342 research outputs found
Abstract numeration systems on bounded languages and multiplication by a constant
A set of integers is -recognizable in an abstract numeration system 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
does not preserve -recognizability, meaning that there always
exists a -recognizable set such that is not
-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
Structural properties of bounded languages with respect to multiplication by a constant
peer reviewedWe consider the preservation of recognizability of a set of integers after multiplication by a constant for numeration systems built over a bounded language. As a corollary we show that any nonnegative integer can be written as a sum of binomial coefficients with some prescribed properties
Numeration systems on a regular language: Arithmetic operations, Recognizability and Formal power series
Generalizations of numeration systems in which N is recognizable by a finite
automaton are obtained by describing a lexicographically ordered infinite
regular language L over a finite alphabet A. For these systems, we obtain a
characterization of recognizable sets of integers in terms of rational formal
series. We also show that, if the complexity of L is Theta (n^q) (resp. if L is
the complement of a polynomial language), then multiplication by an integer k
preserves recognizability only if k=t^{q+1} (resp. if k is not a power of the
cardinality of A) for some integer t. Finally, we obtain sufficient conditions
for the notions of recognizability and U-recognizability to be equivalent,
where U is some positional numeration system related to a sequence of integers.Comment: 34 pages; corrected typos, two sections concerning exponential case
and relation with positional systems adde
Redundancy of minimal weight expansions in Pisot bases
Motivated by multiplication algorithms based on redundant number
representations, we study representations of an integer as a sum , where the digits are taken from a finite alphabet
and is a linear recurrent sequence of Pisot type with
. The most prominent example of a base sequence is the
sequence of Fibonacci numbers. We prove that the representations of minimal
weight are recognised by a finite automaton and obtain an
asymptotic formula for the average number of representations of minimal weight.
Furthermore, we relate the maximal order of magnitude of the number of
representations of a given integer to the joint spectral radius of a certain
set of matrices
Numeration Systems: a Link between Number Theory and Formal Language Theory
We survey facts mostly emerging from the seminal results of Alan Cobham
obtained in the late sixties and early seventies. We do not attempt to be
exhaustive but try instead to give some personal interpretations and some
research directions. We discuss the notion of numeration systems, recognizable
sets of integers and automatic sequences. We briefly sketch some results about
transcendence related to the representation of real numbers. We conclude with
some applications to combinatorial game theory and verification of
infinite-state systems and present a list of open problems.Comment: 21 pages, 3 figures, invited talk DLT'201
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
Automatic sequences: from rational bases to trees
The th term of an automatic sequence is the output of a deterministic
finite automaton fed with the representation of in a suitable numeration
system. In this paper, instead of considering automatic sequences built on a
numeration system with a regular numeration language, we consider these built
on languages associated with trees having periodic labeled signatures and, in
particular, rational base numeration systems. We obtain two main
characterizations of these sequences. The first one is concerned with -block
substitutions where morphisms are applied periodically. In particular, we
provide examples of such sequences that are not morphic. The second
characterization involves the factors, or subtrees of finite height, of the
tree associated with the numeration system and decorated by the terms of the
sequence.Comment: 25 pages, 15 figure
Boundary of central tiles associated with Pisot beta-numeration and purely periodic expansions
This paper studies tilings related to the beta-transformation when beta is a
Pisot number (that is not supposed to be a unit). Then it applies the obtained
results to study the set of rational numbers having a purely periodic
beta-expansion. Special focus is given to some quadratic examples
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
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