Ultimate periodicity problem for linear numeration systems

We address the following decision problem. Given a numeration system U and a U-recognizable subset X of N, i.e. the set of its greedy U-representations is recognized by a finite automaton, decide whether or not X is ultimately periodic. We prove that this problem is decidable for a large class of numeration systems built on linear recurrence sequences. Based on arithmetical considerations about the recurrence equation and on p-adic methods, the DFA given as input provides a bound on the admissible periods to test

Systèmes de numération abstraits : reconnaissabilité, décidabilité, mots S-automatiques multidimensionnels et nombres réels

In this dissertation we study and we solve several questions regarding abstract numeration systems. Each particular problem is the focus of a chapter. The first problem concerns the study of the preservation of recognizability under multiplication by a constant in abstract numeration systems built on polynomial regular languages. The second is a decidability problem, which has been already studied notably by J. Honkala and A. Muchnik and which is studied here for two new cases: the linear positional numeration systems and the abstract numeration systems. Next, we focus on the extension to the multidimensional setting of a result of A. Maes and M. Rigo regarding S-automatic infinite words. Finally, we propose a formalism to represent real numbers in the general framework of abstract numeration systems built on languages that are not necessarily regular. We end by a list of open questions in the continuation of the present work.Dans cette dissertation, nous étudions et résolvons plusieurs questions autour des systèmes de numération abstraits. Chaque problème étudié fait l'objet d'un chapitre. Le premier concerne l'étude de la conservation de la reconnaissabilité par la multiplication par une constante dans des systèmes de numération abstraits construits sur des langages réguliers polynomiaux. Le deuxième est un problème de décidabilité déjà étudié notamment par J. Honkala et A. Muchnik et ici décliné en deux nouvelles versions : les systèmes de numération de position linéaires et les systèmes de numération abstraits. Ensuite, nous nous penchons sur l'extension au cas multidimensionnel d'un résultat d'A. Maes et de M. Rigo à propos des mots infinis S-automatiques. Finalement, nous proposons un formalisme de la représentation des nombres réels dans le cadre général des systèmes de numération abstraits basés sur des langages qui ne sont pas nécessairement réguliers. Nous terminons par une liste de questions ouvertes dans la continuité de ce travail

Enumeration and Decidable Properties of Automatic Sequences

We show that various aspects of k-automatic sequences -- such as having an unbordered factor of length n -- are both decidable and effectively enumerable. As a consequence it follows that many related sequences are either k-automatic or k-regular. These include many sequences previously studied in the literature, such as the recurrence function, the appearance function, and the repetitivity index. We also give some new characterizations of the class of k-regular sequences. Many results extend to other sequences defined in terms of Pisot numeration systems

Decision Algorithms for Ostrowski-Automatic Sequences

We extend the notion of automatic sequences to a broader class, the Ostrowski-automatic sequences. We develop a procedure for computationally deciding certain combinatorial and enumeration questions about such sequences that can be expressed as predicates in first-order logic. In Chapter 1, we begin with topics and ideas that are preliminary to this work, including a small introduction to non-standard positional numeration systems and the relationship between words and automata. In Chapter 2, we define the theoretical foundations for recognizing addition in a generalized Ostrowski numeration system and formalize the general theory that develops our decision procedure. Next, in Chapter 3, we show how to implement these ideas in practice, and provide the implementation as an integration to the automatic theorem-proving software package -- Walnut. Further, we provide some applications of our work in Chapter 4. These applications span several topics in combinatorics on words, including repetitions, pattern-avoidance, critical exponents of special classes of words, properties of Lucas words, and so forth. Finally, we close with open problems on decidability and higher-order numeration systems and discuss future directions for research

Wythoff Wisdom

International audienceSix authors tell their stories from their encounters with the famous combinatorial game Wythoff Nim and its sequences, including a short survey on exactly covering systems

Combinatorics of Pisot Substitutions

Siirretty Doriast