单源Kolmogorov抽取器的最优界
Optimal bounds for single-source Kolmogorov extractors
摘要：
字符串$σ$的随机率（或维数）是比率$C(σ)/|σ|$，其中$C(σ)$是$σ$的柯尔莫哥洛夫复杂度。虽然众所周知，单个可计算变换无法增加所有序列的随机率，但Fortnow、Hitchcock、Pavan、Vinodchandran和Wang表明，对于任何$0<α<β<1$，存在有限数量的可计算变换，使得任何速率至少为$α$的字符串都可以通过其中一种变换变成速率至少为$β$的字符串。然而，他们的证明只给出了变换次数和随机性增加率之间的对应关系的非常宽松的界限。通过将这个问题转化为（超）图上的组合数学，我们提供了一个紧界，即：使用$k$变换，可以得到从速率$α$到任何速率$β< kα/(1+(k-1)α)$的增加，这是最优的。
英文摘要：
The rate of randomness (or dimension) of a string $σ$ is the ratio $C(σ)/|σ|$ where $C(σ)$ is the Kolmogorov complexity of $σ$. While it is known that a single computable transformation cannot increase the rate of randomness of all sequences, Fortnow, Hitchcock, Pavan, Vinodchandran, and Wang showed that for any $0<α<β<1$, there are a finite number of computable transformations such that any string of rate at least $α$ is turned into a string of rate at least $β$ by one of these transformations. However, their proof only gives very loose bounds on the correspondence between the number of transformations and the increase of rate of randomness one can achieve. By translating this problem to combinatorics on (hyper)graphs, we provide a tight bound, namely: Using $k$ transformations, one can get an increase from rate $α$ to any rate $β< kα/(1+(k-1)α)$, and this is optimal.
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。
