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 Tr. Mat. Inst. Steklova, 2011, Volume 274, Pages 103–118 (Mi tm3316)

On joint conditional complexity (entropy)

Nikolay K. Vereshchagina, Andrej A. Muchnik

a Department of Mathematical Logic and Theory of Algorithms, Faculty of Mechanics and Mathematics, Moscow State University, Moscow, Russia

Abstract: The conditional Kolmogorov complexity of a word $a$ relative to a word $b$ is the minimum length of a program that prints $a$ given $b$ as an input. We generalize this notion to quadruples of strings $a,b,c,d$: their joint conditional complexity $K((a\to c)\land(b\to d))$ is defined as the minimum length of a program that transforms $a$ into $c$ and transforms $b$ into $d$. In this paper, we prove that the joint conditional complexity cannot be expressed in terms of the usual conditional (and unconditional) Kolmogorov complexity. This result provides a negative answer to the following question asked by A. Shen on a session of the Kolmogorov seminar at Moscow State University in 1994: Is there a problem of information processing whose complexity is not expressible in terms of the conditional (and unconditional) Kolmogorov complexity? We show that a similar result holds for the classical Shannon entropy. We provide two proofs of both results, an effective one and a “quasi-effective” one. Finally, we present a quasi-effective proof of a strong version of the following statement: there are two strings whose mutual information cannot be extracted. Previously, only a noneffective proof of that statement has been known.

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English version:
Proceedings of the Steklov Institute of Mathematics, 2011, 274, 90–104

Bibliographic databases:

UDC: 510.5

Citation: Nikolay K. Vereshchagin, Andrej A. Muchnik, “On joint conditional complexity (entropy)”, Algorithmic aspects of algebra and logic, Collected papers. Dedicated to Academician Sergei Ivanovich Adian on the occasion of his 80th birthday, Tr. Mat. Inst. Steklova, 274, MAIK Nauka/Interperiodica, Moscow, 2011, 103–118; Proc. Steklov Inst. Math., 274 (2011), 90–104

Citation in format AMSBIB
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\paper On joint conditional complexity (entropy)
\inbook Algorithmic aspects of algebra and logic
\bookinfo Collected papers. Dedicated to Academician Sergei Ivanovich Adian on the occasion of his 80th birthday
\serial Tr. Mat. Inst. Steklova
\yr 2011
\vol 274
\pages 103--118
\publ MAIK Nauka/Interperiodica
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\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2962936}
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\jour Proc. Steklov Inst. Math.
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\pages 90--104
\crossref{https://doi.org/10.1134/S008154381106006X}
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This publication is cited in the following articles:
1. Ewert W., Dembski W.A., Marks Ii R.J., “Measuring Meaningful Information in Images: Algorithmic Specified Complexity”, IET Comput. Vis., 9:6 (2015), 884–894
2. Lukas L., Plevny M., “Using entropy for quantitative measurement of operational complexity of supplier-customer system: case studies”, Cent. Europ. J. Oper. Res., 24:2, SI (2016), 371–387
3. Lukas L., Hofman J., “Operational Complexity of Supplier-Customer Systems Measured by Entropy-Case Studies”, Entropy, 18:4 (2016), UNSP 137
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