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Axel Thue's work on repetitions in words
 Invited Lecture at the 4th Conference on Formal Power Series and Algebraic Combinatorics
, 1992
"... The purpose of this survey is to present, in contemporary terminology, the fundamental contributions of Axel Thue to the study of combinatorial properties of sequences of symbols, insofar as repetitions are concerned. The present state of the art is also sketched. ..."
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Cited by 22 (3 self)
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The purpose of this survey is to present, in contemporary terminology, the fundamental contributions of Axel Thue to the study of combinatorial properties of sequences of symbols, insofar as repetitions are concerned. The present state of the art is also sketched.
Unavoidable Binary Patterns
 Acta Informatica
, 1993
"... Peter Roth proved that there are no binary patterns of length six or more that are unavoidable on the twoletter alphabet. He gave an almost complete description of unavoidable binary patterns. In this paper we prove one of his conjectures: the pattern ff 2 fi 2 ff is 2avoidable. From this we d ..."
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Cited by 10 (1 self)
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Peter Roth proved that there are no binary patterns of length six or more that are unavoidable on the twoletter alphabet. He gave an almost complete description of unavoidable binary patterns. In this paper we prove one of his conjectures: the pattern ff 2 fi 2 ff is 2avoidable. From this we deduce the complete classification of unavoidable binary patterns. We also study the concept of avoidability by iterated morphisms and prove that there are a few 2avoidable patterns which are not avoided by any iterated morphism. 1 Introduction The concept of unavoidable pattern was introduced by Bean, Ehrenfeucht & McNulty [2] and independently by Zimin [10]. They gave a characterization of unavoidable patterns, but there is still no characterization of kunavoidable patterns, i.e. patterns that are unavoidable over a kletter alphabet. However, we can try to find all kunavoidable patterns that can be written with a given alphabet. The case of unary patterns (in other terms: powers of a ...
String NonInclusion Optimization Problems
"... . For every string inclusion relation there are two optimization problems: nd a longest string included in every string of a given nite language, and nd a shortest string including every string of a given nite language. As an example, the two wellknown pairs of problems, the longest common subst ..."
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Cited by 9 (4 self)
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. For every string inclusion relation there are two optimization problems: nd a longest string included in every string of a given nite language, and nd a shortest string including every string of a given nite language. As an example, the two wellknown pairs of problems, the longest common substring (or subsequence) problem and the shortest common superstring (or supersequence) problem, are interpretations of these two problems. In this paper we consider a class of opposite problems connected with string noninclusion relations: nd a shortest string included in no string of a given nite language and nd a longest string including no string of a given nite language. The predicate \string is not included in string " is interpreted either as \ is not a substring of " or as \ is not a subsequence of ". The main purpose is to determine the complexity status of the string noninclusion optimization problems. Using graph approaches we present polynomialtime algorith...
An algorithm to test if a given circular HDOLlanguage avoids a pattern
 in: IFIP World Computer Congress'94
, 1994
"... To prove that a pattern p is avoidable on a given alphabet, one has to construct an infinite language L that avoids p. Usually, L is a DOLlanguage (obtained by iterating a morphism h) or a HDOLlanguage (obtained by coding a DOLlanguage with another morphism g). Our purpose is to find an algorithm ..."
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Cited by 5 (0 self)
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To prove that a pattern p is avoidable on a given alphabet, one has to construct an infinite language L that avoids p. Usually, L is a DOLlanguage (obtained by iterating a morphism h) or a HDOLlanguage (obtained by coding a DOLlanguage with another morphism g). Our purpose is to find an algorithm to test, given a HDOLsystem G, whether the language L(G) generated by this system avoids p. We first define the notions of circular morphism, circular DOLsystem and circular HDOLsystem, and we show how to compute the inverse image of a pattern by a circular morphism. Then we prove that by computing successive inverse images of p, we can decide whether the language L(G) avoids p for any fixed pattern p (which may even contain constants), provided that the HDOLsystem G is circular and expansive. 1 Introduction The theory of avoidable patterns, introduced by Zimin [13] and Bean, Ehrenfeucht and McNulty [2], generalizes problems studied by Axel Thue [12] and many others, such as the ex...