限制复杂性
Limit complexities revisited
摘要：
本文的主要目的是把一些已知的结果在一个共同的角度，并简化其证明。我们从（Vereshchagin，2002）的一个结果的简单证明开始，即$\limsup_n\KS(x|n)$（这里$\KS(x|n)$是当$n$已知时$x$的条件（普通）柯尔莫哥洛夫复杂度）等于$\KS^{\mathbf{0'}(x)$，即$\mathbf{0'$ -oracle的普通柯尔莫哥洛夫复杂度。然后我们用同样的论证证明了前缀复杂度（也改进了Muchnik，1987）关于极限频率的结果）、二叉树上的先验概率和有效开集的测度的类似结果。作为副产品，我们得到了一个$\mathbf{0'}$ Martin-Löf随机性（也称为2-随机性）的判据（Miller，2004）：序列$ω$是2-随机的当且仅当存在$c$使得$ω$的任何前缀$x$是某个字符串$y$的前缀使得$\KS(y)\ge |y|-c$。(In在20世纪60年代，该性质在（Kolmogorov，1968）中被建议作为可能的随机性定义之一;其与2-随机性的等价性在（Miller，2004）中被示出，同时证明另一个2-随机性准则（也参见（Nies等人，2005））：PDMATHAMT是2-随机的当且仅当对于某些$c$和$ω$的无穷多个前缀$x$，$\KS(x)\ge |x|-c$。最后，我们证明了低基定理可以用来得到这些结果的替代证明，并改进了关于有效开集的结果;这个更强的版本包含了前一句中提到的2-随机性准则。
英文摘要：
The main goal of this paper is to put some known results in a common perspective and to simplify their proofs. We start with a simple proof of a result from (Vereshchagin, 2002) saying that $\limsup_n\KS(x|n)$ (here $\KS(x|n)$ is conditional (plain) Kolmogorov complexity of $x$ when $n$ is known) equals $\KS^{\mathbf{0'}(x)$, the plain Kolmogorov complexity with $\mathbf{0'$-oracle. Then we use the same argument to prove similar results for prefix complexity (and also improve results of (Muchnik, 1987) about limit frequencies), a priori probability on binary tree and measure of effectively open sets. As a by-product, we get a criterion of $\mathbf{0'}$ Martin-Löf randomness (called also 2-randomness) proved in (Miller, 2004): a sequence $ω$ is 2-random if and only if there exists $c$ such that any prefix $x$ of $ω$ is a prefix of some string $y$ such that $\KS(y)\ge |y|-c$. (In the 1960ies this property was suggested in (Kolmogorov, 1968) as one of possible randomness definitions; its equivalence to 2-randomness was shown in (Miller, 2004) while proving another 2-randomness criterion (see also (Nies et al. 2005)): $ω$ is 2-random if and only if $\KS(x)\ge |x|-c$ for some $c$ and infinitely many prefixes $x$ of $ω$. Finally, we show that the low-basis theorem can be used to get alternative proofs for these results and to improve the result about effectively open sets; this stronger version implies the 2-randomness criterion mentioned in the previous sentence.
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