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Towards parametrizing word equations

H. Abdulrab, P. Goralčík, G. S. Makanin (2010)

RAIRO - Theoretical Informatics and Applications

Classically, in order to resolve an equation u ≈ v over a free monoid X*, we reduce it by a suitable family ℱ of substitutions to a family of equations uf ≈ vf, f ∈ ℱ , each involving less variables than u ≈ v, and then combine solutions of uf ≈ vf into solutions of u ≈ v. The problem is to get ℱ in a handy parametrized form. The method we propose consists in parametrizing the path traces in the so called graph of prime equations associated to u ≈ v. We carry out such a parametrization in the case the...

Transcendence of numbers with an expansion in a subclass of complexity 2n + 1

Tomi Kärki (2006)

RAIRO - Theoretical Informatics and Applications

We divide infinite sequences of subword complexity 2n+1 into four subclasses with respect to left and right special elements and examine the structure of the subclasses with the help of Rauzy graphs. Let k ≥ 2 be an integer. If the expansion in base k of a number is an Arnoux-Rauzy word, then it belongs to Subclass I and the number is known to be transcendental. We prove the transcendence of numbers with expansions in the subclasses II and III.

Transitive Hall sets.

Duchamp, Gérard, Flouret, Marianne, Luque, Jean-Gabriel (2005)

Séminaire Lotharingien de Combinatoire [electronic only]

Two-variable word equations

Lucian Ilie, Wojciech Plandowski (2000)

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

Two-variable word equations

Lucian Ilie, Wojciech Plandowski (2010)

RAIRO - Theoretical Informatics and Applications

We consider languages expressed by word equations in two variables and give a complete characterization for their complexity functions, that is, the functions that give the number of words of the same length. Specifically, we prove that there are only five types of complexities: constant, linear, exponential, and two in between constant and linear. For the latter two, we give precise characterizations in terms of the number of solutions of Diophantine equations of certain types. In particular,...

Unambiguous erasing morphisms in free monoids

Johannes C. Schneider (2010)

RAIRO - Theoretical Informatics and Applications

This paper discusses the fundamental combinatorial question of whether or not, for a given string α, there exists a morphism σ such that σ is unambiguous with respect to α, i.e. there exists no other morphism τ satisfying τ(α) = σ(α). While Freydenberger et al. [Int. J. Found. Comput. Sci. 17 (2006) 601–628] characterise those strings for which there exists an unambiguous nonerasing morphism σ, little is known about the unambiguity of erasing morphisms, i.e. morphisms that map symbols...

Unavoidable Set: Extension and Reduction

Phan Trung Huy, Nguyen Huong Lam (2010)

RAIRO - Theoretical Informatics and Applications

We give an explicit criterion for unavoidability of word sets. We characterize extendible, finitely and infinitely as well, elements in them. We furnish a reasonable upper bound and an exponential lower bound on the maximum leghth of words in a reduced unavoidable set of a given cardinality.

Undecidability of infinite post correspondence problem for instances of size 8

Jing Dong, Qinghui Liu (2012)

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

The infinite Post Correspondence Problem (ωPCP) was shown to be undecidable by Ruohonen (1985) in general. Blondel and Canterini [Theory Comput. Syst. 36 (2003) 231–245] showed that ωPCP is undecidable for domain alphabets of size 105, Halava and Harju [RAIRO–Theor. Inf. Appl. 40 (2006) 551–557] showed that ωPCP is undecidable for domain alphabets of size 9. By designing a special coding, we delete a letter from Halava and Harju’s construction. So we prove that ωPCP is undecidable for domain alphabets...

Undecidability of infinite post correspondence problem for instances of size 8

Jing Dong, Qinghui Liu (2012)

RAIRO - Theoretical Informatics and Applications

The infinite Post Correspondence Problem (ωPCP) was shown to be undecidable by Ruohonen (1985) in general. Blondel and Canterini [Theory Comput. Syst. 36 (2003) 231–245] showed that ωPCP is undecidable for domain alphabets of size 105, Halava and Harju [RAIRO–Theor. Inf. Appl. 40 (2006) 551–557] showed that ωPCP is undecidable for domain alphabets of size 9. By designing a special coding, we delete a letter from Halava and Harju’s construction. So we prove that ωPCP is undecidable for domain alphabets...

Undecidability of infinite post correspondence problem for instances of Size 9

Vesa Halava, Tero Harju (2006)

RAIRO - Theoretical Informatics and Applications

In the infinite Post Correspondence Problem an instance (h,g) consists of two morphisms h and g, and the problem is to determine whether or not there exists an infinite word ω such that h(ω) = g(ω). This problem was shown to be undecidable by Ruohonen (1985) in general. Recently Blondel and Canterini (Theory Comput. Syst.36 (2003) 231–245) showed that this problem is undecidable for domain alphabets of size 105. Here we give a proof that the infinite Post Correspondence Problem is undecidable...

Une caractérisation simple des nombres de Sturm

Cyril Allauzen (1998)

Journal de théorie des nombres de Bordeaux

Un mot sturmien est la discrétisation d’une droite de pente irrationnelle. Un nombre de Sturm est la pente d’un mot sturmien qui est invariant par une substitution non triviale. Ces nombres sont certains irrationnels quadratiques caractérisés par la forme de leur développement en fraction continue. Nous donnons une caractérisation très simple des nombres de Sturm : un nombre irrationnel positif est de Sturm (de première espèce) si et seulement s’il est quadratique et à conjugué négatif.

Une généralisation du théorème de Cobham

S. Fabre (1994)

Acta Arithmetica

Nous généralisons le théorème de Cobham ([2]), en démontrant qu'une partie infinie de ℕ est reconnaissable en base k (k entier strictement plus grand que un) et reconnaissable dans un système de numération associé à un nombre de Pisot unitaire (ayant une propriété arithmétique supplémentaire) si et seulement si elle est ultimement périodique.

Uniformly bounded duplication codes

Peter Leupold, Victor Mitrana (2007)

RAIRO - Theoretical Informatics and Applications

Duplication is the replacement of a factor w within a word by ww. This operation can be used iteratively to generate languages starting from words or sets of words. By undoing duplications, one can eventually reach a square-free word, the original word's duplication root. The duplication root is unique, if the length of duplications is fixed. Based on these unique roots we define the concept of duplication code. Elementary properties are stated, then the conditions under which infinite duplication...

Weak mixing and eigenvalues for Arnoux-Rauzy sequences

Julien Cassaigne, Sébastien Ferenczi, Ali Messaoudi (2008)

Annales de l’institut Fourier

We define by simple conditions two wide subclasses of the so-called Arnoux-Rauzy systems; the elements of the first one share the property of (measure-theoretic) weak mixing, thus we generalize and improve a counter-example to the conjecture that these systems are codings of rotations; those of the second one have eigenvalues, which was known hitherto only for a very small set of examples.

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