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46
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 66 (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
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
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", ..."
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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
Robust Universal Complete Codes for Transmission and Compression
 Discrete Applied Mathematics
, 1996
"... Several measures are defined and investigated, which allow the comparison of codes as to their robustness against errors. Then new universal and complete sequences of variablelength codewords are proposed, based on representing the integers in a binary Fibonacci numeration system. Each sequence is ..."
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Cited by 10 (4 self)
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Several measures are defined and investigated, which allow the comparison of codes as to their robustness against errors. Then new universal and complete sequences of variablelength codewords are proposed, based on representing the integers in a binary Fibonacci numeration system. Each sequence is constant and need not be generated for every probability distribution. These codes can be used as alternatives to Huffman codes when the optimal compression of the latter is not required, and simplicity, faster processing and robustness are preferred. The codes are compared on several "reallife" examples. 1. Motivation and Introduction Let A = fA 1 ; A 2 ; \Delta \Delta \Delta ; An g be a finite set of elements, called cleartext elements, to be encoded by a static uniquely decipherable (UD) code. For notational ease, we use the term `code' as abbreviation for `set of codewords'; the corresponding encoding and decoding algorithms are always either given or clear from the context. A code i...
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.
The structure of complementary sets of integers: a 3shift theorem
 INTERNAT. J. PURE AND APPL. MATH
"... Let 0 < α < β be two irrational numbers satisfying 1/α +1/β = 1. Then the sequences a ′ n = ⌊nα⌋, b ′ n = ⌊nβ⌋, n ≥ 1, are complementary: 1 ≤ i < n}, n ≥ 1 over Z≥1, thus a ′ n satisfies: a ′ n = mex1{a ′ i, b ′ i (mex1(S), the smallest positive integer not in the set S). Suppose that c = β − α is a ..."
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Cited by 8 (6 self)
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Let 0 < α < β be two irrational numbers satisfying 1/α +1/β = 1. Then the sequences a ′ n = ⌊nα⌋, b ′ n = ⌊nβ⌋, n ≥ 1, are complementary: 1 ≤ i < n}, n ≥ 1 over Z≥1, thus a ′ n satisfies: a ′ n = mex1{a ′ i, b ′ i (mex1(S), the smallest positive integer not in the set S). Suppose that c = β − α is an integer. Then b ′ n = a ′ n + cn for all n ≥ 1. We define the following generalization of sequences a ′ n, b ′ n: Let c, n0 ∈ Z≥1, and let X ⊂ Z≥1 be an arbitrary finite set. Let an = mex1(X ∪{ai, bi: 1 ≤ i < n}), bn = an +cn, n ≥ n0. Let sn = a ′ n −an. We show that no matter how we pick c, n0 and X, from some point on the shift sequence sn assumes either one constant value or three successive values; and if the second case holds, it assumes these values in a very distinct fractallike pattern, which we describe. This work was motivated by a generalization of Wythoff’s game to N ≥ 3 piles.
Heap Games, Numeration Systems and Sequences
 Ann. of Combinatorics
, 1998
"... . We propose and analyze a 2parameter family of 2player games on two heaps of tokens, and present a strategy based on a class of sequences. The strategy looks easy, but it is actually hard. A class of exotic numeration systems is then used, which enables us to decide whether the family has an effi ..."
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Cited by 5 (3 self)
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. We propose and analyze a 2parameter family of 2player games on two heaps of tokens, and present a strategy based on a class of sequences. The strategy looks easy, but it is actually hard. A class of exotic numeration systems is then used, which enables us to decide whether the family has an efficient strategy or not. We introduce yet another class of sequences and demonstrate its equivalence with the class of sequences defined for the strategy of our games. Keywords: heap games, numeration systems, sequences 1. Example Given a 2player game played on two heaps (piles) of finitely many tokens. There are two types of moves: (I) Take any positive number of tokens from one heap, possibly the entire heap. (II) Take from both heaps, k from one and l from the other, with, say, k l. Then the move is constrained by the condition 0 ! k l ! 2k + 2, which is equivalent to 0 l \Gamma k ! k +2; k ? 0. The player making the last move (after which both heaps are empty) wins, and the opponent ...