Results on resource-bounded measure

  • Harry Buhrman
  • Stephen Fenner
  • Lance Fortnow
Session 3: Computational Complexity
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1256)


We construct an oracle relative to which NP has p-measure 0 but Dp has measure 1 in EXP. This gives a strong relativized negative answer to a question posed by Lutz [Lut96]. Secondly, we give strong evidence that BPP is small. We show that BPP has p-measure 0 unless EXP=MA and thus the polynomial-time hierarchy collapses. This contrasts with the work of Regan et. al. [RSC95], where it is shown that P/poly does not have p-measure 0 if exponentially strong pseudorandom generators exist.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [ASMZ96]
    K. Ambos-Spies, E. Mayordomo, and Xizhong Zheng. A comparison of weak completeness notions. In Proeceedings of Eleventh Annual Conference on Computational Complexity, pages 171–178, 1996.Google Scholar
  2. [BFNW93]
    L. Babai, L. Fortnow, N. Nisan, and A. Wigderson. BPP has subexponential simulations unless EXPTIME has publishable proofs. Computational Complexity, 3:307–318, 1993.CrossRefGoogle Scholar
  3. [BM89]
    László Babai and Shlomo Moran. Proving properties of interactive proofs by a generalized counting technique. Information and Computation, 82(2):185–197, August 1989.Google Scholar
  4. [BT94]
    Buhrman and Torenvliet. On the cutting edge of relativization: The resource bounded injury method. In Annual International Colloquium on Automata, Languages and Programming, pages 263–273, 1994.Google Scholar
  5. [FF95]
    S. Fenner and L. Fortnow. Beyond P NP =NEXP. In STACS 95, volume 900 of Lecture Notes in Computer Science, pages 619–627. Springer, 1995.Google Scholar
  6. [LM96]
    J. Lutz and E. Mayordomo. Cook versus Karp-Levin: Separating completeness notions if NP is not small. Theoretical Computer Science, 164(1–2):141–163, 1996.CrossRefGoogle Scholar
  7. [Lut87]
    J. Lutz. Resource-Bounded Category and Measure in Exponential Complexity Classes. PhD thesis, Department of Mathematics, California Institute of Technology, 1987.Google Scholar
  8. [Lut90]
    J. Lutz. Category and measure in complexity classes. SIAM J. Comput., 19(6):1100–1131, December 1990.CrossRefGoogle Scholar
  9. [Lut92]
    J. Lutz. Almost everywhere high nonuniform complexity. J. Computer and System Sciences, 44:220–258, 1992.CrossRefGoogle Scholar
  10. [Lut96]
    J. Lutz. Observations on measure and lowness for Δ 2P. In STACS 96, volume 1046 of Lecture Notes in Computer Science, pages 87–98. Springer, 1996.Google Scholar
  11. [May94a]
    E. Mayordomo. Almost every set in exponential time is p-bi-immune. Theoretical Computer Science, 136(2):487–506, 1994.CrossRefGoogle Scholar
  12. [May94b]
    E. Mayordomo. Contributions to the study of resource-bounded measure. PhD thesis, Universität Politècnica de Catalunya, 1994.Google Scholar
  13. [RSC95]
    K. Regan, D. Sivakumar, and J. Cai. Pseudorandom generators, measure theory, and natural proofs. In 36th Annual Symposium on Foundations of Computer Science, pages 26–35, 1995.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Harry Buhrman
    • 1
  • Stephen Fenner
    • 2
  • Lance Fortnow
    • 3
  1. 1.Centrum voor Wiskunde en InformaticaGermany
  2. 2.University of Southern MaineUSA
  3. 3.CWI & The University of ChicagoUSA

Personalised recommendations