Automatic Kolmogorov Complexity and Normality Revisited

It is well known that normality (all factors of given length appear in an infinite sequence with the same frequency) can be described as incompressibility via finite automata. Still the statement and proof of this result as given by Becher and Heiber [4] in terms of “lossless finite-state compressors” do not follow the standard scheme of Kolmogorov complexity definition (the 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 [9], Calude–Salomaa–Roblot [5]) of description complexity related to finite automata are discussed (see the last section). As a byproduct, we obtain simple proofs of classical results about normality (equivalence of definitions with aligned occurences and all occurencies, Wall’s theorem saying that a normal number remains normal when multiplied by a rational number, and Agafonov’s result saying that normality is preserved by automatic selection rules). 1 Automatic Kolmogorov complexity Let A and B be two finite alphabets. Consider a directed graph G whose edges are labeled by pairs (a,b) of letters (from A and B respectively). We also allow pairs of the form (a,ε), (ε,b), and (ε,ε) where ε is a special symbol (not in A or B) that informally means “no letter”. For such a graph, consider all directed paths in it (no restriction on starting or final point), and for each path concatenate all the first and all the second components of the pairs; ε is replaced by an empty word. For each path we get some pair (u,v) where u ∈ A∗ and v ∈ B∗ (i.e., u and v are words over alphabets A and B). Consider all pairs that can be read in this way along all paths in G. For each labeled graph G we obtain a relation (set of pairs) RG that is a subset of A ∗×B∗. For the purposes of this paper, we call the relations obtained in this way “automatic”. Definition 1. A relation R⊂ A∗×B∗ is automatic if there exists a labeled graph (automaton) G such that R= RG in the sense described above. Now let us recall the definition of algorithmic (Kolmogorov) complexity. It is usually defined in the following way: C(x), the complexity of an object x, is the minimal length of its ∗Supported by ANR-15-CE40-0016 RaCAF grant. Part of the work was done while visiting National Research University High School of Economics, Moscow. E-mail: [email protected] or [email protected]