In the group theory various representations of free groups are used. A representation of a free group of rank two by the so-calledtime-varying Mealy automata over the changing alphabet is given. Two different constructions of such automata are presented.
Considering each occurrence of a word w in a recurrent infinite word, we define the set of return words of w to be the set of all distinct words beginning with an occurrence of w and ending exactly just before the next occurrence of w in the infinite word. We give a simpler proof of the recent result (of the second author) that an infinite word is Sturmian if and only if each of its factors has exactly two return words in it. Then, considering episturmian infinite words, which are a natural generalization...
Six kinds of both of primitivity and periodicity of words, introduced by Ito and Lischke [M. Ito and G. Lischke, Math. Log. Quart. 53 (2007) 91–106; Corrigendum in Math. Log. Quart. 53 (2007) 642–643], give rise to defining six kinds of roots of a nonempty word. For 1 ≤ k ≤ 6, a k-root word is a word which has exactly k different roots, and a k-cluster is a set of k-root words u where the roots of u fulfil a given prefix relationship. We show that out of the 89 different clusters that can be considered...
Arithmetical complexity of a sequence is the number of words of length n that can be extracted from it according to arithmetic progressions. We study uniformly recurrent words of low arithmetical complexity and describe the family of such words having lowest complexity.
We investigate the Sandpile Model and Chip Firing Game and an extension of these models on cycle graphs. The extended model consists of allowing a negative number of chips at each vertex. We give the characterization of reachable configurations and of fixed points of each model. At the end, we give explicit formula for the number of their fixed points.
The complexity of infinite words is considered from the point of view of a transformation with a Mealy machine that is the simplest model of a finite automaton transducer. We are mostly interested in algebraic properties of the underlying partially ordered set. Results considered with the existence of supremum, infimum, antichains, chains and density aspects are investigated.
Episturmian morphisms generalize sturmian morphisms. They are defined as compositions of exchange morphisms and two particular morphisms , and . Epistandard morphisms are the morphisms obtained without considering . In [14], a general study of these morphims and of conjugacy of morphisms is given. Here, given a decomposition of an Episturmian morphism over exchange morphisms and , we consider two problems: how to compute a decomposition of one conjugate of ; how to compute a list of decompositions...
Episturmian morphisms generalize Sturmian morphisms. They are defined as compositions of exchange morphisms and two particular morphisms L, and R. Epistandard morphisms are the morphisms obtained without considering R. In [14], a general study of these morphims and of conjugacy of morphisms is given. Here, given a decomposition of an Episturmian morphism f over exchange morphisms and {L,R}, we consider two problems: how to compute a decomposition of one conjugate of f; how to compute a...
Given a finite set of matrices with integer entries, consider the question of determining whether the semigroup they generated 1) is free; 2) contains the identity matrix; 3) contains the null matrix or 4) is a group. Even for matrices of dimension , questions 1) and 3) are undecidable. For dimension , they are still open as far as we know. Here we prove that problems 2) and 4) are decidable by proving more generally that it is recursively decidable whether or not a given non singular matrix belongs...
Given a finite set of matrices with integer entries, consider the question of determining whether the semigroup they generated 1) is free; 2) contains the identity matrix; 3) contains the null matrix or 4) is a group. Even for matrices of dimension 3, questions 1) and 3) are undecidable. For dimension 2, they are still open as far as we know. Here we prove that problems 2) and 4) are decidable by proving more generally that it is recursively decidable whether or not a given non singular matrix belongs...
We prove that every Sturmian word ω has infinitely many prefixes of the form UnVn3, where |Un| < 2.855|Vn| and limn→∞|Vn| = ∞. In passing, we give a very simple proof of the known fact that every Sturmian word begins in arbitrarily long squares.
We consider the position and number of occurrences of squares in the Thue-Morse sequence, and show that the corresponding sequences are 2-regular. We also prove that changing any finite but nonzero number of bits in the Thue-Morse sequence creates an overlap, and any linear subsequence of the Thue-Morse sequence (except those corresponding to decimation by a power of 2) contains an overlap.
Among the various ways to construct a characteristic Sturmian word, one of the most used consists in defining an infinite sequence of prefixes that are standard. Nevertheless in any characteristic word c, some standard words occur that are not prefixes of c. We characterize all standard words occurring in any characteristic word (and so in any Sturmian word) using firstly morphisms, then standard prefixes and finally palindromes.
Dans cet article, nous introduisons la notion de semi-groupe fortement automatique, qui entraîne la notion d’automaticité des semi-groupes usuelle. On s’intéresse particulièrement aux semi-groupes de développements en base , pour lesquels on obtient un critère de forte automaticité.