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A Generalization of Resource-Bounded Measure, With Application to the BPP vs. EXP Problem Harry Buhrman Dieter van Melkebeek Kenneth W. Regan D. Sivakumar § Martin Strauss December 1999 Abstract We introduce resource-bounded betting games , and propose a general- ization of Lutz’s resource-bounded measure in which the choice of next str ing to bet on is ful ly adapti ve . Lutz’s mar tingal es are equiv ale nt to betting games constra ined to bet on strings in lexicogra phic order. We show that if strong pseudo-random number generators exist, then betting games are equivalen t to martin gales, for measur e on E and EXP. How- ever , we construc t b etting games that succeed on certain class es whose Lutz measures are important open problems: the class of polynomial-t ime Turing-complete languages in EXP, and its superclass of polynomial-time Tu ring-autoreducible languages. If an EXP-martingale succeeds on either of these classes, or if betting games have the “nite union property” pos- sessed by Lutz’s measure, one obtains the non-relativizable consequence CWI, Kruislaa n 413, 1098SJ Amsterdam, The Nethe rlands . Part ially supported by the Dutch foundation for scientic research (NWO) through SION project 612-34-002, and by the Europe an Union through NeuroCOLT ESPRIT Workin g Group Nr. 8556, and HC&M grant nr. ERB405 0PL93- 0516. E-mail: [email protected]. DIMACS Center, Rutgers University, 96 Frelinghuysen Road, CoRE Building - Room 419, Pisca taway , NJ 08854-8018 USA. Partl y support ed by the European Union throug h Marie Curie Research Training Grant ERB-4001-GT-96-0783 at CWI and at the University of Ams- terdam; by NSF Grant CCR 92-53582; and by the Fields Institute of the University of Toronto; research performed mainly at the University of Chicago. E-mail: [email protected] Department of Computer Science and Engineering, State Univ. of NY at Bualo, 226 Bell Hall, Bualo, NY 14260-2000 USA. Supported in part by the National Science Foundation under Grant CCR-94091 04. E-mail : [email protected] § IBM Almaden Research Center, Dept. K-53, 650 Harry Road San Jose, CA 95120 USA. Part of this research was performed while at the State Univ. of N.Y. at Bualo, supported in part by Nation al Scien ce F oundation Grant CCR-9409104. E-mail : [email protected] AT&T Labs Room C216, 180 Park Ave, Florham Park, NJ 07932-0971 USA. Research performed while at Rutgers University and Iowa State University, supported by NSF grants CCR-9204874 and CCR-9157382. E-mail : [email protected] 1

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