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• W2964135541 abstract "The Kučera–Gács theorem is a landmark result in algorithmic randomness asserting that every real is computable from a Martin-Löf random real. If the computation of the first n bits of a sequence requires n + h ( n ) bits of the random oracle, then h is the redundancy of the computation. Kučera implicitly achieved redundancy n log ⁡ n while Gács used a more elaborate coding procedure which achieves redundancy n log ⁡ n . A similar bound is implicit in the later proof by Merkle and Mihailović. In this paper we obtain optimal strict lower bounds on the redundancy in computations from Martin-Löf random oracles. We show that any nondecreasing computable function g such that ∑ n 2 − g ( n ) = ∞ is not a general upper bound on the redundancy in computations from Martin-Löf random oracles. In fact, there exists a real X such that the redundancy g of any computation of X from a Martin-Löf random oracle satisfies ∑ n 2 − g ( n ) < ∞ . Moreover, the class of such reals is comeager and includes a Δ 2 0 real as well as all weakly 2-generic reals. On the other hand, it has been recently shown that any real is computable from a Martin-Löf random oracle with redundancy g , provided that g is a computable nondecreasing function such that ∑ n 2 − g ( n ) < ∞ . Hence our lower bound is optimal, and excludes many slow growing functions such as log ⁡ n from bounding the redundancy in computations from random oracles for a large class of reals. Our results are obtained as an application of a theory of effective betting strategies with restricted wagers which we develop." @default.
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• W2964135541 date "2016-12-01" @default.
• W2964135541 modified "2023-09-29" @default.
• W2964135541 title "Lower bounds on the redundancy in computations from random oracles via betting strategies with restricted wagers" @default.
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• W2964135541 doi "https://doi.org/10.1016/j.ic.2016.09.010" @default.
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