序列的有界图灵约简与数据处理不等式
Bounded Turing Reductions and Data Processing Inequalities for Sequences
摘要：
数据处理不等式指出，当两个实体（例如信号，字符串）中的一个实体被某些类型的变换处理时，两个实体之间共享的信息量不会显着增加。本文证明了序列的几个数据处理不等式，其中变换是有界的图灵泛函，共享信息由序列间的上下互维数度量.我们表明，对于所有的序列$X,Y,$和$Z$，如果$Z$是可计算的Lipschitz可约化$X$，那么\[ mdim(Z:Y) \leq mdim(X:Y) \text{ and } Mdim(Z:Y) \leq Mdim(X:Y). \]我们还展示了如何通过调整使用图灵函数的可计算界限来推导出不同的数据处理不等式。访问oracle $S$的至多$n$位的图灵函数$Φ^S$的产量是最小输入$m \in \mathbb{N}$，使得$Φ^{S \upharpoonright n}(m)\uparrow$。我们展示了如何推导出反向数据处理不等式（即，数据处理不等式，其中变换可以显著增加两个实体之间的共享信息）。
英文摘要：
A data processing inequality states that the quantity of shared information between two entities (e.g. signals, strings) cannot be significantly increased when one of the entities is processed by certain kinds of transformations. In this paper, we prove several data processing inequalities for sequences, where the transformations are bounded Turing functionals and the shared information is measured by the lower and upper mutual dimensions between sequences. We show that, for all sequences $X,Y,$ and $Z$, if $Z$ is computable Lipschitz reducible to $X$, then \[ mdim(Z:Y) \leq mdim(X:Y) \text{ and } Mdim(Z:Y) \leq Mdim(X:Y). \] We also show how to derive different data processing inequalities by making adjustments to the computable bounds of the use of a Turing functional. The yield of a Turing functional $Φ^S$ with access to at most $n$ bits of the oracle $S$ is the smallest input $m \in \mathbb{N}$ such that $Φ^{S \upharpoonright n}(m)\uparrow$. We show how to derive reverse data processing inequalities (i.e., data processing inequalities where the transformation may significantly increase the shared information between two entities) for sequences by applying computable bounds to the yield of a Turing functional.
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。
