可计算近似实数的速度性及其近似
Speedability of computably approximable reals and their approximations
AI总结：
研究可计算近似实数的速度性，证明速度性等价于非马丁-洛夫随机性，并展示每个可计算近似实数都有可计算速度性近似。
摘要：
一个实数的近似是收敛于该实数的有理数序列。一个近似是左c.e.的，如果它是可计算的且非递减；是d.c.e.的，如果它是可计算的且有界变差。一个实数是可计算近似的，如果它有可计算的近似。一个近似{a_s}_{s∈ω}是速度性的，如果存在一个非递减的可计算函数f，使得近似{a_{f(s)}}_{s∈ω}在某种形式意义上比{a_s}_{s∈ω}收敛得更快。这导致了各种实数的速度性概念。Merkle和Titov建立了左c.e.实数速度性概念的等价性。我们扩展这些结果到d.c.e.实数和d.c.e.近似，并证明在这种情况下，速度性等价于非马丁-洛夫随机性。最后，我们展示每个可计算近似实数都有一个可计算速度性近似。
英文摘要：
An approximation of a real is a sequence of rational numbers that converges to the real. An approximation is left-c.e. if it is computable and nondecreasing and is d.c.e. if it is computable and has bounded variation. A real is computably approximable if it has some computable approximation, and left-c.e. and d.c.e. reals are defined accordingly. An approximation $\{a_s\}_{s \in ω}$ is speedable if there exists a nondecreasing computable function $f$ such that the approximation $\{a_{f(s)}\}_{s \in ω}$ converges in a certain formal sense faster than $\{a_s\}_{s \in ω}$. This leads to various notions of speedability for reals, e.g., one may require for a computably approximable real that either all or some of its approximations of a specific type are speedable. Merkle and Titov established the equivalence of several speedability notions for left-c.e. reals that are defined in terms of left-c.e. approximations. We extend these results to d.c.e. reals and d.c.e. approximations, and we prove that in this setting, being speedable is equivalent to not being Martin-Löf random. Finally, we demonstrate that every computably approximable real has a computable approximation that is speedable.
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。
