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.
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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
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DOI: https://doi.org/10.1007/BFb0028558
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