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International Symposium on Fundamentals of Computation Theory

FCT 2019: Fundamentals of Computation Theory pp 80–94Cite as

Two Characterizations of Finite-State Dimension

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11651))

Abstract

In this paper we provide two equivalent characterizations of the notion of finite-state dimension introduced by Dai, Lathrop, Lutz and Mayordomo [7]. One of them uses Shannon’s entropy of non-aligned blocks and generalizes old results of Pillai [12] and Niven – Zuckerman [11]. The second characterizes finite-state dimension in terms of superadditive functions that satisfy some calibration condition (in particular, superadditive upper bounds for Kolmogorov complexity). The use of superadditive bounds allows us to prove a general sufficient condition for normality that easily implies old results of Champernowne [5], Besicovitch [1], Copeland and Erdös [6], and also a recent result of Calude, Staiger and Stephan [4].

A. Shen—On leave from IITP RAS.

Supported by RaCAF ANR-15-CE40-0016-01 grant. The article was prepared within the framework of the HSE University Basic Research Program and funded by the Russian Academic Excellence Project ‘5-100’.

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Notes

  1. 1.

    In fact, Champernowne spoke about decimal notation and sequences of digits, but this does not make a big difference.

  2. 2.

    More precisely, we should speak not about the probability of a given block, since the same k-bit block may appear in several positions, but about the probability of its appearance in a given position. Formally speaking, we use the following obvious fact: if we apply some function to two random variables, the statistical difference between them may only decrease. Here the function forgets the position of a block.

  3. 3.

    This is a technical condition needed to avoid infinities in the logarithms.

References

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  2. Bourke, C., Hitchcock, J.M., Vinodchandran, N.: Entropy rates and finite-state dimension. Theor. Comput. Sci. 349(3), 392–406 (2005). https://doi.org/10.1016/j.tcs.2005.09.040

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  3. Calude, C.S., Salomaa, K., Roblot, T.K.: Finite state complexity. Theor. Comput. Sci. 412(41), 5668–5677 (2011). https://doi.org/10.1016/j.tcs.2011.06.021

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  4. Calude, C.S., Staiger, L., Stephan, F.: Finite state incompressible infinite sequences. Inf. Comput. 247, 23–36 (2016). https://doi.org/10.1016/j.ic.2015.11.003

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Author information

Authors and Affiliations

  1. National Research University Higher School of Economics, Moscow, Russia

    Alexander Kozachinskiy

  2. LIRMM CNRS & University of Montpellier, Montpellier, France

    Alexander Shen

  3. IITP RAS, Moscow, Russia

    Alexander Shen

  4. Lomonosov Moscow State University, Moscow, Russia

    Alexander Kozachinskiy

Authors
  1. Alexander Kozachinskiy
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  2. Alexander Shen
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Correspondence to Alexander Kozachinskiy .

Editor information

Editors and Affiliations

  1. University of Liverpool, Liverpool, UK

    Leszek Antoni Gąsieniec

  2. Hong Kong Polytechnic University, Hong Kong, China

    Jesper Jansson

  3. Lund University, Lund, Sweden

    Christos Levcopoulos

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Kozachinskiy, A., Shen, A. (2019). Two Characterizations of Finite-State Dimension. In: Gąsieniec, L., Jansson, J., Levcopoulos, C. (eds) Fundamentals of Computation Theory. FCT 2019. Lecture Notes in Computer Science(), vol 11651. Springer, Cham. https://doi.org/10.1007/978-3-030-25027-0_6

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  • DOI: https://doi.org/10.1007/978-3-030-25027-0_6

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-25026-3

  • Online ISBN: 978-3-030-25027-0

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