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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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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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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 ...
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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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...
On RepetitionFree Binary Words of Minimal Density
 Theoretical Computer Science
, 1999
"... We study the minimal proportion (density) of one letter in nth powerfree binary words. First, we introduce and analyse a general notion of minimal letter density for any innite set of words which don't contain a specied set of \prohibited" subwords. We then prove that for nth powerfre ..."
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We study the minimal proportion (density) of one letter in nth powerfree binary words. First, we introduce and analyse a general notion of minimal letter density for any innite set of words which don't contain a specied set of \prohibited" subwords. We then prove that for nth powerfree binary words the density function is 1 n + 1 n 3 + 1 n 4 + O( 1 n 5 ). We also consider a generalization of nth powerfree words for fractional powers (exponents): a word is xth powerfree for a real x, if it does not contain subwords of exponent x or more. We study the minimal proportion of one letter in xth powerfree binary words as a function of x and prove, in particular, that this function is discontinuous at 7 3 as well as at all integer points n 3. Finally, we give an estimate of the size of the jumps. Keywords: Unavoidable patterns, powerfree words, exponent, minimal density. 1 Introduction One of classical topics of formal language theory and word combinatorics is th...
The Subword Complexity of a TwoParameter Family of Sequences
"... We determine the subword complexity of the characteristic functions of a twoparameter family fA n g 1 n=1 of infinite sequences which are associated with the winning strategies for a family of 2player games. A special case of the family has the form A n = bnffc for all n 2 Z?0 , where ff is a f ..."
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We determine the subword complexity of the characteristic functions of a twoparameter family fA n g 1 n=1 of infinite sequences which are associated with the winning strategies for a family of 2player games. A special case of the family has the form A n = bnffc for all n 2 Z?0 , where ff is a fixed positive irrational number. The characteristic functions of such sequences have been shown to have subword complexity n + 1. We show that every sequence in the extended family has subword complexity O(n). 1 Introduction Denote by Z0 and Z?0 the set of nonnegative integers and positive integers respectively. Given two heaps of finitely many tokens, we define a 2player heap game as follows. There are two types of moves: 1. Remove any positive number of tokens from a single heap. 2. Remove k ? 0 tokens from one heap and l ? 0 from the other. Here k and l are constrained by the condition: 0 ! k l ! sk + t, where s and t are predetermined positive integers. The player who reaches a stat...
FURTHER APPLICATIONS OF A POWER SERIES METHOD FOR PATTERN AVOIDANCE
, 907
"... Abstract. In combinatorics on words, a word w over an alphabet Σ is said to avoid a pattern p over an alphabet ∆ if there is no factor x of w and no nonerasing morphism h from ∆ ∗ to Σ ∗ such that h(p) = x. Bell and Goh have recently applied an algebraic technique due to Golod to show that for a ..."
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Abstract. In combinatorics on words, a word w over an alphabet Σ is said to avoid a pattern p over an alphabet ∆ if there is no factor x of w and no nonerasing morphism h from ∆ ∗ to Σ ∗ such that h(p) = x. Bell and Goh have recently applied an algebraic technique due to Golod to show that for a certain wide class of patterns p there are exponentially many words of length n over a 4letter alphabet that avoid p. We consider some further consequences of their work. In particular, we show that any pattern with k variables of length at least 4 k is avoidable on the binary alphabet. This improves an earlier bound due to Cassaigne and Roth. 1.
Unavoidable sets of partial words
 Theory of Computing Systems
"... The notion of an unavoidable set of words appears frequently in the fields of mathematics and theoretical computer science, in particular with its connection to the study of combinatorics on words. The theory of unavoidable sets has seen extensive study over the past twenty years. In this paper we e ..."
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The notion of an unavoidable set of words appears frequently in the fields of mathematics and theoretical computer science, in particular with its connection to the study of combinatorics on words. The theory of unavoidable sets has seen extensive study over the past twenty years. In this paper we extend the definition of unavoidable sets of words to unavoidable sets of partial words. Partial words, or finite sequences that may contain a number of “do not know ” symbols or “holes,” appear naturally in several areas of current interest such as molecular biology, data communication, and DNA computing. We demonstrate the utility of the notion of unavoidability of sets of partial words by making use of it to identify several new classes of unavoidable sets of full words. Along the way we begin work on classifying the unavoidable sets of partial words of small cardinality. We pose a conjecture, and show that affirmative proof of this conjecture gives a sufficient condition for classifying all the unavoidable sets of partial words of size two. We give a result which makes the conjecture easy to verify for a significant number of cases. We characterize many forms of unavoidable sets of partial words of size three over a binary alphabet, and completely characterize such sets over a ternary alphabet. Finally, we extend our results to unavoidable sets of partial words of size k over a kletter alphabet. ∗This material is based upon work supported by the National Science Foun
Minimal Letter Frequency in NTh PowerFree Binary Words
 in Mathematical Foundations of Computer Science 1997, Lecture Notes in Comput. Sci., 1295, eds. I. Privara and P. Ru˘zička
, 1997
"... We show that the minimal proportion of one letter in an nth powerfree binary word is asymptotically 1=n. We also consider a generalization of nth powerfree words defined through the notion of exponent: a word is xth powerfree for a real x, if it does not contain subwords of exponent x or more. ..."
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We show that the minimal proportion of one letter in an nth powerfree binary word is asymptotically 1=n. We also consider a generalization of nth powerfree words defined through the notion of exponent: a word is xth powerfree for a real x, if it does not contain subwords of exponent x or more. We study the minimal proportion of one letter in an xth powerfree binary word as a function of x and prove, in particular, that this function is discontinuous. 1 Introduction One of classical topics of formal language theory and word combinatorics is the construction of infinite words verifying certain restrictions. A typical restriction is the requirement that the word does not contain a subword of the form specified by some general pattern. Results of this kind find their applications in different areas such as algebra, number theory, game theory (see [12, 16]). The oldest results of this kind, dating back to the beginning of the century, are Thue's famous constructions of infinite squ...
Words strongly avoiding fractional powers
 Europ. J. of Combinatorics
, 1999
"... Abstract Let k be fixed, 1! k! 2. There exists an infinite word over a finite alphabet which contains no subword of the form xyz with jxyzj=jxyj * k and where z is a permutation of x. 1 Introduction Nonrepetitive words have been studied since Thue [34]. A word is nonrepetitive if it cannot be writt ..."
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Abstract Let k be fixed, 1! k! 2. There exists an infinite word over a finite alphabet which contains no subword of the form xyz with jxyzj=jxyj * k and where z is a permutation of x. 1 Introduction Nonrepetitive words have been studied since Thue [34]. A word is nonrepetitive if it cannot be written xyyz, where x; y; z are words, and y is nonempty. Infinite