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The Monadic Theory of Morphic Infinite Words and Generalizations
"... We present new examples of infinite words which have a decidable monadic theory. Formally, we consider structures hN; <; P i which expand the ordering hN;
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We present new examples of infinite words which have a decidable monadic theory. Formally, we consider structures hN; <; P i which expand the ordering hN; <i of the natural numbers by a unary predicate P ; the corresponding infinite word is the characteristic 01sequence xP of P . We show that for a morphic predicate P the associated monadic secondorder theory MThhN; <; P i is decidable, thus extending results of Elgot and Rabin (1966) and Maes (1999). The solution is obtained in the framework of semigroup theory, which is then connected to the known automata theoretic approach of Elgot and Rabin. Finally, a large class of predicates P is exhibited such that the monadic theory MThhN; <; P i is decidable, which unifies and extends the previously known examples.
Finding approximate repetitions under Hamming distance
 Theoretical Computer Science
, 2001
"... The problem of computing tandem repetitions with K possible mismatches is studied. Two main definitions are considered, and for both of them an O(nK log K + S) algorithm is proposed (S the size of the output). This improves, in particular, the bound obtained in [LS93]. Finally, other possible defini ..."
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Cited by 25 (1 self)
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The problem of computing tandem repetitions with K possible mismatches is studied. Two main definitions are considered, and for both of them an O(nK log K + S) algorithm is proposed (S the size of the output). This improves, in particular, the bound obtained in [LS93]. Finally, other possible definions are briefly analyzed.
Periodicity on Partial Words
 Computers and Mathematics with Applications 47
, 2004
"... Codes play an important role in the study of combinatorics on words. Recently, we introduced pcodes that play a role in the study of combinatorics on partial words. Partial words are strings over a finite alphabet that may contain a number of “do not know ” symbols. In this paper, the theory of code ..."
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Cited by 22 (8 self)
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Codes play an important role in the study of combinatorics on words. Recently, we introduced pcodes that play a role in the study of combinatorics on partial words. Partial words are strings over a finite alphabet that may contain a number of “do not know ” symbols. In this paper, the theory of codes of words is revisited starting from pcodes of partial words. We present some important properties of pcodes. We give several equivalent definitions of pcodes and the monoids they generate. We investigate in particular the Defect Theorem for partial words. We describe an algorithm to test whether or not a finite set of partial words is a pcode. We also discuss twoelement pcodes, complete pcodes, maximal pcodes, and the class of circular pcodes. A World Wide Web server interface has been established at
Polynomial versus exponential growth in repetitionfree binary words
 To appear, J. Combinatorial Theory Ser. A
, 2003
"... It is known that the number of overlapfree binary words of length n grows polynomially, while the number of cubefree binary words grows exponentially. We show that the dividing line between polynomial and exponential growth is 7 3. More precisely, there are only polynomially many binary words of le ..."
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Cited by 18 (4 self)
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It is known that the number of overlapfree binary words of length n grows polynomially, while the number of cubefree binary words grows exponentially. We show that the dividing line between polynomial and exponential growth is 7 3. More precisely, there are only polynomially many binary words of length n that avoid 7 3powers, but there are exponentially many binary words of length n that avoid 7+ 3powers. This answers an open question of Kobayashi from 1986. 1
Characterizations of finite and infinite episturmian words via lexicographic orderings
 European J. Combin. (in
, 2006
"... In this paper, we characterize by lexicographic order all finite Sturmian and episturmian words, i.e., all (finite) factors of such infinite words. Consequently, we obtain a characterization of infinite episturmian words in a wide sense (episturmian and episkew infinite words). That is, we character ..."
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Cited by 15 (8 self)
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In this paper, we characterize by lexicographic order all finite Sturmian and episturmian words, i.e., all (finite) factors of such infinite words. Consequently, we obtain a characterization of infinite episturmian words in a wide sense (episturmian and episkew infinite words). That is, we characterize the set of all infinite words whose factors are (finite) episturmian. Similarly, we characterize by lexicographic order all balanced infinite words over a 2letter alphabet; in other words, all Sturmian and skew infinite words, the factors of which are (finite) Sturmian. Key words: combinatorics on words; lexicographic order; episturmian word; Sturmian word; ArnouxRauzy sequence; balanced word; skew word; episkew word 2000 MSC: 68R15 1
Languages of DotDepth 3/2
 In Proceedings 17th Symposium on Theoretical Aspects of Computer Science
, 2000
"... . We prove an effective characterization of languages having dotdepth 3=2. Let B 3=2 denote this class, i.e., languages that can be written as finite unions of languages of the form u0L1u1L2u2 \Delta \Delta \Delta Lnun , where u i 2 A and L i are languages of dotdepth one. Let F be a determi ..."
