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Resource-bounded randomness and compressibility with respect to nonuniform measures

  • Steven M. Kautz
Complexity
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1269)

Abstract

Most research on resource-bounded measure and randomness has focused on the uniform probability density, or Lebesgue measure, on {0,1}; the study of resource-bounded measure theory with respect to a nonuniform underlying measure was recently initiated by Breutzmann and Lutz [1]. In this paper we prove a series of fundamental results on the role of nonuniform measures in resource-bounded measure theory. These results provide new tools for analyzing and constructing martingales and, in particular, offer new insight into the compressibility characterization of randomness given recently by Buhrman and Longpré [2]. We give several new characterizations of resource-bounded randomness with respect to an underlying measure μ: the first identifies those martingales whose rate of success is asymptotically optimal on the given sequence; the second identifies martingales which induce a maximal compression of the sequence; the third is a (nontrivial) extension of the compressibility characterization to the nonuniform case. In addition we prove several technical results of independent interest, including an extension to resource-bounded measure of the classical theorem of Kakutani on the equivalence of product measures; this answers an open question in [1].

Keywords

Lebesgue Measure Initial Segment Maximal Compression Arithmetic Code Dyadic Interval 
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.
    J.M. Breutzmann and J. H. Lutz. Equivalence of measures of complexity classes. 1996. To appear.Google Scholar
  2. 2.
    H. Buhrman and L. Longpré. Compressibility and resource bounded measure. 1995 STACS.Google Scholar
  3. 3.
    W. Feller. An Introduction to Probability Theory and its Applications. Volume 2, John Wiley and Sons, Inc., 1971.Google Scholar
  4. 4.
    P.R. Halmos. Measure Theory. Springer-Verlag, 1974.Google Scholar
  5. 5.
    S. Kakutani. On the equivalence of infinite product measures. Annals of Mathematics, 49:214–224, 1948.Google Scholar
  6. 6.
    S. M. Kautz. Degrees of Random Sets. PhD thesis, Cornell University, 1991.Google Scholar
  7. 7.
    M. Li and P. Vitányi. An Introduction to Kolmogorov Complexity and Its Applications. Springer-Verlag, 1993.Google Scholar
  8. 8.
    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
  9. 9.
    J. H. Lutz and E. Mayordomo. Genericity, measure, and inseparable pairs. 1996. In preparation.Google Scholar
  10. 10.
    J.H. Lutz. Almost everywhere high nonuniform complexity. Journal of Computer and System Sciences, 44:220–258, 1992.Google Scholar
  11. 11.
    E. Mayordomo. Contributions to the Study of Resource-Bounded Measure. PhD thesis, Universitat Politècnica de Catalunya, 1994.Google Scholar
  12. 12.
    Noam Nisan. Extracting randomness: how and why. A survey. In Proceedings of the 11th IEEE Conference on Computational Complexity, 1996.Google Scholar
  13. 13.
    M. Santha and U.V. Vazirani. Generating quasi-random sequences from slightly-random sources. In Proc. 25th Ann. Symp. on the Theory of Computing, 1984.Google Scholar
  14. 14.
    A. Kh. Shen. Algorithmic complexity and randomness: recent developments. Theory Probab. Appl., 37(3):92–97, 1993.Google Scholar
  15. 15.
    A. Kh. Shen. On relations between different algorithmic definitions of randomness. Soviet Math. Dokl., 38(2):316–319, 1989.Google Scholar
  16. 16.
    M. van Lambalgen. Random Sequences. PhD thesis, University of Amsterdam, 1987.Google Scholar
  17. 17.
    M. van Lambalgen. Von Mises' definition of random sequences reconsidered. Journal of Symbolic Logic, 52(3):725–755, 1987.Google Scholar
  18. 18.
    U.V. Vazirani and V.V. Vazirani. Random polynomial time is equal to slightly-random polynomial time. In Proceedings of the 26th IEEE Symposium on Foundations of Computer Science, 1985.Google Scholar
  19. 19.
    I. H. Witten, R. M. Neal, and J. G. Cleary. Arithmetic coding for data compression. Communications of the Association for Computing Machinery, 30:520–540, 1987.Google Scholar
  20. 20.
    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, 25:83–123, 1970.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Steven M. Kautz
    • 1
  1. 1.Department of MathematicsRandolph-Macon Woman's CollegeLynchburg

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