Abstract
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.
Partly supported by the Dutch foundation for scientific research (NWO), by SION project 612-34-002, and by the European Union through NeuroCOLT ESPRIT Working Group Nr: 8556 and HC&M grant HC&M ERB4050PL93-0516.
Party supported by the European Union through Marie Curie Research Training Grant ERB-4001-GT-96-0783 at CWI and by NSF Grant CCR 92-53582.
Partly supported at Buffalo by NSF Grant CCR-9409104.
Research performed at Rutgers University and Iowa State University, supported by NSF grants CCR-9204874 and CCR-9157382.
Preview
Unable to display preview. Download preview PDF.
References
Allender, E., Strauss, M.: Measure on small complexity classes, with applications for BPP. DIMACS TR 94-18, Rutgers University and DIMACS, April 1994.
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.
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.
Ambos-Spies, K., Lempp, S.: Presentation at a Schloss Dagstuhl workshop on “Algorithmic Information Theory and Randomness,” July 1996.
Babai, L.: Trading group theory for randomness. In Proc. 17th Annual ACM Symposium on the Theory of Computing, pages 421–429, 1985.
Babai, L., Fortnow, L., Nisan, N., Wigderson, A.: BPP has subexponential time simulations unless EXPTIME has publishable proofs. Computational Complexity, 3, 1993.
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.
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.
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.
Heller, F.: On relativized exponential and probabilistic complexity classes. Inform. and Control, 71:231–243, 1986.
Loveland, D. W.: A variant of the Kolmogorov concept of complexity. Inform. and Control, 15:510–526, 1969.
Lutz, J.: Almost everywhere high nonuniform complexity. J. Comp. Sys. Sci., 44:220–258, 1992.
Lutz, J.: The quantitative structure of exponential time. In L. Hemaspaandra and A. Selman, eds., Complexity Theory Retrospective II. Springer Verlag, 1997.
Mayordomo, E.: Contributions to the Study of Resource-Bounded Measure. PhD thesis, Universidad Polytécnica de Catalunya, Barcelona, April 1994.
Nisan, N., Wigderson, A.: Hardness versus randomness. J. Comp. Sys. Sci., 49:149–167, 1994.
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.
Author information
Authors and Affiliations
Editor information
Copyright information
© 1998 Springer-Verlag
About this paper
Cite this paper
Buhrman, H., van Melkebeek, D., Regan, K.W., Sivakumar, D., Strauss, M. (1998). A generalization of resource-bounded measure, with an application (Extended abstract). In: Morvan, M., Meinel, C., Krob, D. (eds) STACS 98. STACS 1998. Lecture Notes in Computer Science, vol 1373. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0028558
Download citation
DOI: https://doi.org/10.1007/BFb0028558
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-64230-5
Online ISBN: 978-3-540-69705-3
eBook Packages: Springer Book Archive