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Numeration systems, linear recurrences, and regular sets
 Inform. and Comput
, 1994
"... A numeration system based on a strictly increasing sequence of positive integers u0 = 1, u1, u2,... expresses a nonnegative integer n as a sum n = � i j=0 ajuj. In this case we say the string aiai−1 · · · a1a0 is a representation for n. If gcd(u0, u1,...) = g, then every sufficiently large mult ..."
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Cited by 35 (4 self)
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A numeration system based on a strictly increasing sequence of positive integers u0 = 1, u1, u2,... expresses a nonnegative integer n as a sum n = � i j=0 ajuj. In this case we say the string aiai−1 · · · a1a0 is a representation for n. If gcd(u0, u1,...) = g, then every sufficiently large multiple of g has some representation. If the lexicographic ordering on the representations is the same as the usual ordering of the integers, we say the numeration system is orderpreserving. In particular, if u0 = 1, then the greedy representation, obtained via the greedy algorithm, is orderpreserving. We prove that, subject to some technical assumptions, if the set of all representations in an orderpreserving numeration system is regular, then the sequence u = (uj)j≥0 satisfies a linear recurrence. The converse, however, is not true. The proof uses two lemmas about regular sets that may be of independent interest. The first shows that if L is regular, then the set of lexicographically greatest strings of every length in L is also regular. The second shows that the number of strings of length n in a regular language L is bounded by a constant (independent of n) iff L is the finite union of sets of the form xy ∗ z. 1
A Construction on Finite Automata That Has Remained Hidden
, 1998
"... We show how a construction on matrix representations of two tape automata proposed by Schutzenberger to prove that rational function are unambiguous can be given a central role in the theory of relations and functions realized by finite automata, in such a way that the other basic results such as th ..."
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Cited by 11 (5 self)
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We show how a construction on matrix representations of two tape automata proposed by Schutzenberger to prove that rational function are unambiguous can be given a central role in the theory of relations and functions realized by finite automata, in such a way that the other basic results such as the "CrossSection Theorem", its dual the theorem of rational uniformisation, or the decomposition theorem of rational functions into sequential functions, appear as direct and formal consequences of it.
Radix enumeration of rational languages is almost cosequential
, 2008
"... We define and study here the class of rational functions that are finite union of sequential functions. These functions can be realized by cascades of sequential transducers. After showing that cascades of any height are equivalent to cascades of height at most two and that this class strictly conta ..."
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We define and study here the class of rational functions that are finite union of sequential functions. These functions can be realized by cascades of sequential transducers. After showing that cascades of any height are equivalent to cascades of height at most two and that this class strictly contains sequential functions and is strictly contained in the class of rational functions, we prove the result whose statement gives the paper its title.
REDUNDANCY OF MINIMAL WEIGHT EXPANSIONS IN PISOT BASES
"... Abstract. Motivated by multiplication algorithms based on redundant number representations, we study representations of an integer n as a sum n = ∑ k εkUk, where the digits εk are taken from a finite alphabet Σ and (Uk)k is a linear recurrent sequence of Pisot type with U0 = 1. The most prominent ex ..."
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Abstract. Motivated by multiplication algorithms based on redundant number representations, we study representations of an integer n as a sum n = ∑ k εkUk, where the digits εk are taken from a finite alphabet Σ and (Uk)k is a linear recurrent sequence of Pisot type with U0 = 1. The most prominent example of a base sequence (Uk)k is the sequence of Fibonacci numbers. We prove that the representations of minimal weight k εk  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 number of representations of a given integer to the joint spectral radius of a certain set of matrices. 1.
EFFICIENT ALGORITHMS FOR ZECKENDORF ARITHMETIC
"... Abstract. We study the problem of addition and subtraction using the Zeckendorf representation of integers. We show that both operations can be performed in linear time; in fact they can be performed by combinational logic networks with linear size and logarithmic depth. The implications of these re ..."
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Abstract. We study the problem of addition and subtraction using the Zeckendorf representation of integers. We show that both operations can be performed in linear time; in fact they can be performed by combinational logic networks with linear size and logarithmic depth. The implications of these results for multiplication, division and squareroot extraction are also discussed. 1.
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"... This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal noncommercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or sel ..."
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This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal noncommercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier’s archiving and manuscript policies are encouraged to visit:
Minimal Digit Sets for Parallel Addition in NonStandard Numeration Systems
"... Dedicated to JeanPaul Allouche for his Sixtieth Birthday We study parallel algorithms for addition of numbers having finite representation in a positional numeration system defined by a base β in C and a finite digit set A of contiguous integers containing 0. For a fixed base β, we focus on the que ..."
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Dedicated to JeanPaul Allouche for his Sixtieth Birthday We study parallel algorithms for addition of numbers having finite representation in a positional numeration system defined by a base β in C and a finite digit set A of contiguous integers containing 0. For a fixed base β, we focus on the question of the size of the alphabet that permits addition in constant time, independently of the length of representation of the summands. We produce lower bounds on the size of such an alphabet A. For several types of wellstudied bases (negative integer, complex 1 numbers −1 + ı, 2ı, and ı √ 2, quadratic Pisot units, and noninteger rational bases), we give explicit parallel algorithms performing addition in constant time. Moreover we show that digit sets used by these algorithms are the smallest possible. 1
Geometric study of the betaintegers for a Perron number and mathematical quasicrystals
, 2003
"... Résumé. Nous étudions géométriquement les ensembles de points de R obtenus par la betanumération que sont les βentiers Z β ⊂ Z[β] où β est un nombre de Perron. Nous montrons qu’il existe deux schémas de coupeetprojection canoniques associés à la βnumération, où les βentiers se relèvent en certa ..."
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Résumé. Nous étudions géométriquement les ensembles de points de R obtenus par la betanumération que sont les βentiers Z β ⊂ Z[β] où β est un nombre de Perron. Nous montrons qu’il existe deux schémas de coupeetprojection canoniques associés à la βnumération, où les βentiers se relèvent en certains points du réseau Z m (m = degré de β) , situés autour du sousespace propre dominant de la matrice compagnon de β. Lorsque β est en particulier un nombre de Pisot, nous redonnons une preuve du fait que Z β est un ensemble de Meyer. Dans les espaces internes les fenêtres d’acceptation canoniques sont des fractals dont l’une est le fractal de Rauzy (à quasihomothétie près). Nous le montrons sur un exemple. Nous montrons que Z β ∩ R + est de type fini sur N, faisons le lien avec la classification de Lagarias des ensembles de Delaunay et donnons une borne supérieure effective de l’entier q dans la relation: x, y ∈ Z β = ⇒ x + y (respectivement x − y) ∈ β −q Z β lorsque x + y (respectivement x − y) a un βdéveloppement de Rényi fini. Abstract. We investigate in a geometrical way the point sets of R obtained by the βnumeration that are the βintegers Z β ⊂ Z[β] where β is a Perron number. We show that there exist two canonical cutandproject schemes associated with the βnumeration, allowing to lift up the βintegers to some points of the lattice Z m (m = degree of β) lying about the dominant eigenspace of the companion matrix of β. When β is in particular a Pisot number, this framework gives another proof of the fact that Z β is a Meyer set. In the internal spaces, the canonical acceptance windows are fractals and one of them is the Rauzy fractal (up to quasidilation). We show it on an example. We show that Z β∩R + is finitely generated over N and make a link with the classification of Delone sets proposed by Lagarias. Finally we give an effective upper bound for the integer q taking place in the relation: x, y ∈ Z β = ⇒ x + y (respectively x − y) ∈ β −q Z β if x + y (respectively x − y) has a finite Rényi β expansion. 1.