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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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