通过2-随机序列的变换实现连续随机性
Continuous Randomness via Transformations of 2-Random Sequences
AI总结：
本研究通过分析2-随机序列的多种变换，证明了相关序列及其Turing跳、经多种有效算子作用后的结果均关于∅'-可计算连续测度满足Martin-Löf随机性，给出了非正当连续随机序列的实例。
摘要：
Reimann和Slaman开创了对关于连续测度满足Martin-Löf随机性的序列的研究，确立了关于NCR（即不关于任何连续测度满足Martin-Löf随机性的序列的集合）的基本结论。对于关于可计算连续测度满足随机性的序列，相关图景已相当明晰：这类序列与一个Martin-Löf随机序列是真值表等价的。然而，对于一个关于某连续测度满足随机性、但不关于任何可计算测度满足随机性的序列，我们可以提出问题：使其满足连续随机性的测度，其有效性程度如何？\n在本研究中，我们通过考察2-随机序列（即相对于停机集$\boldsymbol{\u2205}'$满足Martin-Löf随机性的序列）的各类变换来探讨这一问题，得到了若干关于“关于可在$\boldsymbol{\u2205}'$中计算的连续测度满足连续随机性的序列”的结论。具体而言，我们证明：(i) 每一个可由2-随机序列计算得到的不可计算序列，都关于一个$\boldsymbol{\u2205}'$-可计算的连续测度满足Martin-Löf随机性；(ii) 每一个2-随机序列的Turing跳，都关于一个$\boldsymbol{\u2205}'$-可计算的连续测度满足Martin-Löf随机性。由这些结论，我们得到了若干序列实例：它们不具备正当性（即不关于任何可计算测度满足随机性），但关于一个$\boldsymbol{\u2205}'$-可计算的连续测度满足随机性。最后，我们考察了2-随机性在更广泛的一类有效算子（c.e.算子、伪跳算子，以及通过伪跳反演定义的算子）下的行为，证明这些算子作用后得到的序列同样关于一个$\boldsymbol{\u2205}'$-可计算的连续测度满足Martin-Löf随机性。
英文摘要：
Reimann and Slaman initiated the study of sequences that are Martin-Löf random with respect to a continuous measure, establishing fundamental facts about NCR, the collection of sequences that are not Martin-Löf random with respect to any continuous measure. In the case of sequences that are random with respect to a computable, continuous measure, the picture is fairly well-understood: such sequences are truth-table equivalent to a Martin-Löf random sequence. However, given a sequence that is random with respect to a continuous measure but not with respect to any computable measure, we can ask: how close to effective is the measure with respect to which it is continuously random? In this study, we take up this question by examining various transformations of 2-random sequences (sequences that are Martin-Löf random relative to the halting set $\emptyset'$) to establish several results on sequences that are continuously random with respect to a measure that is computable in $\emptyset'$. In particular, we show that (i) every noncomputable sequence that is computable from a 2-random sequence is Martin-Löf random with respect to a continuous, $\emptyset'$-computable measure and (ii) the Turing jump of every 2-random sequence is Martin-Löf random with respect to a continuous, $\emptyset'$-computable measure. From these results, we obtain examples of sequences that are not proper, i.e., not random with respect to any computable measure, but are random with respect to a continuous, $\emptyset'$-computable measure. Lastly, we consider the behavior of 2-randomness under a wider class of effective operators (c.e. operators, pseudojump operators, and operators defined in terms of pseudojump inversion), showing that these too yield sequences that are Martin-Löf random with respect to a continuous, $\emptyset'$-computable measure.
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