  For a finite binary string $x$ its logical depth $d$ for significance $b$ is
the shortest running time of a program for $x$ of length $K(x)+b$. There is
another definition of logical depth. We give a new proof that the two versions
are close. There is an infinite sequence of strings of consecutive lengths such
that for every string there is a $b$ such that incrementing $b$ by 1 makes the
associated depths go from incomputable to computable. The maximal gap between
depths resulting from incrementing appropriate $b$'s by 1 is incomputable. The
size of this gap is upper bounded by the Busy Beaver function. Both the upper
and the lower bound hold for the depth with significance 0. As a consequence,
the minimal computation time of the associated shortest programs rises faster
than any computable function but not so fast as the Busy Beaver function.
