简单停止时间和条件复杂性
Plain stopping time and conditional complexities revisited
摘要：
在本文中，我们分析了“停止时间复杂度”的概念，非正式地定义为在读取无限序列时指定何时停止所需的信息量。这个概念是由Vovk和Pavlovic（2016）提出的。结果表明，二进制字符串$x$的纯停止时间复杂度可以等价地定义为：（a）在单向输入带上读取$x$后停止的图灵机的最小纯复杂度;（b）枚举包含$x$的无前缀集合的算法的最小纯复杂度;（c）~条件复杂度$C(x|x*)$，其中条件中的$x$被理解为无限二进制序列的前缀，而第一个$x$被理解为终止的二进制串;（d）作为最小上半可计算函数$K$，使得每个二进制序列至多具有$2^n$前缀$z$，使得$K(z)<n$ ;（e）作为$\max C^X(x)$，其中$C^X(z)$是$z$相对于oracle $X$的普通柯尔莫哥洛夫复杂度，并且最大值被取在$x$的所有扩展$X$上。我们还表明，这些等价的定义中的一些成为不等价的条件$y$和对象$x$可能不同的更一般的设置。我们还回答了一个开放性的问题，从Zerov，Hutter和~Schmidhuber。
英文摘要：
In this paper we analyze the notion of "stopping time complexity", informally defined as the amount of information needed to specify when to stop while reading an infinite sequence. This notion was introduced by Vovk and Pavlovic (2016). It turns out that plain stopping time complexity of a binary string $x$ could be equivalently defined as (a) the minimal plain complexity of a Turing machine that stops after reading $x$ on a one-directional input tape; (b) the minimal plain complexity of an algorithm that enumerates a prefix-free set containing $x$; (c)~the conditional complexity $C(x|x*)$ where $x$ in the condition is understood as a prefix of an infinite binary sequence while the first $x$ is understood as a terminated binary string; (d) as a minimal upper semicomputable function $K$ such that each binary sequence has at most $2^n$ prefixes $z$ such that $K(z)<n$; (e) as $\max C^X(x)$ where $C^X(z)$ is plain Kolmogorov complexity of $z$ relative to oracle $X$ and the maximum is taken over all extensions $X$ of $x$. We also show that some of these equivalent definitions become non-equivalent in the more general setting where the condition $y$ and the object $x$ may differ. We also answer an open question from Chernov, Hutter and~Schmidhuber.
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