伯努利随机性的有效方面
Effective Aspects of Bernoulli Randomness
摘要：
本文研究了Bernoulli随机序列，即，序列是马丁-洛夫随机相对于伯努利措施$μ_p$为一些$p\in[0,1]$，其中我们允许的可能性，$p$是不可计算的。我们特别关注的情况下，基本的伯努利参数$p$是适当的（即马丁-勒夫随机相对于一些可计算的措施）。我们证明了对于每一个伯努利参数$p$，如果存在一个序列，这是正确的和马丁-洛夫随机关于$μ_p$，那么$p$本身必须是正确的，并探讨这一结果的进一步后果。我们还研究了伯努利随机序列的图灵度，例如，包含伯努利随机序列的图灵度与包含马丁-洛夫随机序列的图灵度不一致。最后，我们考虑了几种可能的方法来表征盲伯努利随机性，其中相应的马丁-洛夫测试没有访问伯努利参数$p$，并表明这些无法表征盲伯努利随机性。
英文摘要：
In this paper, we study Bernoulli random sequences, i.e., sequences that are Martin-Löf random with respect to a Bernoulli measure $μ_p$ for some $p\in[0,1]$, 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 (that is, 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 $μ_p$, 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
