Abstract
It is well known that normality (all factors of a given length appear in an infinite sequence with the same frequency) can be described as incompressibility via finite automata. Still the statement and the proof of this result as given by Becher and Heiber [9] in terms of "lossless finite-state compressors" do not follow the standard scheme of Kolmogorov complexity definition (an automaton is used for compression, not decompression). We modify this approach to make it more similar to the traditional Kolmogorov complexity theory (and simpler) by explicitly defining the notion of automatic Kolmogorov complexity and using its simple properties. Other known notions (Shallit-Wang [50], Calude-Salomaa-Roblot [16]) of description complexity related to finite automata are discussed (see the last section). Using this characterization, we provide easy proofs for most of the classical results about normal sequences, including the equivalence between aligned and non-aligned definitions of normality, the Piatetski-Shapiro sufficient condition for normality (in a strong form), and Wall's theorem saying that a normal real number remains normal when multiplied by a rational number or when a rational number is added. Using the same characterization, we prove a sufficient condition for normality of a sequence in terms of Kolmogorov complexity. This condition implies the normality of Champernowne's sequence as well as some generalizations of this result (provided by Champernowne himself, Besicovitch, Copeland and Erdös). It can be also used to give a simple proof of the result of Calude-Staiger-Stephan [17] saying that normality cannot be characterized in terms of the automatic complexity notion introduced by Calude-Salomaa-Roblot [16]. Then we extend this approach to finite state dimension showing that automatic Kolmogorov complexity can be used to characterize the finite state dimension (defined by Dai, Lathrop, Lutz and Mayordomo in [23]). We start with the block entropy definition of the finite state dimension given by Bourke, Hitchcock and Vinogradchandran [14] and show that one may use non-aligned blocks in this definition.