Results 1  10
of
20
Logic and precognizable sets of integers
 Bull. Belg. Math. Soc
, 1994
"... We survey the properties of sets of integers recognizable by automata when they are written in pary expansions. We focus on Cobham’s theorem which characterizes the sets recognizable in different bases p and on its generalization to N m due to Semenov. We detail the remarkable proof recently given ..."
Abstract

Cited by 68 (4 self)
 Add to MetaCart
We survey the properties of sets of integers recognizable by automata when they are written in pary expansions. We focus on Cobham’s theorem which characterizes the sets recognizable in different bases p and on its generalization to N m due to Semenov. We detail the remarkable proof recently given by Muchnik for the theorem of CobhamSemenov, the original proof being published in Russian. 1
Numeration systems on a regular language
 Theory Comput. Syst
"... Generalizations of linear numeration systems in which IN is recognizable by finite automata are obtained by describing an arbitrary infinite regular language following the lexicographic ordering. For these systems of numeration, we show that ultimately periodic sets are recognizable. We also study t ..."
Abstract

Cited by 14 (7 self)
 Add to MetaCart
Generalizations of linear numeration systems in which IN is recognizable by finite automata are obtained by describing an arbitrary infinite regular language following the lexicographic ordering. For these systems of numeration, we show that ultimately periodic sets are recognizable. We also study the translation and the multiplication by constants as well as the orderdependence of the recognizability. 1
Additive And Multiplicative Properties Of Point Sets Based On BetaIntegers
"... .  To each number fi ? 1 correspond abelian groups in R d , of the form fi = P d i=1 Z fi e i , which obey fi fi ae fi . The set Z fi of betaintegers is a countable set of numbers : it is precisely the set of real numbers which are polynomial in fi when they are written in "basis fi", ..."
Abstract

Cited by 12 (0 self)
 Add to MetaCart
.  To each number fi ? 1 correspond abelian groups in R d , of the form fi = P d i=1 Z fi e i , which obey fi fi ae fi . The set Z fi of betaintegers is a countable set of numbers : it is precisely the set of real numbers which are polynomial in fi when they are written in "basis fi", and Z fi = Zwhen fi 2 N. We prove here a list of arithmetic properties of Z fi : addition, multiplication, relation with integers, when fi is a quadratic PisotVijayaraghavan unit (quasicrystallographic inflation factors are particular examples). We also consider the case of a cubic PisotVijayaraghavan unit associated with the sevenfold cyclotomic ring. At the end, we show how the point sets fi are vertices of ddimensional tilings. R'esum'e.  ` A chaque nombre fi ? 1 correspondent des groupes ab'eliens dans R d , de la forme fi = P d i=1 Z fi e i , et qui satisfont fi fi ae fi . L'ensemble Z fi des betaentiers est un ensemble d'enombrable de nombres, qui est form'e de t...
On the Sequentiality of the Successor Function
, 1997
"... Let U be a strictly increasing sequence of integers. By a greedy algorithm, every nonnegative integer has a greedy Urepresentation. The successor function maps the greedy Urepresentation of N onto the greedy Urepresentation of N+1. We characterize the sequences U such that the successor functi ..."
Abstract

Cited by 10 (1 self)
 Add to MetaCart
Let U be a strictly increasing sequence of integers. By a greedy algorithm, every nonnegative integer has a greedy Urepresentation. The successor function maps the greedy Urepresentation of N onto the greedy Urepresentation of N+1. We characterize the sequences U such that the successor function associated to U is a left, resp. a right sequential function. We also show that the odometer associated to U is continuous if and only if the successor function is right sequential.
Systèmes De Numération Independants Et Syndéticité
 Theoret. Comp. Sci
, 1998
"... Introduction Ce travail est dedie au sourire de M. P. Schutzenberger, sourire incomparable, inoubliable, expression de qualites rarement rencontrees ensemble, genie et gentillesse, humour et rigueur intellectuelle, le tout appuye sur un courage indomptable face aux epreuves de l'histoire ou de la v ..."
Abstract

