arXiv subjects

Rodney Downey

Publications and source records attributed to Rodney Downey.

2 recordsLinked to original sources

arXiv2024

Low$_2$ computably enumerable sets have hyperhypersimple supersets

A longstanding question is to characterize the lattice of supersets (modulo finite sets), $\mathcal{L}^*(A)$, of a low$_2$ computably enumerable (c.e.) set. The conjecture is that $\mathcal{L}^*(A)\cong {\mathcal E}^*$. In spite of claims in the literature, this longstanding question/conjecture remains open. We contribute to this problem by solving one of the main test cases. We show that if c.e.\ $A$ is low$_2$ then $A$ has an atomless hyperhypersimple superset. In fact, if $A$ is c.e.\ and low$_2$, then for any $\Sigma_3$-Boolean algebra~$B$ there is some c.e.\ $H\supseteq A$ such that $\mathcal{L}^*(H)\cong B$.

math.LO↗

arXiv2022

Limit Complexities, Minimal Descriptions, and $n$-Randomness

Let $K$ denote prefix-free Kolmogorov Complexity, and $K^A$ denote it relative to an oracle $A$. We show that for any $n$, $K^{\emptyset^{(n)}}$ is definable purely in terms of the unrelativized notion $K$. It was already known that 2-randomness is definable in terms of $K$ (and plain complexity $C$) as those reals which infinitely often have maximal complexity. We can use our characterization to show that $n$-randomness is definable purely in terms of $K$. To do this we extend a certain ``limsup'' formula from the literature, and apply Symmetry of Information. This extension entails a novel use of semilow sets, and a more precise analysis of the complexity of $\Delta_2^0$ sets of mimimal descriptions.

math.LO↗