Information measures for infinite sequences
Theoretical Computer Science
Effective complexity and its relation to logical depth
IEEE Transactions on Information Theory
CiE'12 Proceedings of the 8th Turing Centenary conference on Computability in Europe: how the world computes
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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.