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.

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 two-element pcodes, complete pcodes, maximal pcodes, and the class of circular pcodes. A World Wide Web server interface has been established at

It is known that the number of overlap-free 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 3-powers, but there are exponentially many binary words of length n that avoid 7+ 3-powers. This answers an open question of Kobayashi from 1986. 1

We give an elementary short proof for a well-known 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.

. We prove an effective characterization of languages having dot--depth 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 dot--depth 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 NL--algorithm for the membership problem for B 3=2 . Due to known relations between the dot--depth hierarchy and symbolic logic, the decidability of the class of languages definable by \Sigma 2--formulas 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...

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

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

. It is shown that it is decidable whether an equation over a free partially commutative monoid has a solution. We give a proof of this result using normal forms. Our method is a direct reduction of a trace equation system to a word equation system with regular constraints. Hereby we use the extension of Makanin's theorem on the decidability of word equations to word equations with regular constraints, which is due to Schulz. Keywords: word equations, partial commutations, traces, word equations with regular constraints 1 Introduction Solving equations is a central topic in various elds of computer science, especially concerning unication, as required by automated theorem proving or logic programming. A famous result of Makanin [15] states that there exists an algorithm deciding for a given equation L = R, where L; R 2 ( [ ) contain both unknowns from and constants from , whether an assignment : ! exists satisfying (L) = (R). In a more general setting it is k...

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 2-letter 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; Arnoux-Rauzy sequence; balanced word; skew word; episkew word 2000 MSC: 68R15 1