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20
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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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.
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...
A Generalization of Repetition Threshold
, 2005
"... Brandenburg and (implicitly) Dejean introduced the concept of repetition threshold: the smallest real number α such that there exists an infinite word over a kletter alphabet that avoids βpowers for all β > α. We generalize this concept to include the lengths of the avoided words. We give some ..."
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Brandenburg and (implicitly) Dejean introduced the concept of repetition threshold: the smallest real number α such that there exists an infinite word over a kletter alphabet that avoids βpowers for all β > α. We generalize this concept to include the lengths of the avoided words. We give some conjectures supported by numerical evidence and prove some of these conjectures. As a consequence of one of our results, we show that the pattern ABCBABC is 2avoidable. This resolves a question left open in Cassaigne’s thesis.
No iterated morphism generates any Arshon sequence of odd order
 Discr. Math
"... We show that no Arshon sequence of odd order can be generated by an iterated morphism. This solves a problem of Kitaev and generalizes results of Berstel and of Kitaev. Keywords: Combinatorics on words, nonrepetitive words, Arshon’s sequence, DOL systems 1 ..."
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We show that no Arshon sequence of odd order can be generated by an iterated morphism. This solves a problem of Kitaev and generalizes results of Berstel and of Kitaev. Keywords: Combinatorics on words, nonrepetitive words, Arshon’s sequence, DOL systems 1
Avoiding patterns in the Abelian sense
 Canadian J. Math
, 2001
"... Abstract. We classify all 3 letter patterns that are avoidable in the abelian sense. A short list of four letter patterns for which abelian avoidance is undecided is given. Using a generalization of Zimin words we deduce some properties of ωwords avoiding these patterns. 1 ..."
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Abstract. We classify all 3 letter patterns that are avoidable in the abelian sense. A short list of four letter patterns for which abelian avoidance is undecided is given. Using a generalization of Zimin words we deduce some properties of ωwords avoiding these patterns. 1
Patterns in Words Versus Patterns in Trees: A Brief Survey and New Results
 International Conference &quot;Perspectives of Systems Informatics
, 1999
"... In this paper we study some natural problems related to specifying sets of words and trees by patterns. ..."
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In this paper we study some natural problems related to specifying sets of words and trees by patterns.
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...
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.