On the rate of decrease in logical depth

The logical depth with significance b of a string x is the shortest running time of a program for x that can be compressed by at most b bits. Another definition is based on algorithmic probability. We give a simple new proof for the known relation between the two definitions. We also prove the following: Given a string we can consider the maximal decrease in logical depth when the significance parameter increases by 1. There exists a sequence of strings of lengths n=1,2,…, such that this maximal decrease as a function of n rises faster than any computable function but not as fast as the Busy Beaver function. This holds also for the computation times of the shortest programs of these strings.