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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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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
Code and parse trees for lossless source encoding
 Communications in Information and Systems
, 2001
"... This paper surveys the theoretical literature on fixedtovariablelength lossless source code trees, called code trees, and on variablelengthtofixed lossless sounce code trees, called parse trees. Huffman coding [ l] is the most well known code tree problem, but there are a number of interestin ..."
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Cited by 63 (1 self)
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This paper surveys the theoretical literature on fixedtovariablelength lossless source code trees, called code trees, and on variablelengthtofixed lossless sounce code trees, called parse trees. Huffman coding [ l] is the most well known code tree problem, but there are a number of interesting variants of the problem formulation which lead to other combinatorial optimization problems. Huffman coding as an
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 40 (5 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
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 23 (9 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 18 (7 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...
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 18 (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...
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 13 (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.