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Randomness, relativization and Turing degrees. (English) Zbl 1090.03013
The paper at hand makes a substantial contribution to our understanding of the relationship between the Turing degree of a real and its relative randomness. Let \(K\) denote prefix-free Kolmogorov complexity and let \(C\) denote plain complexity. By old work of Kolmogorov, Martin-Löf, Levin and Chaitin, we know that the highest plain complexity a string \(\sigma\) can have is \(C(\sigma)=^+ | \sigma| \) where the \(=^+\) denotes equality up to some absolute additive constant. The highest prefix-free complexity is \(K(\sigma)=^+ | \sigma| +K(| \sigma| ).\) On the other hand, no real \(\alpha\) can have \(C(\alpha\upharpoonright n)=^+ n\) for all \(n\), and the best that is possible is that for a real \(C(\alpha\upharpoonright n)=^+ n\) for infinitely many \(n\) (such reals are called Kolmogorov random), this being implied by \(K(\alpha\upharpoonright n)\) being maximal for infinitely many \(n\) (this being called strongly Chaitin random). The authors use an ingenious method based around the low basis theorem to show that Kolmogorov random reals coincide with the 2-random reals; where a real is called 2-random iff it avoids all \(\Sigma_2^0\) classes of measure effectively converging to 0. One direction had been independently proven by J. S. Miller [J. Symb. Log. 69, 907–913 (2004; Zbl 1090.03012), reviewed above]. The authors then investigate reals that are low for Chaitin’s halting probability \(\Omega\). This means that \(\Omega\) is 1-\(A\)-random. The nicest result is to show that if \(A\) is a c.e. low for \(\omega\) set then \(A\) is \(K\)-trivial. This uses a “KC”-set construction which is surprisingly easy. Finally, in the last section, the authors investigate the relationship between jump classes and restricted kinds of randomness. They show that outside the high degrees, Schnorr randomness and Martin-Löf randomness coincide, and within each high degree Schnorr, computable and Martin-Löf randomness may be separated. This argument is rather delicate.

MSC:
03D80 Applications of computability and recursion theory
03D28 Other Turing degree structures
68Q30 Algorithmic information theory (Kolmogorov complexity, etc.)
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