通过开放式覆盖的随机性和低性概念
Randomness and lowness notions via open covers
摘要：
算法随机性研究的一条主线是低度概念。给定一个随机性概念R，我们问哪些序列A相对于A使R保持不变（即，R^A = R）？这样的序列对于R来说是低的。这个问题延伸到一对随机性概念R和S，其中S更弱：对于哪一个A是S^A仍然弱于R？在过去的几年中，许多结果的特点是低随机性的序列，其低计算强度。一些结果也给出了低序列的测度论特征。例如，Kjos-Hanssen证明了对于Martin-L，A是低的随机性当且仅当每个A-c. e.测度小于1的开集可以用c.e.开测度集小于1。在本文中，我们给出了一系列结果，表明各种各样的低概念可以用类似的方式表达，即，通过某种类型的开集覆盖某种类型的开集的能力。这提供了一个统一的框架，澄清了随机性概念的低研究，并允许我们给出一些已知结果的简单证明。我们还使用这个框架来证明新的结果，包括表明类低（MLR;SR）和低（W2 R;SR）相吻合，回答了Nies的问题。其他的应用包括随机性概念的特征，一个广泛适用的解释，为什么随机性的低与测试的低是相同的，以及一个简单的证明，低（W2 R;S）=低（MLR;S），其中S是马丁-L的类可计算或Schnorr随机序列。最后一节使用可和函数和收敛测度机代替开覆盖给出了低度概念的刻画。最后，我们简单证明Nies的一个结果，即Low（MLR）= Low（MLR; CR）。
英文摘要：
One of the main lines of research in algorithmic randomness is that of lowness notions. Given a randomness notion R, we ask for which sequences A does relativization to A leave R unchanged (i.e., R^A = R)? Such sequences are call low for R. This question extends to a pair of randomness notions R and S, where S is weaker: for which A is S^A still weaker than R? In the last few years, many results have characterized the sequences that are low for randomness by their low computational strength. A few results have also given measure-theoretic characterizations of low sequences. For example, Kjos-Hanssen proved that A is low for Martin-Löf randomness if and only if every A-c.e. open set of measure less than 1 can be covered by a c.e. open set of measure less than 1. In this paper, we give a series of results showing that a wide variety of lowness notions can be expressed in a similar way, i.e., via the ability to cover open sets of a certain type by open sets of some other type. This provides a unified framework that clarifies the study of lowness for randomness notions, and allows us to give simple proofs of a number of known results. We also use this framework to prove new results, including showing that the classes Low(MLR;SR) and Low(W2R;SR) coincide, answering a question of Nies. Other applications include characterizations of highness notions, a broadly applicable explanation for why low for randomness is the same as low for tests, and a simple proof that Low(W2R;S)=Low(MLR;S), where S is the class of Martin-Löf, computable, or Schnorr random sequences. The final section gives characterizations of lowness notions using summable functions and convergent measure machines instead of open covers. We finish with a simple proof of a result of Nies, that Low(MLR) = Low(MLR; CR).
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