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Uspekhi Mat. Nauk, 1970, Volume 25, Issue 6(156), Pages 85–127 (Mi umn5428)  

This article is cited in 169 scientific papers (total in 169 papers)

The complexity of finite objects and the development of the concepts of information and randomness by means of the theory of algorithms

A. K. Zvonkin, L. A. Levin

Abstract: In 1964 Kolmogorov introduced the concept of the complexity of a finite object (for instance, the words in a certain alphabet). He defined complexity as the minimum number of binary signs containing all the information about a given object that are sufficient for its recovery (decoding). This definition depends essentially on the method of decoding. However, by means of the general theory of algorithms, Kolmogorov was able to give an invariant (universal) definition of complexity. Related concepts were investigated by Solomonoff (U.S.A.) and Markov. Using the concept of complexity, Kolmogorov gave definitions of the quantity of information in finite objects and of the concept of a random sequence (which was then defined more precisely by Martin-Löf). Afterwards, this circle of questions developed rapidly. In particular, an interesting development took place of the ideas of Markov on the application of the concept of complexity to the study of quantitative questions in the theory of algorithms. The present article is a survey of the fundamental results connected with the brief remarks above.

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English version:
Russian Mathematical Surveys, 1970, 25:6, 83–124

Bibliographic databases:

UDC: 519.24+517.11+519.9
MSC: 68Q30, 94B35
Received: 07.08.1970

Citation: A. K. Zvonkin, L. A. Levin, “The complexity of finite objects and the development of the concepts of information and randomness by means of the theory of algorithms”, Uspekhi Mat. Nauk, 25:6(156) (1970), 85–127; Russian Math. Surveys, 25:6 (1970), 83–124

Citation in format AMSBIB
\by A.~K.~Zvonkin, L.~A.~Levin
\paper The complexity of finite objects and the development of the concepts of information and randomness by means of the theory of algorithms
\jour Uspekhi Mat. Nauk
\yr 1970
\vol 25
\issue 6(156)
\pages 85--127
\jour Russian Math. Surveys
\yr 1970
\vol 25
\issue 6
\pages 83--124

