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 Funktsional. Anal. i Prilozhen.: Year: Volume: Issue: Page: Find

 Funktsional. Anal. i Prilozhen., 2014, Volume 48, Issue 2, Pages 51–66 (Mi faa3141)

Zipf's Law and L. Levin Probability Distributions

Yu. I. Manin

Max Planck Institute for Mathematics

Abstract: Zipf's law in its basic incarnation is an empirical probability distribution governing the frequency of usage of words in a language. As Terence Tao recently remarked, it still lacks a convincing and satisfactory mathematical explanation.
In this paper I suggest that, at least in certain situations, Zipf's law can be explained as a special case of the a priori distribution introduced and studied by L. Levin. The Zipf ranking corresponding to diminishing probability appears then as the ordering by growing Kolmogorov complexity.
One argument justifying this assertion is the appeal to a recent interpretation by Yu. Manin and M. Marcolli of asymptotic bounds for error-correcting codes in terms of phase transition. In the respective partition function, the Kolmogorov complexity of a code plays the role of its energy.

Keywords: Zipf's law, Kolmogorov complexity

DOI: https://doi.org/10.4213/faa3141

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English version:
Functional Analysis and Its Applications, 2014, 48:2, 116–127

Bibliographic databases:

UDC: 519.1+519.2

Citation: Yu. I. Manin, “Zipf's Law and L. Levin Probability Distributions”, Funktsional. Anal. i Prilozhen., 48:2 (2014), 51–66; Funct. Anal. Appl., 48:2 (2014), 116–127

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/faa3141
• https://doi.org/10.4213/faa3141
• http://mi.mathnet.ru/eng/faa/v48/i2/p51

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Y.I. Manin, “Complexity vs energy: theory of computation and theoretical physics”, 3Quantum: Algebra Geometry Information, J. Phys.: Conf. Ser., 532, IOP Publishing Ltd, 2014, 012018
2. Yuri I. Manin, “Neural codes and homotopy types: mathematical models of place field recognition”, Mosc. Math. J., 15:4 (2015), 741–748
3. Manin Yu.I., Marcolli M., “Semantic Spaces”, Math. Comput. Sci., 10:4 (2016), 459–477
4. C. S. Calude, M. Dumitrescu, “A probabilistic anytime algorithm for the halting problem”, Computability, 7:2-3 (2018), 259–271
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