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A characterization of Sturmian words by return words
"... : We present a new characterization of Sturmian words using return words. Considering each occurrence of a word w in a recurrent word, we define the set of return words over w to be the set of all distinct words beginning with an occurrence of w and ending exactly before the next occurrence of w in ..."
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: We present a new characterization of Sturmian words using return words. Considering each occurrence of a word w in a recurrent word, we define the set of return words over w to be the set of all distinct words beginning with an occurrence of w and ending exactly before the next occurrence of w in the infinite word. It is shown that an infinite word is a Sturmian word if and only if for each nonempty word w appearing in the infinite word, the cardinality of the set of return words over w is equal to two. 2 1 Introduction Sturmian words are infinite words over a binary alphabet with exactly n+1 factors of length n for each n 0 (see [2, 6, 12]). In fact, the study of the Sturmian words appears in many areas like combinatorics on words ([6]), symbolic dynamics ([3, 1, 7, 21]), theoretical computer science ([5, 17]) and tilings ([8, 14, 20, 22, 24]). The Sturmian words have many equivalent characterizations (see for a complete presentation of Sturmian words [6]) using complexity func...
Repetitive Delone sets and quasicrystals
, 1999
"... This paper considers the problem of characterizing the simplest discrete point sets that are aperiodic, using invariants based on topological dynamics. A Delone set of finite type is a Delone set X such that X − X is locally finite. Such sets are characterized by their patchcounting function NX(T) ..."
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Cited by 18 (0 self)
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This paper considers the problem of characterizing the simplest discrete point sets that are aperiodic, using invariants based on topological dynamics. A Delone set of finite type is a Delone set X such that X − X is locally finite. Such sets are characterized by their patchcounting function NX(T) of radius T being finite for all T. A Delone set X of finite type is repetitive if there is a function MX(T) such that every closed ball of radius MX(T)+T contains a complete copy of each kind of patch of radius T that occurs in X. This is equivalent to the minimality of an associated topological dynamical system with R naction. There is a lower bound for MX(T) in terms of NX(T), namely MX(T) ≥ c(NX(T)) 1/n for some positive constant c depending on the Delone set constants r,R, but there is no general upper bound for MX(T) purely in terms of NX(T). The complexity of a repetitive Delone set of finite type is measured by the growth rate of its repetitivity function MX(T). For example, the function MX(T) is bounded if and only if X is a periodic crystal. A set X is linearly repetitive if MX(T) = O(T) as T → ∞ and is densely repetitive if MX(T) = O(NX(T)) 1/n as T → ∞. We show that linearly repetitive sets
Sequences With Grouped Factors
 Developments in Language Theory III, Publications of Aristotle University of Thessaloniki
, 1998
"... We define a new class of sequences, sequences with grouped factors, in which all factors of a given length occur consecutively at some position. This class contains periodic sequences, Sturmian sequences, but also sequences with maximal complexity. As a particular case, we obtain a new characteristi ..."
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Cited by 14 (0 self)
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We define a new class of sequences, sequences with grouped factors, in which all factors of a given length occur consecutively at some position. This class contains periodic sequences, Sturmian sequences, but also sequences with maximal complexity. As a particular case, we obtain a new characteristic property of Sturmian words. 1 Introduction The recurrence function R(n) is a classical tool associated to symbolic sequences [6, 8]
Special Factors of Sequences With Linear Subword Complexity
 In Developments in Language Theory
, 1996
"... In this paper, we prove the following result, which was conjectured by S. Ferenczi: if the complexity function of a sequence has linear growth, then its differences are bounded by a constant. ..."
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Cited by 12 (2 self)
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In this paper, we prove the following result, which was conjectured by S. Ferenczi: if the complexity function of a sequence has linear growth, then its differences are bounded by a constant.
Generalization of automatic sequences for numeration systems on a regular language, preprint
, 1999
"... Let L be an infinite regular language on a totally ordered alphabet (Σ, <). Feeding a finite deterministic automaton (with output) with the words of L enumerated lexicographically with respect to < leads to an infinite sequence over the output alphabet of the automaton. This process generalizes the ..."
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Cited by 11 (5 self)
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Let L be an infinite regular language on a totally ordered alphabet (Σ, <). Feeding a finite deterministic automaton (with output) with the words of L enumerated lexicographically with respect to < leads to an infinite sequence over the output alphabet of the automaton. This process generalizes the concept of kautomatic sequence for abstract numeration systems on a regular language (instead of systems in base k). Here, I study the first properties of these sequences and their relations with numeration systems. 1
Complexity of sequences and dynamical systems
 Discr. Math
, 1999
"... In recent years, there has been a number of papers about the combinatorial notion of symbolic complexity: this is the function counting the number of factors of length n for a sequence. The complexity is an indication of the degree of randomness of the sequence: a periodic sequence has a bounded com ..."
