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Lowness for the class of random sets. (English) Zbl 0954.68080
The original Martin-Löf definition defines a random sequence as a sequence which passes all computable randomness tests. For every non-recursive set (binary sequence) $$A$$, we can consider, instead, tests which are computable relative to the oracle $$A$$, and thus get a relativized version of randomness. Since by adding an oracle we add new tests, the class $$\text{RAND}^A$$ of all resulting random sequences usually decreases: $$\text{RAND}^A\subset \text{RAND}$$, and $$\text{RAND}^A\neq \text{RAND}$$. The authors prove, however, that there exists an r.e. non-recursive oracle $$A$$ for which $$\text{RAND}^A= \text{RAND}$$. This proof provides an answer to a longstanding open question by M. van Lambalgen and D. Zambella.

##### MSC:
 68Q30 Algorithmic information theory (Kolmogorov complexity, etc.)
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##### References:
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