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.
Research done during an academic internship at NEC. This author is partially supported by funds granted to LIACC through the Programa de Financiamento Plurianual, Fundação para a Ciência e Tecnologia and Programa POSI.
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Antunes, L., Fortnow, L. (2003). Sophistication Revisited. In: Baeten, J.C.M., Lenstra, J.K., Parrow, J., Woeginger, G.J. (eds) Automata, Languages and Programming. ICALP 2003. Lecture Notes in Computer Science, vol 2719. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45061-0_23
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DOI: https://doi.org/10.1007/3-540-45061-0_23
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