A Generalization of Resource-Bounded Measure, with Application to the BPP vs. EXP Problem

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 ∗CWI, Kruislaan 413, 1098SJ Amsterdam, The Netherlands. Partially supported by the Dutch foundation for scientific research (NWO) through SION project 612-34-002, and by the European Union through NeuroCOLT ESPRIT Working Group Nr. 8556, and HC&M grant nr. ERB4050PL93-0516. E-mail: [email protected]. †DIMACS Center, Rutgers University, 96 Frelinghuysen Road, CoRE Building Room 419, Piscataway, NJ 08854-8018 USA. Partly supported by the European Union through Marie Curie Research Training Grant ERB-4001-GT-96-0783 at CWI and at the University of Amsterdam; 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 Buffalo, 226 Bell Hall, Buffalo, NY 14260-2000 USA. Supported in part by the National Science Foundation under Grant CCR-9409104. 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 Buffalo, supported in part by National Science Foundation 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]