Skip to main content
SpringerLink
Log in
Book cover

Annual Symposium on Theoretical Aspects of Computer Science

STACS 1997: STACS 97 pp 535–545Cite as

Equivalence of measures of complexity classes

  • Structural Complexity III
  • Conference paper
  • First Online:

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1200))

Abstract

The resource-bounded measures of complexity classes are shown to be robust with respect to certain changes in the underlying probability measure. Specifically, for any real number δ > 0, any uniformly polynomial-time computable sequence β=(β 0,β 1,β 2), ... of real numbers (biases) β i ε [δ, 1−δ], and any complexity class C (such as P, NP, BPP, P/Poly, PH, PSPACE, etc.) that is closed under positive, polynomial-time, truth-table reductions with queries of at most linear length, it is shown that the following two conditions are equivalent.

  1. (1)

    C has p-measure 0 (respectively, measure 0 in E, measure 0 in E2) relative to the coin-toss probability measure given by the sequence β.

  2. (2)

    C has p-measure 0 (respectively, measure 0 in E, measure 0 in E2) relative to the uniform probability measure.

The proof introduces three techniques that may be useful in other contexts, namely, (i) the transformation of an efficient martingale for one probability measure into an efficient martingale for a “nearby” probability measure; (ii) the construction of a positive bias reduction, a truthtable reduction that encodes a positive, efficient, approximate simulation of one bias sequence by another; and (iii) the use of such a reduction to dilate an efficient martingale for the simulated probability measure into an efficient martingale for the simulating probability measure.

This research was supported in part by National Science Foundation Grant CCR-9157382, with matching funds from Rockwell, Microware Systems Corporation, and Amoco Foundation.

This is a preview of subscription content, log in via an institution.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. K. Ambos-Spies, E. Mayordomo, and X. Zheng. A comparison of weak completeness notions. In Proceedings of the Eleventh IEEE Conference on Computational Complexity. IEEE Computer Society Press, 1996. To appear.

    Google Scholar 

  2. K. Ambos-Spies, S. A. Terwijn, and X. Zheng. Resource bounded randomness and weakly complete problems. Theoretical Computer Science, 1996. To appear. See also Proceedings of the Fifth Annual International Symposium on Algorithms and Computation, 1994, pp. 369–377. Springer-Verlag.

    Google Scholar 

  3. J. Cai and A. L. Selman. Fine separation of average time complexity classes. In Proceedings of the Thirteenth Symposium on Theoretical Aspects of Computer Science, pages 331–343. Springer-Verlag, 1996.

    Google Scholar 

  4. R. I. Freidzon. Families of recursive predicates of measure zero. translated in Journal of Soviet Mathematics, 6(1976):449–455, 1972.

    Article  Google Scholar 

  5. P. R. Halmos. Measure Theory. Springer-Verlag, 1950.

    Google Scholar 

  6. D. W. Juedes. Weakly complete problems are not rare. Computational Complexity, 5:267–283, 1995.

    Article  Google Scholar 

  7. D. W. Juedes and J. H. Lutz. The complexity and distribution of hard problems. SIAM Journal on Computing, 24(2):279–295, 1995.

    Article  Google Scholar 

  8. D. W. Juedes and J. H. Lutz. Weak completeness in E and E2. Theoretical Computer Science, 143:149–158, 1995.

    Article  Google Scholar 

  9. S. Kakutani. On the equivalence of infinite product measures. Annals of Mathematics, 49:214–224, 1948.

    Google Scholar 

  10. S. M. Kautz. Personal communication, 1996.

    Google Scholar 

  11. J. H. Lutz. Observations on measure and lowness for Δ P2 . Mathematical Systems Theory. To appear. See also Proceedings of the Thirteenth Symposium on Theoretical Aspects of Computer Science, pages 87–97. Springer-Verlag, 1996.

