Kolmogorov复杂性理论中独立性放大的不可能性
Impossibility of independence amplification in Kolmogorov complexity theory
摘要：
本文在Kolmogorov复杂性框架下，研究了有界独立信源的随机性提取和信源的独立性放大问题。字符串$x$和$y$的依赖关系是${\rm dep}(x,y) = \max\{C(x) - C(x \mid y), C(y) - C(y\mid x)\}$，其中$C(\cdot)$表示柯尔莫哥洛夫复杂度。它表明，存在一个可计算的柯尔莫哥洛夫提取器$f$，这样，对于任何两个$n$ -位串的复杂性$s(n)$和依赖$α(n)$，它输出一个字符串的长度$s(n)$与复杂性$s(n)- α(n)$的条件下的任何一个输入字符串。证明了上述参数是Kolmogorov提取器可以实现的最佳参数。结果表明，独立性放大不能有效地实现。具体地说，如果（在排除一个平凡的情况之后）存在可计算函数$f_1$和$f_2$，使得对于所有的$n$，${\rm dep}(f_1(x,y), f_2(x,y)) \leq β(n)$-位串$x$和$y$与${\rm dep}(x,y) \leq α(n)$，则$β(n) \geq α(n) - O(\log n)$ .
英文摘要：
The paper studies randomness extraction from sources with bounded independence and the issue of independence amplification of sources, using the framework of Kolmogorov complexity. The dependency of strings $x$ and $y$ is ${\rm dep}(x,y) = \max\{C(x) - C(x \mid y), C(y) - C(y\mid x)\}$, where $C(\cdot)$ denotes the Kolmogorov complexity. It is shown that there exists a computable Kolmogorov extractor $f$ such that, for any two $n$-bit strings with complexity $s(n)$ and dependency $α(n)$, it outputs a string of length $s(n)$ with complexity $s(n)- α(n)$ conditioned by any one of the input strings. It is proven that the above are the optimal parameters a Kolmogorov extractor can achieve. It is shown that independence amplification cannot be effectively realized. Specifically, if (after excluding a trivial case) there exist computable functions $f_1$ and $f_2$ such that ${\rm dep}(f_1(x,y), f_2(x,y)) \leq β(n)$ for all $n$-bit strings $x$ and $y$ with ${\rm dep}(x,y) \leq α(n)$, then $β(n) \geq α(n) - O(\log n)$.
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