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Relations between varieties of kolmogorov complexities

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Abstract

There are several sorts of Kolmogorov complexity, better to say several Kolmogorov complexities: decision complexity, simple complexity, prefix complexity, monotonie complexity,a priori complexity. The last three can and the first two cannot be used for defining randomness of an infinite binary sequence. All those five versions of Kolmogorov complexity were considered, from a unified point of view, in a paper by the first author which appeared in Watanabe's book [23]. Upper and lower bounds for those complexities and also for their differences were announced in that paper without proofs. (Some of those bounds are mentioned in Section 4.4.5 of [16].) The purpose of this paper (which can be read independently of [23]) is to give proofs for the bounds from [23].

The terminology used in this paper is somehow nonstandard: we call “Kolmogorov entropy” what is usually called “Kolmogorov complexity.” This is a Moscow tradition suggested by Kolmogorov himself. By this tradition the term “complexity” relates toany mode of description and “entropy” is the complexity related to anoptimal mode (i.e., to a mode that, roughly speaking, gives theshortest descriptions).

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Authors and Affiliations

  1. Division of Mathematical Logic and the Theory of Algorithms, Fakultet of Mechanics and Mathematics, Lomonosov Moscow State University, V-234 Moscow, GSP-3, 119899, Russia

    V. A. Uspensky

  2. Laboratory 13, Institute for Problems of Information Transmission, Ermolovoi 19, K-51 Moscow, GSP-4, 101447, Russia

    A. Shen

Authors
  1. V. A. Uspensky
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  2. A. Shen
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Additional information

The research described in this publication was made possible in part by Grant No. MQ3000 from the International Science Foundation. A preliminary version of this paper appeared in April, 1993, as CWI's Technical Report CS-R9329 (CWI is the Dutch National Research Institute for Mathematics and Computer Science). That version was supported by the Netherlands Organization for Scientific Research (NWO). The second author was also supported by AMS fSU Aid fund and Volkswagen Stiftung.

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Uspensky, V.A., Shen, A. Relations between varieties of kolmogorov complexities. Math. Systems Theory 29, 271–292 (1996). https://doi.org/10.1007/BF01201280

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