1
  • Jack H. Lutz
    • 1
  1. 1.Department of Computer ScienceIowa State UniversityAmesUSA

Personalised recommendations