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

TL;DR

This paper demonstrates that higher-level relativized Kolmogorov complexities are definable using unrelativized complexity and extends the characterization of n-randomness through new formulas and techniques involving semilow sets.

Contribution

It provides a new characterization of relativized Kolmogorov complexities and n-randomness purely in terms of unrelativized complexity, using novel formulas and set-theoretic analysis.

Findings

Relativized complexities $K^{ ext{oracle}}$ are definable from unrelativized $K$.

n-randomness can be characterized without oracles, using extended formulas.

Introduces a novel use of semilow sets and analyzes complexity of minimal descriptions.

Abstract

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.