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Demuth randomness and computational complexity. (English) Zbl 1223.03026
Demuth tests are a generalization of Martin-Löf tests \(\{G_m\}_{m\in \omega}\) so that one can exchange the \(m\)-th component a computably bounded number of times. A set \(Z\subseteq \omega\) fails a Demuth test if \(Z\) is in infinitely many final versions of the \(G_m\). One has weak Demuth randomness if \(G_m\supseteq G_{m+1}\) for each \(m\).
It is shown that, different from weak-2-randomness, which is a well-known randomness notion stronger than Martin-Löf randomness, each \(\Pi^0_1\) class of positive measure contains a weakly Demuth random set which is \(\Delta^0_2\) and high, but no weakly Demuth random set is superhigh. Moreover, any c.e. set Turing-below a Demuth random set is strongly jump-traceable.
Reviewer: Liang Yu (Nanjing)

MSC:
03D32 Algorithmic randomness and dimension
68Q30 Algorithmic information theory (Kolmogorov complexity, etc.)
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