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Eighth International Conference on Computability, Complexity and Randomness (CCR 2013)

September 23–27, 2013, Moscow, Russia

Probabilistic algorithms in computability theory

Laurent Bienvenu
Can we compute with randomness something we cannot without? While complexity theorists still struggle with the fascinating question BPP =? P, the analogue problem in computability theory has seemingly been solved a long time ago: in 1954, de Leeuw, Moore, Shannon, and Shapiro proved that if a function can be computed by a probabilistic algorithm (= Turing machine), then it can be deterministically computed. However, the situation changes if one wishes to probabilistically generate \textbf{some} function in a class. For example, Martin showed that there exists a probabilistic algorithm which, with positive probability, computes some function which is not dominated by any computable one (obviously there is no way to computably and deterministically compute such a function). In fact, Martin's proof was quite abstract, but the best way to understand it is via an explicit probabilistic algorithm. I will present the proof of this result and several others classical theorems of the same type. I will then present some more recent work connecting the idea of probabilistic algorithm to reverse mathematics.