Cited by 9 (0 self)
 Add to MetaCart
Introduction Ce travail est dedie au sourire de M. P. Schutzenberger, sourire incomparable, inoubliable, expression de qualites rarement rencontrees ensemble, genie et gentillesse, humour et rigueur intellectuelle, le tout appuye sur un courage indomptable face aux epreuves de l'histoire ou de la vie. Dans les lignes qui vont suivre, il aurait reconnu  ou peutetre reconnatil  la trace de quelquesunes de ses innombrables et profondes idees. Dans un article fondamental [6], Cobham a montre le theoreme suivant : soient p et q deux entiers positifs independants (i.e. il n'existe pas de relation non triviale de la forme p m = q n ) ; une partie X ` N est reconnaissable dans les systemes de numeration en bases p et q si et seulement si elle est ultimement periodique. Ce resultat a depuis donne lieu a de nombreux travaux, essentie
Automatic conversion from Fibonacci representation to representation in base qhi, and a generalization
, 1998
"... Every positive integer can be written as a sum of Fibonacci numbers; it can also be written as a (finite) sum of (positive and negative) powers of the golden mean '. We show that there exists a lettertoletter finite twotape automaton that maps the Fibonacci representation of any positive integer ..."
Abstract

Cited by 9 (2 self)
 Add to MetaCart
Every positive integer can be written as a sum of Fibonacci numbers; it can also be written as a (finite) sum of (positive and negative) powers of the golden mean '. We show that there exists a lettertoletter finite twotape automaton that maps the Fibonacci representation of any positive integer onto its 'expansion, provided the latter is folded around the radix point. As a corollary, the set of 'expansions of the positive integers is a linear contextfree language. These results are actually proved in the more general case of quadratic Pisot units. R'esum'e Tout nombre entier positif peut s"ecrire comme une somme de nombres de Fibonacci; tout entier peut 'egalement s"ecrire comme une somme (finie) de puissances (positives et n'egatives) du "nombre d'or" '. Nous montrons qu'il existe un automate `a deux bandes, fini et lettre`alettre, qui envoie la repr'esentation d'un entier en base de Fibonacci sur sa repr'esentation dans la base ' modulo le fait qu'on a repli'e cette derni`e...
Purely periodic βexpansions in the Pisot nonunit case
, 2005
"... Abstract. It is well known that real numbers with a purely periodic decimal expansion are the rationals having, when reduced, a denominator coprime with 10. The aim of this paper is to extend this result to betaexpansions with a Pisot base beta which is not necessarily a unit: we characterize real ..."
Abstract

Cited by 6 (5 self)
 Add to MetaCart
Abstract. It is well known that real numbers with a purely periodic decimal expansion are the rationals having, when reduced, a denominator coprime with 10. The aim of this paper is to extend this result to betaexpansions with a Pisot base beta which is not necessarily a unit: we characterize real numbers having a purely periodic expansion in such a base; this characterization is given in terms of an explicit set, called generalized Rauzy fractal, which is shown to be a graphdirected selfaffine compact subset of nonzero measure which belongs to the direct product of Euclidean and padic spaces.
On the cost and complexity of the successor function
 In Proc. WORDS 2007
, 2009
"... Abstract. For a given numeration system, the successor function maps the representation of an integer n onto the representation of its successor n+1. In a general setting, the successor function maps the nth word of a genealogically ordered language L onto the (n+1)th word of L. We show that, if t ..."
Abstract

Cited by 4 (3 self)
 Add to MetaCart
Abstract. For a given numeration system, the successor function maps the representation of an integer n onto the representation of its successor n+1. In a general setting, the successor function maps the nth word of a genealogically ordered language L onto the (n+1)th word of L. We show that, if the ratio of the number of elements of length n +1overthenumber of elements of length n of the language has a limit β>1, then the amortized cost of the successor function is equal to β/(β − 1). From this, we deduce the value of the amortized cost for several classes of numeration systems (integer base systems, canonical numeration systems associated with a Parry number, abstract numeration systems built on a rational language, and rational base numeration systems). 1
BetaIntegers As A Group
, 1999
"... this paper. The fiintegers are defined via a numeration system with base fi, see below. In ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
this paper. The fiintegers are defined via a numeration system with base fi, see below. In