Abstract.
Let ω be a Kolmogorov–Chaitin random sequence with ω1: n denoting the first n digits of ω. Let P be a recursive predicate defined on all finite binary strings such that the Lebesgue measure of the set {ω|∃nP(ω1: n )} is a computable real α. Roughly, P holds with computable probability for a random infinite sequence. Then there is an algorithm which on input indices for any such P and α finds an n such that P holds within the first n digits of ω or not in ω at all. We apply the result to the halting probability Ω and show that various generalizations of the result fail.
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Received: 1 December 1998 / Published online: 3 October 2001
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Davie, G. Recursive events in random sequences. Arch. Math. Logic 40, 629–638 (2001). https://doi.org/10.1007/s001530100075
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DOI: https://doi.org/10.1007/s001530100075