A Variant Of Chaitin's Omega function
Yuxuan Li, Shuheng Zhang, Xiaoyan Zhang, Xuanheng Zhao
TL;DR
This paper introduces a continuous variant of Chaitin's Omega, defined by $f(x)=\sum_{\sigma\le_L x}2^{-K(\sigma)}$, and analyzes it through analysis, computability, and algorithmic randomness. It establishes that $f$ is differentiable precisely at density random points with derivative $0$ when it exists, and that $f(x)$ is $x$-random iff $x$ is weakly low for $K$. The range $f(2^\omega)$ is a null, nowhere dense, perfect $Π^0_1(\emptyset')$ class with Hausdorff dimension $1$, and the function satisfies $f(x)\oplus x\ge_T\emptyset'$ for all $x$, while being non-invariant under Turing reductions in general but invariant on the $K$-trivial ideal. The authors connect $f$ to other Omega variants and show that a rich set of randomness and dimension properties carry over to this variant, including the existence of continuum many $x$ with non-1-random $f(x)$ and a detailed analysis of left-c.e. tests related to $2$-randomness.
Abstract
We investigate the continuous function $f$ defined by $$x\mapsto \sum_{σ\le_L x }2^{-K(σ)}$$ as a variant of Chaitin's Omega from the perspective of analysis, computability, and algorithmic randomness. Among other results, we obtain that: (i) $f$ is differentiable precisely at density random points; (ii) $f(x)$ is $x$-random if and only if $x$ is weakly low for $K$ (low for $Ω$); (iii) the range of $f$ is a null, nowhere dense, perfect $Π^0_1(\emptyset')$ class with Hausdorff dimension $1$; (iv) $f(x)\oplus x\ge_T\emptyset'$ for all $x$; (v) there are $2^{\aleph_0}$ many $x$ such that $f(x)$ is not 1-random; (vi) $f$ is not Turing invariant but is Turing invariant on the ideal of $K$-trivial reals. We also discuss the connection between $f$ and other variants of Omega.