  It is shown that the length of the algorithmic minimal sufficient statistic
of a binary string x, either in a representation of a finite set, computable
semimeasure, or a computable function, has a length larger than the
computational depth of x, and can solve the Halting problem for all programs
with length shorter than the m-depth of x. It is also shown that there are
strings for which the algorithmic minimal sufficient statistics can contain a
substantial amount of information that is not Halting information. The weak
sufficient statistic is introduced, and it is shown that a minimal weak
sufficient statistic for x is equivalent to a minimal typical model of x, and
to the Halting problem for all strings shorter than the BB-depth of x.
