The purpose of this paper is to answer two questions left open in Durand et al. (2001) [2]. Namely, we consider the following two complexities of an infinite computable 0-1-sequence α: $C^{0^{\prime }}(\alpha )$, defined as the minimal length of a program with oracle 0′ that prints α, and $M_{\infty }(\alpha )$, defined as $\lim \sup C({\alpha }_{1:n}|n)$, where ${\alpha }_{1:n}$ denotes the length-n prefix of α and $C(x|y)$ stands for conditional Kolmogorov complexity. We show that $C^{0^{\prime }}(\alpha )⩽M_{\infty }(\alpha )+O(1)$ and $M_{\infty }(\alpha )$ is not bounded by any computable function of $C^{0^{\prime }}(\alpha )$, even on the domain of computable sequences.

中文翻译：

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重述可计算序列的描述复杂性

本文的目的是回答Durand等人悬而未决的两个问题。（2001）[2]。即，我们考虑无限可计算的0-1序列α的以下两个复杂度：$C^{0^{\prime }}（\alpha ）$，定义为带有oracle 0'且输出α的程序的最小长度，以及${\mathrm{中号}}_{\infty }（\alpha ）$， 定义为 $\mathrm{林}\mathrm{SUP}C（{\alpha }_{\mathrm{1个}：ñ}|ñ）$，在哪里 ${\alpha }_{\mathrm{1个}：ñ}$表示长度- Ñ的前缀α和$C（X|ÿ）$代表条件Kolmogorov复杂度。我们证明$C^{0^{\prime }}（\alpha ）⩽{\mathrm{中号}}_{\infty }（\alpha ）+Ø（\mathrm{1个}）$ 和 ${\mathrm{中号}}_{\infty }（\alpha ）$ 不受任何可计算函数的限制 $C^{0^{\prime }}（\alpha ）$，甚至在可计算序列的域上。