  The information in an individual finite object (like a binary string) is
commonly measured by its Kolmogorov complexity. One can divide that information
into two parts: the information accounting for the useful regularity present in
the object and the information accounting for the remaining accidental
information. There can be several ways (model classes) in which the regularity
is expressed. Kolmogorov has proposed the model class of finite sets,
generalized later to computable probability mass functions. The resulting
theory, known as Algorithmic Statistics, analyzes the algorithmic sufficient
statistic when the statistic is restricted to the given model class. However,
the most general way to proceed is perhaps to express the useful information as
a recursive function. The resulting measure has been called the
``sophistication'' of the object. We develop the theory of recursive functions
statistic, the maximum and minimum value, the existence of absolutely
nonstochastic objects (that have maximal sophistication--all the information in
them is meaningful and there is no residual randomness), determine its relation
with the more restricted model classes of finite sets, and computable
probability distributions, in particular with respect to the algorithmic
(Kolmogorov) minimal sufficient statistic, the relation to the halting problem
and further algorithmic properties.
