We prove that the continuous function${\rm{\hat \Omega }}:2^\omega \to $ that is defined via$X \mapsto \mathop \sum \limits_n 2^{ - K\left( {Xn} \right)} $ for all $X \in {2^\omega }$ is differentiable exactly at the Martin-Löf random reals with the derivative having value 0; that it is nowhere monotonic; and that $\mathop \smallint \nolimits _0^1{\rm{\hat{\Omega }}}\left( X \right)\,{\rm{d}}X$ is a left-c.e. $wtt$-complete real having effective Hausdorff dimension ${1 / 2}$.We further investigate the algorithmic properties of ${\rm{\hat{\Omega }}}$. For example, we show that the maximal value of ${\rm{\hat{\Omega }}}$ must be random, the minimal value must be Turing complete, and that ${\rm{\hat{\Omega }}}\left( X \right) \oplus X{ \ge _T}\emptyset \prime$ for every X. We also obtain some machine-dependent results, including that for every $\varepsilon > 0$, there is a universal machine V such that ${{\rm{\hat{\Omega }}}_V}$ maps every real X having effective Hausdorff dimension greater than ε to a real of effective Hausdorff dimension 0 with the property that $X{ \le _{tt}}{{\rm{\hat{\Omega }}}_V}\left( X \right)$; and that there is a real X and a universal machine V such that ${{\rm{\Omega }}_V}\left( X \right)$ is rational.

中文翻译：

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CHAITIN 的 Ω 作为一个连续函数

我们证明了连续函数${\rm{\hat \Omega }}:2^\omega \to $这是通过定义的$X \mapsto \mathop \sum \limits_n 2^{ - K\left( {Xn} \right)} $对所有人$X \in {2^\omega }$可在 Martin-Löf 随机实数上精确微分，导数为 0；它不是单调的；然后$\mathop \smallint \nolimits _0^1{\rm{\hat{\Omega }}}\left( X \right)\,{\rm{d}}X$是左撇子$wt$- 具有有效 Hausdorff 维数的完全实数${1 / 2}$.我们进一步研究了算法的性质${\rm{\hat{\Omega }}}$. 例如，我们证明了最大值${\rm{\hat{\Omega }}}$必须是随机的，最小值必须是图灵完备的，并且${\rm{\hat{\Omega }}}\left( X \right) \oplus X{ \ge _T}\emptyset \prime$对于每个X. 我们还获得了一些与机器相关的结果，包括对于每个$\伐普西隆 > 0$，有万能机五这样${{\rm{\hat{\Omega }}}_V}$映射每一个真实的X有效豪斯多夫维数大于ε到有效 Hausdorff 维数 0 的实数，其性质为$X{ \le _{tt}}{{\rm{\hat{\Omega }}}_V}\left( X \right)$; 并且有一个真实的X和一台通用机器五这样${{\rm{\Omega }}_V}\left( X \right)$是理性的。