# Vol. 9 no. 2

### 1. Baire and automata

In his thesis Baire defined functions of Baire class 1. A function f is of Baire class 1 if it is the pointwise limit of a sequence of continuous functions. Baire proves the following theorem. A function f is not of class 1 if and only if there exists a closed nonempty set F such that the restriction of f to F has no point of continuity. We prove the automaton version of this theorem. An ω-rational function is not of class 1 if and only if there exists a closed nonempty set F recognized by a Büchi automaton such that the restriction of f to F has no point of continuity. This gives us the opportunity for a discussion on Hausdorff's analysis of Δ°2, ordinals, transfinite induction and some applications of computer science.

### 2. Gray code order for Lyndon words

At the 4th Conference on Combinatorics on Words, Christophe Reutenauer posed the question of whether the dual reflected order yields a Gray code on the Lyndon family. In this paper we give a positive answer. More precisely, we present an O(1)-average-time algorithm for generating length n binary pre-necklaces, necklaces and Lyndon words in Gray code order.

### 3. Addition and multiplication of beta-expansions in generalized Tribonacci base

We study properties of β-numeration systems, where β > 1 is the real root of the polynomial x3 - mx2 - x - 1, m ∈ ℕ, m ≥ 1. We consider arithmetic operations on the set of β-integers, i.e., on the set of numbers whose greedy expansion in base β has no fractional part. We show that the number of fractional digits arising under addition of β-integers is at most 5 for m ≥ 3 and 6 for m = 2, whereas under multiplication it is at most 6 for all m ≥ 2. We thus generalize the results known for Tribonacci numeration system, i.e., for m = 1. We summarize the combinatorial properties of infinite words naturally defined by β-integers. We point out the differences between the structure of β-integers in cases m = 1 and m ≥ 2.

### 4. The \v Cerný conjecture for aperiodic automata

A word w is called a synchronizing (recurrent, reset, directable) word of a deterministic finite automaton (DFA) if w brings all states of the automaton to some specific state; a DFA that has a synchronizing word is said to be synchronizable. Cerny conjectured in 1964 that every n-state synchronizable DFA possesses a synchronizing word of length at most (n-1)2. We consider automata with aperiodic transition monoid (such automata are called aperiodic). We show that every synchronizable n-state aperiodic DFA has a synchronizing word of length at most n(n-1)/2. Thus, for aperiodic automata as well as for automata accepting only star-free languages, the Cerny conjecture holds true.

### 5. Foreword to the special issue dedicated to the tenth ''Journées montoises d'informatique théorique''

This special issue of Discrete Mathematics & Theoretical Computer Science is dedicated to the tenth "Journées montoises d'informatique théorique" conference (Mons theoretical computer science days) which was held, for the first time, at the Institute of Mathematics of the University of Liège, Belgium, From 8th to 11th September 2004. Previous editions of this conference took place in Mons 1990, 1992, 1994, 1998, in Rouen 1991, in Bordeaux 1993, Marseille 1995, Marne-La-Vallée 2000 and Montpellier 2002.<p> This tenth edition can be considered as a widely international one. We were lucky to have almost 85 participants from fourteen different countries: Austria, Belgium, Burkina Faso, Canada, Czech republic, Finland, France, Germany, Israel, Italy, Japan, Norway, Poland and Portugal. The main proportion of researchers participating to this event was coming from France and Italy where a long tradition of combinatorics on words is well established. During four days, 42 contributed talks and 7 invited talks were given, the main topics being combinatorics on words, numeration systems, automata and formal languages theory, coding theory, verification, bio-informatics, number theory, grammars, text algorithms, symbolic dynamics and tilings. The invited speakers were: J. Cassaigne (CNRS, Luminy-Marseille), D. Caucal (IRISIA-CNRS, Rennes), C. Frougny (LIAFA, Université Paris 8), T. Helleseth (University of Bergen), S. Langerman (FNRS, Université Libre de […]

### 6. Words and bisimulations of dynamical systems

In this paper we study bisimulations on dynamical systems through a given partition. Our aim is to give a new vision of the notion of bisimulation by using words. To achieve this goal, we encode the trajectories of the transition system as words. This method was introduced in our paper ''On o-minimal hybrid systems'' in order to give a new proof of the existence of a finite bisimulation for o-minimal hybrid systems (as previously proved in a paper by Lafferriere G., Pappas G.J. and Sastry S.). Here we want to provide a systematic study of this method in order to obtain a procedure for building finite bisimulations based on words.

### 7. Computation of L_⊕ for several cubic Pisot numbers

In this article, we are dealing with β-numeration, which is a generalization of numeration in a non-integer base. We consider the class of simple Parry numbers such that dβ(1) = 0.k1d-1 kd with d ∈ ℕ, d ≥ 2 and k1 ≥ kd ≥ 1. We prove that these elements define Rauzy fractals that are stable under a central symmetry. We use this result to compute, for several cases of cubic Pisot units, the maximal length among the lengths of the finite β-fractional parts of sums of two β-integers, denoted by L_⊕. In particular, we prove that L_⊕ = 5 in the Tribonacci case.

