  The logical depth with significance $b$ of a finite binary string $x$ is the
shortest running time of a binary program for $x$ that can be compressed by at
most $b$ bits. There is another definition of logical depth. We give two
theorems about the quantitative relation between these versions: the first
theorem concerns a variation of a known fact with a new proof, the second
theorem and its proof are new. We select the above version of logical depth and
show the following. There is an infinite sequence of strings of increasing
length such that for each $j$ there is a $b$ such that the logical depth of the
$j$th string as a function of $j$ is incomputable (it rises faster than any
computable function) but with $b$ replaced by $b+1$ the resuling function is
computable. Hence the maximal gap between the logical depths resulting from
incrementing appropriate $b$'s by 1 rises faster than any computable function.
All functions mentioned are upper bounded by the Busy Beaver function. Since
for every string its logical depth is nonincreasing in $b$, the minimal
computation time of the shortest programs for the sequence of strings as a
function of $j$ rises faster than any computable function but not so fast as
the Busy Beaver function.
