Abstract numeration systems on bounded languages and multiplication by a constant

A set of integers is $S$-recognizable in an abstract numeration system $S$ if the language made up of the representations of its elements is accepted by a finite automaton. For abstract numeration systems built over bounded languages with at least three letters, we show that multiplication by an integer $\lambda\ge2$ does not preserve $S$-recognizability, meaning that there always exists a $S$-recognizable set $X$ such that $\lambda X$ is not $S$-recognizable. The main tool is a bijection between the representation of an integer over a bounded language and its decomposition as a sum of binomial coefficients with certain properties, the so-called combinatorial numeration system

Generalization of automatic sequences for numeration systems on a regular language

Let L be an infinite regular language on a totally ordered alphabet (A,<). Feeding a finite deterministic automaton (with output) with the words of L enumerated lexicographically with respect to < leads to an infinite sequence over the output alphabet of the automaton. This process generalizes the concept of k-automatic sequence for abstract numeration systems on a regular language (instead of systems in base k). Here, I study the first properties of these sequences and their relations with numeration systems.Comment: 10 pages, 3 figure

Dynamical Directions in Numeration

International audienceWe survey definitions and properties of numeration from a dynamical point of view. That is we focuse on numeration systems, their associated compactifications, and the dynamical systems that can be naturally defined on them. The exposition is unified by the notion of fibred numeration system. A lot of examples are discussed. Various numerations on natural, integral, real or complex numbers are presented with a special attention payed to beta-numeration and its generalisations, abstract numeration systems and shift radix systems. A section of applications ends the paper

Syndeticity and independent substitutions

We associate in a canonical way a substitution to any abstract numeration system built on a regular language. In relationship with the growth order of the letters, we define the notion of two independent substitutions. Our main result is the following. If a sequence $x$ is generated by two independent substitutions, at least one being of exponential growth, then the factors of $x$ appearing infinitely often in $x$ appear with bounded gaps. As an application, we derive an analogue of Cobham's theorem for two independent substitutions (or abstract numeration systems) one with polynomial growth, the other being exponential

Abstract numeration systems

In this talk, I will introduce abstract numeration systems in general and present some results I have regarding operation preserving regularity. In particular, I will focus on multiplication by a constant

Abstract numeration systems

Abstract numeration systems were introduced in 2001 by P. Lecomte and M. Rigo. This new way to represent numbers generalizes that of usual positional numeration systems such as integer base numeration systems and linear numeration systems. Some standard properties are preserved in this wider framework though some others are not. Yet, the advantages of these systems stem from their great generality: current research on this subject strives to highlight the properties that are independent of the target numeration system, such as properties related to the complexity of the numeration language. In this talk I will introduce this topic. In particular, I will present many open questions in the area and highlight the connections with combinatorics on words

An introduction to numeration systems: Cobham-like theorems, first-order logic and regular sequences

In this talk, I will present an introduction to numeration systems (mostly for representing integers), and show how first-order logic can be used in order to solve problems in combinatorics on words. The base idea is to translate properties of numbers into combinatorial properties of their representations. As far as I can, I will try to determine which results depends on the numeration systems involved and which do not. On the one hand, Cobham’s theorem and its generalizations tell us that most properties of numbers strongly depend on the chosen numeration system. On the other hand, the use of very general numeration systems, such as abstract numeration systems, allows us to understand how far we can exploit techniques from logic and automata theory. Along the way, we will define the notions of recognizable and definable sets of integers, automatic and regular sequences, morphic sequences and abstract numeration systems. If time allows me to do so, I will also present results generalizing these considerations to real numbers