  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.