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Cited by 14 (6 self)
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. We prove an effective characterization of languages having dotdepth 3=2. Let B 3=2 denote this class, i.e., languages that can be written as finite unions of languages of the form u0L1u1L2u2 \Delta \Delta \Delta Lnun , where u i 2 A and L i are languages of dotdepth one. Let F be a deterministic finite automaton accepting some language L. Resulting from a detailed study of the structure of B 3=2 , we identify a pattern P (cf. Fig. 2) such that L belongs to B 3=2 if and only if F does not have pattern P in its transition graph. This yields an NLalgorithm for the membership problem for B 3=2 . Due to known relations between the dotdepth hierarchy and symbolic logic, the decidability of the class of languages definable by \Sigma 2formulas of the logic FO[!; min; max; S; P ] follows. We give an algebraic interpretation of our result. 1 Introduction We contribute to the theory of finite automata and regular languages, with consequences in logic as well as in algeb...
Periods and Binary Words
 J. Combin. Theory Ser. A
, 2000
"... We give an elementary short proof for a wellknown theorem of Guibas and Odlyzko stating that the sets of periods of words are independent of the alphabet size. As a consequence of our constructing proof, we give a linear time algorithm which, given a word, computes a binary one with the same period ..."
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Cited by 13 (6 self)
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We give an elementary short proof for a wellknown theorem of Guibas and Odlyzko stating that the sets of periods of words are independent of the alphabet size. As a consequence of our constructing proof, we give a linear time algorithm which, given a word, computes a binary one with the same periods. We give also a very short proof for the famous Fine and Wilf's periodicity lemma.
Combinatorics of Periods in Strings
"... We consider the set (n) of all period sets of strings of length n over a nite alphabet. We show that there is redundancy in period sets and introduce the notion of an irreducible period set. We prove that (n) is a lattice under set inclusion and does not satisfy the JordanDedekind condition. We ..."
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Cited by 12 (4 self)
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We consider the set (n) of all period sets of strings of length n over a nite alphabet. We show that there is redundancy in period sets and introduce the notion of an irreducible period set. We prove that (n) is a lattice under set inclusion and does not satisfy the JordanDedekind condition. We propose the rst enumeration algorithm for (n) and improve upon the previously known asymptotic lower bounds on the cardinality of (n). Finally, we provide a new recurrence to compute the number of strings sharing a given period set. 1
On a Special Class of Primitive Words
"... Abstract. When representing DNA molecules as words, it is necessary to take into account the fact that a word u encodes basically the same information as its WatsonCrick complement θ(u), where θ denotes the WatsonCrick complementarity function. Thus, an expression which involves only a word u and ..."
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Cited by 12 (10 self)
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Abstract. When representing DNA molecules as words, it is necessary to take into account the fact that a word u encodes basically the same information as its WatsonCrick complement θ(u), where θ denotes the WatsonCrick complementarity function. Thus, an expression which involves only a word u and its complement can be still considered as a repeating sequence. In this context, we define and investigate the properties of a special class of primitive words, called θprimitive, which cannot be expressed as such repeating sequences. For instance, we prove the existence of a unique θprimitive root of a given word, and we give some constraints forcing two distinct words to share their θprimitive root. Also, we present an extension of the wellknown Fine and Wilf Theorem, for which we give an optimal bound. 1
Partial Words and the Critical Factorization Theorem
 J. Combin. Theory Ser. A
, 2007
"... The study of combinatorics on words, or finite sequences of symbols from a finite alphabet, finds applications in several areas of biology, computer science, mathematics, and physics. Molecular biology, in particular, has stimulated considerable interest in the study of combinatorics on partial word ..."
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Cited by 10 (6 self)
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The study of combinatorics on words, or finite sequences of symbols from a finite alphabet, finds applications in several areas of biology, computer science, mathematics, and physics. Molecular biology, in particular, has stimulated considerable interest in the study of combinatorics on partial words that are sequences that may have a number of “do not know ” symbols also called “holes”. This paper is devoted to a fundamental result on periods of words, the Critical Factorization Theorem, which states that the period of a word is always locally detectable in at least one position of the word resulting in a corresponding critical factorization. Here, we describe precisely the class of partial words w with one hole for which the weak period is locally detectable in at least one position of w. Our proof provides an algorithm which computes a critical factorization when one exists. A World Wide Web server interface at