关于逻辑深度函数
On the logical depth function
摘要：
对于一个有限的二进制串$x$，它的逻辑深度$d$对于有效性$b$是长度为$K(x)+b$的$x$的程序的最短运行时间。逻辑深度还有另一种定义。我们给出了一个新的证明，这两个版本是接近的。存在连续长度的字符串的无限序列，使得对于每个字符串，存在$b$，使得将$b$递增1使得相关联的深度从不可计算变为可计算。将适当的$b$递增1所产生的深度之间的最大间隙是不可计算的。该间隙的大小由Busy Beaver函数限定上限。对于显著性为0的深度，上界和下界都成立。因此，相关的最短程序的最小计算时间比任何可计算函数都要快，但不如Busy Beaver函数那么快。
英文摘要：
For a finite binary string $x$ its logical depth $d$ for significance $b$ is the shortest running time of a program for $x$ of length $K(x)+b$. There is another definition of logical depth. We give a new proof that the two versions are close. There is an infinite sequence of strings of consecutive lengths such that for every string there is a $b$ such that incrementing $b$ by 1 makes the associated depths go from incomputable to computable. The maximal gap between depths resulting from incrementing appropriate $b$'s by 1 is incomputable. The size of this gap is upper bounded by the Busy Beaver function. Both the upper and the lower bound hold for the depth with significance 0. As a consequence, the minimal computation time of the associated shortest programs rises faster than any computable function but not so fast as the Busy Beaver function.
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