Publication Date:
2019-10-01
Description:
In this paper, we study Bernoulli random sequences, i.e. sequences that are Martin-Löf random with respect to a Bernoulli measure $mu _p$ for some $pin [0,1]$, where we allow for the possibility that $p$ is noncomputable. We focus in particular on the case in which the underlying Bernoulli parameter $p$ is proper (i.e. Martin-Löf random with respect to some computable measure). We show for every Bernoulli parameter $p$, if there is a sequence that is both proper and Martin-Löf random with respect to $mu _p$, then $p$ itself must be proper, and explore further consequences of this result. We also study the Turing degrees of Bernoulli random sequences, showing, for instance, that the Turing degrees containing a Bernoulli random sequence do not coincide with the Turing degrees containing a Martin-Löf random sequence. Lastly, we consider several possible approaches to characterizing blind Bernoulli randomness, where the corresponding Martin-Löf tests do not have access to the Bernoulli parameter $p$, and show that these fail to characterize blind Bernoulli randomness.
Print ISSN:
0955-792X
Electronic ISSN:
1465-363X
Topics:
Computer Science
,
Mathematics
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