A generalization of resource-bounded measure, with an application (Extended abstract)

  • Harry Buhrman
  • Dieter van Melkebeek
  • Kenneth W. Regan
  • D. Sivakumar
  • Martin Strauss
Complexity II
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1373)


We introduce resource-bounded betting games, and propose a generalization of Lutz's resource-bounded measure in which the choice of next string to bet on is fully adaptive. Lutz's martingales are equivalent to betting games constrained to bet on strings in lexicographic order. We show that if strong pseudo-random number generators exist, then betting games are equivalent to martingales, for measure on E and EXP. However, we construct betting games that succeed on certain classes whose Lutz measures are important open problems: the class of polynomial-time Turing-complete languages in EXP, and its superclass of polynomial-time Turing-autoreducible languages. If an EXP-martingale succeeds on either of these classes, or if betting games have the “finite union property” possessed by Lutz's measure, one obtains the non-relativizable consequence BPP ≠ EXP. We also show that if EXP ≠ MA, then the polynomial-time truth-table-autoreducible languages have Lutz measure zero, whereas if EXP = BPP, they have measure one.


Complexity Class Lexicographic Order Important Open Problem Infinite Process Computable Martingale 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Allender, E., Strauss, M.: Measure on small complexity classes, with applications for BPP. DIMACS TR 94-18, Rutgers University and DIMACS, April 1994.Google Scholar
  2. 2.
    Allender, E., Strauss, M.: Measure on P: Robustness of the notion. In Proc. 20th International Symposium on Mathematical Foundations of Computer Science, volume 969 of Lect. Notes in Comp. Sci., pages 129–138. Springer Verlag, 1995.Google Scholar
  3. 3.
    Ambos-Spies, K.: P-mitotic sets. In E. Börger, G. Hasenjäger, and D. Roding, editors, Logic and Machines, Lecture Notes in Computer Science 177, pages 1–23. Springer-Verlag, 1984.Google Scholar
  4. 4.
    Ambos-Spies, K., Lempp, S.: Presentation at a Schloss Dagstuhl workshop on “Algorithmic Information Theory and Randomness,” July 1996.Google Scholar
  5. 5.
    Babai, L.: Trading group theory for randomness. In Proc. 17th Annual ACM Symposium on the Theory of Computing, pages 421–429, 1985.Google Scholar
  6. 6.
    Babai, L., Fortnow, L., Nisan, N., Wigderson, A.: BPP has subexponential time simulations unless EXPTIME has publishable proofs. Computational Complexity, 3, 1993.Google Scholar
  7. 7.
    Buhrman, H., Fortnow, L., Torenvliet, L.: Using autoreducibility to separate complexity classes. In 36th Annual Symposium on Foundations of Computer Science, pages 520–527, Milwaukee, Wisconsin, 23–25 October 1995. IEEE.Google Scholar
  8. 8.
    Buhrman, H., Longpré, L.: Compressibility and resource bounded measure. In 13th Annual Symposium on Theoretical Aspects of Computer Science, volume 1046 of lncs, pages 13–24, Grenoble, France, 22–24 February 1996. Springer.Google Scholar
  9. 9.
    Babai, L., Moran, S.: Arthur-Merlin games: A randomized proof system, and a hierarchy of complexity classes. J. Comp. Sys. Sci., 36:254–276, 1988.zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Heller, F.: On relativized exponential and probabilistic complexity classes. Inform. and Control, 71:231–243, 1986.zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Loveland, D. W.: A variant of the Kolmogorov concept of complexity. Inform. and Control, 15:510–526, 1969.zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Lutz, J.: Almost everywhere high nonuniform complexity. J. Comp. Sys. Sci., 44:220–258, 1992.zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Lutz, J.: The quantitative structure of exponential time. In L. Hemaspaandra and A. Selman, eds., Complexity Theory Retrospective II. Springer Verlag, 1997.Google Scholar
  14. 14.
    Mayordomo, E.: Contributions to the Study of Resource-Bounded Measure. PhD thesis, Universidad Polytécnica de Catalunya, Barcelona, April 1994.Google Scholar
  15. 15.
    Nisan, N., Wigderson, A.: Hardness versus randomness. J. Comp. Sys. Sci., 49:149–167, 1994.zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Stockmeyer, L.: The complexity of approximate counting. In Proc. 15th Annual ACM Symposium on the Theory of Computing, pages 118–126, Baltimore, USA, April 1983. ACM Press.Google Scholar

Copyright information

© Springer-Verlag 1998

Authors and Affiliations

  • Harry Buhrman
    • 1
  • Dieter van Melkebeek
    • 2
  • Kenneth W. Regan
    • 3
  • D. Sivakumar
    • 4
  • Martin Strauss
    • 5
  1. 1.CWIAmsterdamThe Netherlands
  2. 2.Department of Computer ScienceUniv. of ChicagoChicagoUSA
  3. 3.Computer ScienceUniversity at BuffaloBuffaloUSA
  4. 4.Department of Computer ScienceUniversity of HoustonHoustonUSA
  5. 5.AT&T LabsFlorham ParkUSA

Personalised recommendations