Kolmogorov Complexity in perspective. Part I: Information Theory and Randomnes

We survey diverse approaches to the notion of information: from Shannon entropy to Kolmogorov complexity. Two of the main applications of Kolmogorov complexity are presented: randomness and classification. The survey is divided in two parts in the same volume. Part I is dedicated to information theory and the mathematical formalization of randomness based on Kolmogorov complexity. This last application goes back to the 60's and 70's with the work of Martin-L\"of, Schnorr, Chaitin, Levin, and has gained new impetus in the last years.Comment: 40 page

Kolmogorov complexity in perspective

We survey the diverse approaches to the notion of information content: from Shannon entropy to Kolmogorov complexity. The main applications of Kolmogorov complexity are presented namely, the mathematical notion of randomness (which goes back to the 60's with the work of Martin-Lof, Schnorr, Chaitin, Levin), and classification, which is a recent idea with provocative implementation by Vitanyi and Cilibrasi.Comment: 37 page

Systems of equations over a free monoid and Ehrenfeucht's conjecture

AbstractEhrenfeucht's conjecture states that every language L has a finite subset F such that, for any pair (g, h) of morphisms, g and h agree on every word of L if and only if they agree on every word of F. We show that it holds if and only if every infinite system of equations (with a finite number of unknowns) over a free monoid has an equivalent finite subsystem. It is shown that this holds true for rational (regular) systems of equations.The equivalence and inclusion problems for finite and rational systems of equations are shown to be decidable and, consequently, the validity of Ehrenfeucht's conjecture implies the decidability of the HDOL and DTOL sequence equivalence problems. The simplicity degree of a language is introduced and used to argue in support of Ehrenfeucht's conjecture

A new approach to the 22-regularity of the ℓ\ell-abelian complexity of 22-automatic sequences

We prove that a sequence satisfying a certain symmetry property is $2$-regular in the sense of Allouche and Shallit, i.e., the $\mathbb{Z}$-module generated by its $2$-kernel is finitely generated. We apply this theorem to develop a general approach for studying the $\ell$-abelian complexity of $2$-automatic sequences. In particular, we prove that the period-doubling word and the Thue--Morse word have $2$-abelian complexity sequences that are $2$-regular. Along the way, we also prove that the $2$-block codings of these two words have $1$-abelian complexity sequences that are $2$-regular.Comment: 44 pages, 2 figures; publication versio

Substitutive systems and a finitary version of Cobham's theorem

We study substitutive systems generated by nonprimitive substitutions and show that transitive subsystems of substitutive systems are substitutive. As an application we obtain a complete characterisation of the sets of words that can appear as common factors of two automatic sequences defined over multiplicatively independent bases. This generalises the famous theorem of Cobham.Comment: 23 pages. v2: incorporates referee's comments, updated references, to appear in Combinatoric

Subword complexity and power avoidance

We begin a systematic study of the relations between subword complexity of infinite words and their power avoidance. Among other things, we show that -- the Thue-Morse word has the minimum possible subword complexity over all overlap-free binary words and all $(\frac 73)$-power-free binary words, but not over all $(\frac 73)^+$-power-free binary words; -- the twisted Thue-Morse word has the maximum possible subword complexity over all overlap-free binary words, but no word has the maximum subword complexity over all $(\frac 73)$-power-free binary words; -- if some word attains the minimum possible subword complexity over all square-free ternary words, then one such word is the ternary Thue word; -- the recently constructed 1-2-bonacci word has the minimum possible subword complexity over all \textit{symmetric} square-free ternary words.Comment: 29 pages. Submitted to TC

Nested quasicrystalline discretisations of the line

One-dimensional cut-and-project point sets obtained from the square lattice in the plane are considered from a unifying point of view and in the perspective of aperiodic wavelet constructions. We successively examine their geometrical aspects, combinatorial properties from the point of view of the theory of languages, and self-similarity with algebraic scaling factor $\theta$. We explain the relation of the cut-and-project sets to non-standard numeration systems based on $\theta$. We finally examine the substitutivity, a weakened version of substitution invariance, which provides us with an algorithm for symbolic generation of cut-and-project sequences

Factor Complexity of S-adic sequences generated by the Arnoux-Rauzy-Poincar\'e Algorithm

The Arnoux-Rauzy-Poincar\'e multidimensional continued fraction algorithm is obtained by combining the Arnoux-Rauzy and Poincar\'e algorithms. It is a generalized Euclidean algorithm. Its three-dimensional linear version consists in subtracting the sum of the two smallest entries to the largest if possible (Arnoux-Rauzy step), and otherwise, in subtracting the smallest entry to the median and the median to the largest (the Poincar\'e step), and by performing when possible Arnoux-Rauzy steps in priority. After renormalization it provides a piecewise fractional map of the standard $2$-simplex. We study here the factor complexity of its associated symbolic dynamical system, defined as an $S$-adic system. It is made of infinite words generated by the composition of sequences of finitely many substitutions, together with some restrictions concerning the allowed sequences of substitutions expressed in terms of a regular language. Here, the substitutions are provided by the matrices of the linear version of the algorithm. We give an upper bound for the linear growth of the factor complexity. We then deduce the convergence of the associated algorithm by unique ergodicity.Comment: 36 pages, 16 figure