从有限独立源中提取串和序列的Kolmogorov复杂度
Extracting the Kolmogorov Complexity of Strings and Sequences from Sources with Limited Independence
摘要：
一个无限二进制序列的随机率至少为$σ$，如果对于几乎每个$n$，其长度为$n$的前缀的柯尔莫哥洛夫复杂度至少为$σn$。已知对于每个有理$σ\in (0,1)$，一方面存在随机率为$σ$的序列不能有效地变换为随机率高于$σ$的序列，另一方面任意两个随机率为$σ$的独立序列都可以变换为随机率高于$σ$的序列。我们表明，后者的结果，即使两个输入序列具有线性依赖性（这，非正式地说，意味着所有前缀的长度$n$的两个序列有共同的一个常数部分的信息）。研究了有限弦的类似问题。它表明，从任何两个字符串具有足够大的柯尔莫哥洛夫复杂性和足够小的依赖性，可以有效地构建一个字符串是随机的，甚至由任何一个输入字符串的条件。
英文摘要：
An infinite binary sequence has randomness rate at least $σ$ if, for almost every $n$, the Kolmogorov complexity of its prefix of length $n$ is at least $σn$. It is known that for every rational $σ\in (0,1)$, on one hand, there exists sequences with randomness rate $σ$ that can not be effectively transformed into a sequence with randomness rate higher than $σ$ and, on the other hand, any two independent sequences with randomness rate $σ$ can be transformed into a sequence with randomness rate higher than $σ$. We show that the latter result holds even if the two input sequences have linear dependency (which, informally speaking, means that all prefixes of length $n$ of the two sequences have in common a constant fraction of their information). The similar problem is studied for finite strings. It is shown that from any two strings with sufficiently large Kolmogorov complexity and sufficiently small dependence, one can effectively construct a string that is random even conditioned by any one of the input strings.
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