    Google Scholar 

  12. J. H. Lutz. Almost everywhere high nonuniform complexity. Journal of Computer and System Sciences, 44:220–258, 1992.

    Article  Google Scholar 

  13. J. H. Lutz. The quantitative structure of exponential time. In Proceedings of the Eighth Annual Structure in Complexity Theory Conference, pages 158–175, 1993. Updated version to appear in L.A. Hemaspaandra and A.L. Selman (eds.), Complexity Theory Retrospective II, Springer-Verlag, 1996.

    Google Scholar 

  14. J. H. Lutz. Weakly hard problems. SIAM Journal on Computing, 24:1170–1189, 1995.

    Article  Google Scholar 

  15. J. H. Lutz and E. Mayordomo. Measure, stochasticity, and the density of hard languages. SIAM Journal on Computing, 23:762–779, 1994.

    Article  Google Scholar 

  16. J. H. Lutz and E. Mayordomo. Cook versus Karp-Levin: Separating completeness notions if NP is not small. Theoretical Computer Science, 164:141–163, 1996.

    Article  Google Scholar 

  17. J. H. Lutz and E. Mayordomo. Genericity, measure, and inseparable pairs, 1996. In preparation.

    Google Scholar 

  18. E. Mayordomo. Almost every set in exponential time is P-bi-immune. Theoretical Computer Science, 136(2):487–506, 1994.

    Article  Google Scholar 

  19. K. Mehlhorn. The “almost all” theory of subrecursive degrees is decidable. In Proceedings of the Second Colloquium on Automata, Languages, and Programming, pages 317–325. Springer Lecture Notes in Computer Science, vol. 14, 1974.

    Google Scholar 

  20. J. C. Oxtoby. Measure and Category. Springer-Verlag, second edition, 1980.

    Google Scholar 

  21. K. W. Regan, D. Sivakumar, and J. Cai. Pseudorandom generators, measure theory, and natural proofs. In 36th IEEE Symposium on Foundations of Computer Science, pages 26–35. IEEE Computer Society Press, 1995.

    Google Scholar 

  22. H. Rogers, Jr. Theory of Recursive Functions and Effective Computability. McGraw — Hill, New York, 1967.

    Google Scholar 

  23. C. P. Schnorr. Klassifikation der Zufallsgesetze nach Komplexität und Ordnung. Z. Wahrscheinlichkeitstheorie verw. Geb., 16:1–21, 1970.

    Article  Google Scholar 

  24. C. P. Schnorr. A unified approach to the definition of random sequences. Mathematical Systems Theory, 5:246–258, 1971.

    Article  Google Scholar 

  25. C. P. Schnorr. Zufälligkeit und Wahrscheinlichkeit. Lecture Notes in Mathematics, 218, 1971.

    Google Scholar 

  26. C. P. Schnorr. Process complexity and effective random tests. Journal of Computer and System Sciences, 7:376–388, 1973.

    Google Scholar 

  27. M. van Lambalgen. Random Sequences. PhD thesis, Department of Mathematics, University of Amsterdam, 1987.

    Google Scholar 

  28. V. G. Vovk. On a randomness criterion. Soviet Mathematics Doklady, 35:656–660, 1987.

    Google Scholar 

  29. A. K. Zvonkin and L. A. Levin. The complexity of finite objects and the development of the concepts of information and randomness by means of the theory of algorithms. Russian Mathematical Surveys, 25:83–124, 1970.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Science, Wartburg College, 50677, Waverly, Iowa, USA

    Josef M. Breutzmann

  2. Department of Computer Science, Iowa State University, 50011, Ames, Iowa, USA

    Jack H. Lutz

Authors
  1. Josef M. Breutzmann
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jack H. Lutz
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Rüdiger Reischuk Michel Morvan

Rights and permissions

Reprints and permissions

Copyright information

© 1997 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Breutzmann, J.M., Lutz, J.H. (1997). Equivalence of measures of complexity classes. In: Reischuk, R., Morvan, M. (eds) STACS 97. STACS 1997. Lecture Notes in Computer Science, vol 1200. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0023487

Download citation

  • DOI: https://doi.org/10.1007/BFb0023487

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-62616-9

  • Online ISBN: 978-3-540-68342-1

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation