数学信息距离的精确表示
Precise Expression for the Algorithmic Information Distance
摘要：
我们考虑Bennett、Gács、Li、Vitányi和Zurek在1998年提出的两个对象$x$和$y$之间的信息距离的概念，作为从$y$计算$x$以及从$x$计算$y$的程序的最小长度。本文证明了在加性对数项下距离等于$\max (K(x|y),K(y|x))$，并证明了这不能提高到$O(1)$精度。我们重新审视定义中的微妙问题，并证明了这个猜想。我们表明，如果距离至少是对数的长度，那么这个等式并不持有$O(1)$精度为字符串的长度相等。因此，对于这样的字符串，三角不等式和特征都以最佳精度成立。最后，我们将结果推广到有界大小的集合$S$。我们表明，对于每个常数~ $s$，打印$s$元素集$S \subseteq \{0,1\}^n$给定其任何元素的最短的程序，具有最多$\max_{w \in S} K(S|w) + O(1)$的长度，只要这个最大值至少是对数的~ $n$。
英文摘要：
We consider the notion of information distance between two objects $x$ and $y$ introduced by Bennett, Gács, Li, Vitányi, and Zurek in 1998 as the minimal length of a program that computes $x$ from $y$ as well as computing $y$ from $x$. In this paper, it was proven that the distance is equal to $\max (K(x|y),K(y|x))$ up to additive logarithmic terms, and it was conjectured that this could not be improved to $O(1)$ precision. We revisit subtle issues in the definition and prove this conjecture. We show that if the distance is at least logarithmic in the length, then this equality does hold with $O(1)$ precision for strings of equal length. Thus for such strings, both the triangle inequality and the characterization hold with optimal precision. Finally, we extend the result to sets $S$ of bounded size. We show that for each constant~$s$, the shortest program that prints an $s$-element set $S \subseteq \{0,1\}^n$ given any of its elements, has length at most $\max_{w \in S} K(S|w) + O(1)$, provided this maximum is at least logarithmic in~$n$.
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