### 8. Regular languages and associative language descriptions

The Associative Language Description model (ALD) is a combination of locally testable and constituent structure ideas. It is consistent with current views on brain organization and can rather conveniently describe typical technical languages such as Pascal or HTML. ALD languages are strictly enclosed in context-free languages but in practice the ALD model equals CF grammars in explanatory adequacy. Various properties of ALD have been investigated, but many theoretical questions are still open. For instance, it is unknown, at the present, whether the ALD family includes the regular languages. Here it is proved that several known classes of regular languages are ALD: threshold locally testable languages, group languages, positive commutative languages and commutative languages on 2-letter alphabets. Moreover, we show that there is an ALD language in each level of (restricted) star height hierarchy. These results seem to show that ALD languages are well-distributed over the class of regular languages.

### 9. A geometrical approach of palindromic factors of standard billiard words

Many results are already known, concerning the palindromic factors and the palindomic prefixes of Standard billiard words, i.e., Sturmian words and billiard words in any dimension, starting at the origin. We give new geometrical proofs of these results, especially for the existence in any dimension of Standard billiard words with arbitrary long palindromic prefixes.

### 10. On morphisms preserving infinite Lyndon words

In a previous paper, we characterized free monoid morphisms preserving finite Lyndon words. In particular, we proved that such a morphism preserves the order on finite words. Here we study morphisms preserving infinite Lyndon words and morphisms preserving the order on infinite words. We characterize them and show relations with morphisms preserving Lyndon words or the order on finite words. We also briefly study morphisms preserving border-free words and those preserving the radix order.

### 11. Properties of the extremal infinite smooth words

Smooth words are connected to the Kolakoski sequence. We construct the maximal and the minimal in nite smooth words, with respect to the lexicographical order. The naive algorithm generating them is improved by using a reduction of the De Bruijn graph of their factors. We also study their Lyndon factorizations. Finally, we show that the minimal smooth word over the alphabet f1; 3g belongs to the orbit of the Fibonacci word.

### 12. Tiling a Pyramidal Polycube with Dominoes

The notion of pyramidal polycubes, namely the piling-up of bricks of a non-increasing size, generalizes in R^n the concept of trapezoidal polyominoes. In the present paper, we prove that n-dimensional dominoes can tile a pyramidal polycube if and only if the latter is balanced, that is, if the number of white cubes is equal to the number of black ones for a chessboard-like coloration, generalizing the result of [BC92] when n=2.

### 13. Infinite special branches in words associated with beta-expansions

A Parry number is a real number β > 1 such that the Rényi β-expansion of 1 is finite or infinite eventually periodic. If this expansion is finite, β is said to be a simple Parry number. Remind that any Pisot number is a Parry number. In a previous work we have determined the complexity of the fixed point uβ of the canonical substitution associated with β-expansions, when β is a simple Parry number. In this paper we consider the case where β is a non-simple Parry number. We determine the structure of infinite left special branches, which are an important tool for the computation of the complexity of uβ. These results allow in particular to obtain the following characterization: the infinite word uβ is Sturmian if and only if β is a quadratic Pisot unit.

### 14. Tag-systems for the Hilbert curve

Hilbert words correspond to finite approximations of the Hilbert space filling curve. The Hilbert infinite word H is obtained as the limit of these words. It gives a description of the Hilbert (infinite) curve. We give a uniform tag-system to generate automatically H and, by showing that it is almost cube-free, we prove that it cannot be obtained by simply iterating a morphism.

### 15. NONCOMMUTATIVE SYMMETRIC FUNCTIONS ASSOCIATED WITH A CODE, LAZARD ELIMINATION, AND WITT VECTORS

The construction of the universal ring of Witt vectors is related to Lazard's factorizations of free monoids by means of a noncommutative analogue. This is done by associating to a code a specialization of noncommutative symmetric functions.

### 16. An algorithm for generating ribbon tableaux and spin polynomials

We describe a general algorithm for generating various families of ribbon tableaux and computing their spin polynomials. This algorithm is derived from a new matricial coding. An advantage of this new notation lies in the fact that it permits one to generate ribbon tableaux with skew shapes.

### 17. Coding partitions

Motivated by the study of decipherability conditions for codes weaker than Unique Decipherability (UD), we introduce the notion of coding partition. Such a notion generalizes that of UD code and, for codes that are not UD, allows to recover the ''unique decipherability" at the level of the classes of the partition. By tacking into account the natural order between the partitions, we define the characteristic partition of a code X as the finest coding partition of X. This leads to introduce the canonical decomposition of a code in at most one unambiguous component and other (if any) totally ambiguous components. In the case the code is finite, we give an algorithm for computing its canonical partition. This, in particular, allows to decide whether a given partition of a finite code X is a coding partition. This last problem is then approached in the case the code is a rational set. We prove its decidability under the hypothesis that the partition contains a finite number of classes and each class is a rational set. Moreover we conjecture that the canonical partition satisfies such a hypothesis. Finally we consider also some relationships between coding partitions and varieties of codes.

### 18. Ambiguity in the m-bonacci numeration system

Résumé comportant des formules mathématiques, disponible sur le ficher pdf / Abstract with mathematical formulas, available on pdf file.