Sophistication revisited

Abstract

The Kolmogorov structure function divides the smallest program producing a string in two parts: the useful information present in the string, called sophistication if based on total functions, and the remaining accidental information. We revisit the notion of sophistication due to Koppel, formalize a connection between sophistication and a variation of computational depth (intuitively the useful or nonrandom information in a string), prove the existence of strings with maximum sophistication and show that they encode solutions of the halting problem, i.e., they are the deepest of all strings. © Springer-Verlag Berlin Heidelberg 2003.