  Sophistication and logical depth are two measures that express how
complicated the structure in a string is. Sophistication is defined as the
minimal complexity of a computable function that defines a two-part description
for the string that is shortest within some precision; the second can be
defined as the minimal computation time of a program that is shortest within
some precision. We show that the Busy Beaver function of the sophistication of
a string exceeds its logical depth with logarithmically bigger precision, and
that logical depth exceeds the Busy Beaver function of sophistication with
logarithmically bigger precision. We also show that this is not true if the
precision is only increased by a constant (when the notions are defined with
plain Kolmogorov complexity). Finally we show that sophistication is unstable
in its precision: constant variations can change its value by a linear term in
the length of the string.
