Autoreducibility of Random Sets: A Sharp Bound on the Density of Guessed Bits
MFCS '02 Proceedings of the 27th International Symposium on Mathematical Foundations of Computer Science
Theoretical Computer Science
Comparing notions of randomness
Theoretical Computer Science
Randomness and completeness in computational complexity
Randomness and completeness in computational complexity
Exact learning algorithms, betting games, and circuit lower bounds
ICALP'11 Proceedings of the 38th international colloquim conference on Automata, languages and programming - Volume Part I
Kolmogorov-Loveland randomness and stochasticity
STACS'05 Proceedings of the 22nd annual conference on Theoretical Aspects of Computer Science
Exact Learning Algorithms, Betting Games, and Circuit Lower Bounds
ACM Transactions on Computation Theory (TOCT)
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We introduce resource-bounded betting games and propose a generalization of Lutz's resource-bounded measure in which the choice of the 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 pseudorandom 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 nonrelativizable consequence $\mbox{BPP} \neq \mbox{EXP}$. We also show that if $\mbox{EXP} \neq \mbox{MA}$, then the polynomial-time truth-table-autoreducible languages have Lutz measure zero, whereas if $\mbox{EXP} = \mbox{BPP}$, they have measure one.