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    1. R.P.. Daley, “Noncomplex sequences: characterizations and examples”, J. symb. log, 41:03 (1976), 626  crossref
    2. Robert Daley, “On the inference of optimal descriptions”, Theoretical Computer Science, 4:3 (1977), 301  crossref
    3. A. A. Brudno, “The complexity of the trajectories of a dynamical system”, Russian Math. Surveys, 33:1 (1978), 197–198  mathnet  crossref  mathscinet  zmath
    4. J.M. Maciejowski, “Model discrimination using an algorithmic information criterion”, Automatica, 15:5 (1979), 579  crossref
    5. Peter Gács, “Exact Expressions for Some Randomness Tests”, Z Math Logik Grundlagen Math, 26:25-27 (1980), 385  crossref  mathscinet  zmath  isi
    6. J. P. Crutchfield, N. H. Packard, “Symbolic dynamics of one-dimensional maps: Entropies, finite precision, and noise”, Int J Theor Phys, 21:6-7 (1982), 433  crossref  mathscinet  zmath  isi
    7. Charles H. Bennett, “The thermodynamics of computation—a review”, Int J Theor Phys, 21:12 (1982), 905  crossref  isi
    8. A. Muir, M. W. Warner, “The dynamics of symmetric nets”, Bull Math Biol, 45:5 (1983), 781  crossref  mathscinet  zmath  isi
    9. A. N. Kolmogorov, “Combinatorial foundations of information theory and the calculus of probabilities”, Russian Math. Surveys, 38:4 (1983), 29–40  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    10. M. Milgram, H. Atlan, “Probabilistic automata as a model for epigenesis of cellular networks”, Journal of Theoretical Biology, 103:4 (1983), 523  crossref
    11. Péter Gács, “On the relation between descriptional complexity and algorithmic probability”, Theoretical Computer Science, 22:1-2 (1983), 71  crossref
    12. Stuart A. Kurtz, “On the random oracle hypothesis”, Information and Control, 57:1 (1983), 40  crossref
    13. M. A. Jiménez-Montaño, “On the syntactic structure of protein sequences and the concept of grammar complexity”, Bull Math Biol, 46:4 (1984), 641  crossref  mathscinet  zmath  isi
    14. E. A. Asarin, “On convergence of uniform approximations of continuous functions”, Russian Math. Surveys, 39:3 (1984), 179–193  mathnet  crossref  mathscinet  zmath  adsnasa  isi
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    18. H Atlan, “Self creation of meaning”, Phys Scr, 36:3 (1987), 563  crossref  adsnasa  isi
    19. A. N. Kolmogorov, V. A. Uspenskii, “Algorithms and Randomness”, Theory Probab Appl, 32:3 (1987), 389  mathnet  crossref  mathscinet  zmath  isi
    20. Kristian Lindgren, “Microscopic and macroscopic entropy”, Phys Rev A, 38:9 (1988), 4794  crossref  isi
    21. P. Vitani, M. Li, “Kolmogorovskaya slozhnost: dvadtsat let spustya”, UMN, 43:6(264) (1988), 129–166  mathnet  mathscinet  zmath
    22. J. Higgins, “Embedding recursive functions in universal algorithms”, International Journal of Computer Mathematics, 24:3-4 (1988), 273  crossref
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    27. P R Baldwin, J Phys A Math Gen, 24:16 (1991), L941  crossref  mathscinet  zmath  adsnasa  isi
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    147. Uspensky V.A., V'yugin V.V., “Development of the algorithmic information theory in Russia”, Journal of Communications Technology and Electronics, 56:6 (2011), 739–747  crossref  isi
    148. Gidlevskii A.V., “Paradoksy entropii (spekulyativnyi kharakter sovremennoi nauchnoi metodologii)”, Vestnik omskogo universiteta, 2011, no. 4, 199–201  elib
    149. Verónica Becher, Pablo Ariel Heiber, “A linearly computable measure of string complexity”, Theoretical Computer Science, 2012  crossref
    150. Boris Ryabko, Zhanna Reznikova, Alexey Druzyaka, Sofia Panteleeva, “Using Ideas of Kolmogorov Complexity for Studying Biological Texts”, Theory Comput Syst, 2012  crossref
    151. Adam R. Day, “Process and truth-table characterisations of randomness”, Theoretical Computer Science, 2012  crossref
    152. Luís Antunes, Armando Matos, Alexandre Pinto, André Souto, Andreia Teixeira, “One-Way Functions Using Algorithmic and Classical Information Theories”, Theory Comput Syst, 2012  crossref
    154. Frank Stephan, Jason Teutsch, “An incomplete set of shortest descriptions”, J. symb. log, 77:01 (2012), 291  crossref
    155. José Hernández-Orallo, David L. Dowe, “On Potential Cognitive Abilities in the Machine Kingdom”, Minds & Machines, 2013  crossref
    156. Sergio Romano, Mariano Sigman, Santiago Figueira, “$LT^2C^2$ : A language of thought with Turing-computable Kolmogorov complexity”, Pap. Phys, 5 (2013)  crossref
    157. Dusko Pavlovic, “Monoidal computer I: Basic computability by string diagrams”, Information and Computation, 2013  crossref
    159. P.M..B. Vitányi, “Conditional Kolmogorov complexity and universal probability”, Theoretical Computer Science, 2013  crossref
    160. Nicolas Gauvrit, Hector Zenil, Jean-Paul Delahaye, Fernando Soler-Toscano, “Algorithmic complexity for short binary strings applied to psychology: a primer”, Behav Res, 2013  crossref
    161. Frank Stephan, Jason Teutsch, “Things that can be made into themselves”, Information and Computation, 2014  crossref
    162. Kenshi Miyabe, “Algorithmic randomness over general spaces”, Math. Log. Quart, 60:3 (2014), 184  crossref
    163. Jason Teutsch, “Short lists for shortest descriptions in short time”, comput. complex, 2014  crossref
    164. Yu. I. Manin, “Zipf's Law and L. Levin Probability Distributions”, Funct. Anal. Appl., 48:2 (2014), 116–127  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    165. Tor Lattimore, Marcus Hutter, “On Martin-Löf (non-)convergence of Solomonoff's universal mixture”, Theoretical Computer Science, 2014  crossref
    166. Manin Y.I., “Complexity Vs Energy: Theory of Computation and Theoretical Physics”, 3Quantum: Algebra Geometry Information, Journal of Physics Conference Series, 532, IOP Publishing Ltd, 2014, 012018  crossref  isi
    167. Manin Yu., Marcolli M., “Kolmogorov Complexity and the Asymptotic Bound For Error-Correcting Codes”, J. Differ. Geom., 97:1 (2014), 91–108  isi
    168. Mikhail Andreev, Akim Kumok, “The Sum 2 KM(x)−K(x) Over All Prefixes x of Some Binary Sequence Can be Infinite”, Theory Comput Syst, 2015  crossref
    169. J. Math. Sci. (N. Y.), 226:5 (2017), 667–693  mathnet  crossref  mathscinet
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