Abstract
Assume that a program p produces an output string b for an input string a: p(a) = b. We look for a “reduction” (simplification) of p, i.e., a program q such that q(a) = b but q has Kolmogorov complexity smaller than p and contains no additional information as compared to p (this means that the conditional complexity K(q|p) is negligible). We show that, for any two strings a and b (except for some degenerate cases), one can find a nonreducible program p of any arbitrarily large complexity (any value larger than K(a) + K(b|a) is possible).
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REFERENCES
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Muchnik, A., Shen, A., Vereshchagin, N., and Vyugin, M., Non-reducible Descriptions for Conditional Kolmogorov Complexity, Electron. Colloq. Comput. Complexity, 2004, vol. 11, Technical Report TR04-54. Available from ftp://ftp.eccc.uni-trier.de/pub/eccc/reports/2004/TR04-054/Paper.pdf.
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Translated from Problemy Peredachi Informatsii, No. 3, 2005, pp. 58–63.
Original Russian Text Copyright © 2005 by Ustinov.
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Ustinov, M.A. Nonreducible Descriptions for the Conditional Kolmogorov Complexity. Probl Inf Transm 41, 237–242 (2005). https://doi.org/10.1007/s11122-005-0028-0
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DOI: https://doi.org/10.1007/s11122-005-0028-0