归一化的算法信息距离不能近似
The normalized algorithmic information distance can not be approximated
摘要：
已知归一化算法信息距离$N$不可计算且不可半计算。我们表明，对于所有的$ε< 1/2$，有没有半可计算的功能，从$N$不同的最多~ $ε$。此外，对于任何可计算函数$f$使得$|\lim_t f(x,y,t) - N(x,y)| \le ε$和对于所有$n$，存在长度为$n$使得$\sum_t |f(x,y,t+1) - f(x,y,t)| \ge Ω(\log n)$的字符串$x,y$。这是最佳的常数因子。我们还证明了$N$的极限近似的最大振荡数是$Ω(n/\log n)$。这加强了来自[K]的$ω(1)$下界。Ambos-Spies，W. Merkle和S.A. Terwijn，2019，归一化信息距离和振荡层次，见arXiv：1708.03583。
英文摘要：
It is known that the normalized algorithmic information distance $N$ is not computable and not semicomputable. We show that for all $ε< 1/2$, there exist no semicomputable functions that differ from $N$ by at most~$ε$. Moreover, for any computable function $f$ such that $|\lim_t f(x,y,t) - N(x,y)| \le ε$ and for all $n$, there exist strings $x,y$ of length $n$ such that $\sum_t |f(x,y,t+1) - f(x,y,t)| \ge Ω(\log n)$. This is optimal up to constant factors. We also show that the maximal number of oscillations of a limit approximation of $N$ is $Ω(n/\log n)$. This strengthens the $ω(1)$ lower bound from [K. Ambos-Spies, W. Merkle, and S.A. Terwijn, 2019, Normalized information distance and the oscillation hierarchy], see arXiv:1708.03583 .
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