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26
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 ..."
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Cited by 64 (4 self)
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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
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 ..."
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Cited by 12 (0 self)
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.  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...
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 ..."
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Cited by 11 (5 self)
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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
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 ..."
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Cited by 10 (1 self)
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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.
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 in ..."
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Cited by 9 (2 self)
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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
"... It is well known that real numbers with a purely periodic decimal expansion are 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 havi ..."
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Cited by 5 (4 self)
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It is well known that real numbers with a purely periodic decimal expansion are 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 a 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
"... 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 ..."
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Cited by 4 (3 self)
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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).
BetaIntegers As A Group
, 1999
"... this paper. The fiintegers are defined via a numeration system with base fi, see below. In ..."
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Cited by 4 (0 self)
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this paper. The fiintegers are defined via a numeration system with base fi, see below. In