论文标题
算法上最佳的外部措施
Algorithmically Optimal Outer Measures
论文作者
论文摘要
我们研究算法分形维度与欧几里得空间外部测量的经典局部分形维度之间的关系。我们为较低的半度计算外部措施介绍了全球和局部最佳条件。我们证明存在全球最佳外部措施。我们的主要定理指出，任何局部最佳外部度量的经典局部分形维度与算法分形尺寸完全一致。我们的证明使用特定方便的本地最佳外部措施$ \BOLDSYMBOLκ$定义在Kolmogorov复杂性方面。我们讨论对点对点原则的影响。
We investigate the relationship between algorithmic fractal dimensions and the classical local fractal dimensions of outer measures in Euclidean spaces. We introduce global and local optimality conditions for lower semicomputable outer measures. We prove that globally optimal outer measures exist. Our main theorem states that the classical local fractal dimensions of any locally optimal outer measure coincide exactly with the algorithmic fractal dimensions. Our proof uses an especially convenient locally optimal outer measure $\boldsymbolκ$ defined in terms of Kolmogorov complexity. We discuss implications for point-to-set principles.
