  Logical depth and sophistication are two quantitative measures of the
non-trivial organization of an object. Although apparently different, these
measures have been proven equivalent, when the logical depth is renormalized by
the busy beaver function. In this article, the measures are relativized to
auxiliary information and re-compared to one another. The ability of auxiliary
information to solve the halting problem introduces a distortion between the
measures. Finally, similar to algorithmic complexity, sophistication and
logical depth (renormalized) each offer a relation between their expression of
$(x, y)$, $(x)$ and $(y | x)$.
