Abstract
Kolmogorov complexity measures the amount of information in a string as the size of the shortest program that computes the string. 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 formalize a connection between sophistication (due to Koppel) 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 are the deepest of all strings.
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