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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- 1.
In fact, Champernowne spoke about decimal notation and sequences of digits, but this does not make a big difference.
- 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.
This is a technical condition needed to avoid infinities in the logarithms.
References
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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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