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Cited by 8 (0 self)
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In recent years, there has been a number of papers about the combinatorial notion of symbolic complexity: this is the function counting the number of factors of length n for a sequence. The complexity is an indication of the degree of randomness of the sequence: a periodic sequence has a bounded complexity, the expansion of a normal number has an exponential complexity. For a given sequence, the complexity function is generally not of easy access, and it is a rich and instructive work to compute it; a survey of this kind of results can be found in [ALL]. We are interested here in further results in the theory of symbolic complexity, somewhat beyond the simple question of computing the complexity of various sequences. These lie mainly in two directions; first, we give a survey of an open question which is still very much in progress, namely: to determine which functions can be the symbolic complexity function of a sequence. Then, we investigate the links between the complexity of a sequence and its associated dynamical system, and insist on the cases where the knowledge of
Palindromes and twodimensional Sturmian sequences
 J. Autom. Lang. Comb
"... This paper introduces a twodimensional notion of palindrome for rectangular factors of double sequences: these palindromes are defined as centrosymmetric factors. This notion provides a characterization of twodimensional Sturmian sequences in terms of twodimensional palindromes, generalizing to d ..."
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Cited by 8 (2 self)
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This paper introduces a twodimensional notion of palindrome for rectangular factors of double sequences: these palindromes are defined as centrosymmetric factors. This notion provides a characterization of twodimensional Sturmian sequences in terms of twodimensional palindromes, generalizing to double sequences the results in [13]. Keywords Palindromes, double sequences, generalized Sturmian sequences, symbolic dynamics, combinatorics on words. 1 Introduction This paper studies some properties of symmetry for the rectangular factors of a family of twodimensional sequences obtained as a binary coding of a Z 2 action defined on the onedimensional torus T 1 (= R=Z) by two irrational rotations. More precisely, such a sequence (Um;n ) (m;n)2Z 2 is defined on the alphabet f0; 1g as follows: consider a partition of the unit circle into two halfopen intervals I 0 and I 1 ; let ff; fi; fl 2 R with ff 62 Q; we have 8(m; n) 2 Z 2 ; (Um;n = 0 () mff + nfi + fl 2 I 0 modulo 1): We...
Toeplitz Words, Generalized Periodicity and Periodically Iterated Morphisms
 European J. of Combinatorics
, 1997
"... We consider socalled Toeplitz words which can be viewed as generalizations of oneway infinite periodic words. We compute their subword complexity, and show that they can always be generated by iterating periodically a finite number of morphisms. Moreover, we define a structural classification of T ..."
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Cited by 7 (2 self)
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We consider socalled Toeplitz words which can be viewed as generalizations of oneway infinite periodic words. We compute their subword complexity, and show that they can always be generated by iterating periodically a finite number of morphisms. Moreover, we define a structural classification of Toeplitz words which is reflected in the way how they can be generated by iterated morphisms.
Second Preimage Attacks on Dithered Hash Functions
"... Abstract. We develop a new generic longmessage second preimage attack, based on combining the techniques in the second preimage attacks of Dean [8] and Kelsey and Schneier [16] with the herding attack of Kelsey and Kohno [15]. We show that these generic attacks apply to hash functions using the Mer ..."
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Cited by 7 (1 self)
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Abstract. We develop a new generic longmessage second preimage attack, based on combining the techniques in the second preimage attacks of Dean [8] and Kelsey and Schneier [16] with the herding attack of Kelsey and Kohno [15]. We show that these generic attacks apply to hash functions using the MerkleDamgård construction with only slightly more work than the previously known attack, but allow enormously more control of the contents of the second preimage found. Additionally, we show that our new attack applies to several hash function constructions which are not vulnerable to the previously known attack, including the dithered hash proposal of Rivest [25], Shoup’s UOWHF[26] and the ROX hash construction [2]. We analyze the properties of the dithering sequence used in [25], and develop a timememory tradeoff which allows us to apply our second preimage attack to a wide range of dithering sequences, including sequences which are much stronger than those in Rivest’s proposals. Finally, we show that both the existing second preimage attacks [8,16] and our new attack can be applied even more efficiently to multiple target messages; in general, given a set of many target messages with a total of 2 R message blocks, these second preimage attacks can find a second preimage for one of those target messages with no more work than would be necessary to find a second preimage for a single target message of 2 R message blocks.