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On Uniform DOL Words
 STACS'98, LNCS 1373
, 1998
"... . We introduce the wide class of marked uniform DOL words and study their structure. The criterium of circularity of a marked uniform DOL word is given, and the subword complexity function is found for the uncircular case as well as for the circular one. The same technique is valid for a wider c ..."
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. We introduce the wide class of marked uniform DOL words and study their structure. The criterium of circularity of a marked uniform DOL word is given, and the subword complexity function is found for the uncircular case as well as for the circular one. The same technique is valid for a wider class of uniform DOL sequences which includes (p; 1)Toeplitz words (see [4]). 1 Introduction DOL word w is an infinite word on a finite alphabet \Sigma which is a fixed point of a morphism ' : \Sigma ! \Sigma ; i. e., w = lim i!1 ' i (a) for a 2 \Sigma . The class of DOL words has been extensively studied and contains famous examples concerning pattern avoidance, such as the cubefree ThueMorse word on the twoletter alphabet and a squarefree word on the threeletter alphabet. A DOL word w is called uniform if all the words '(a i ); a i 2 \Sigma , are of the same length. In this paper, we deal with marked DOL words; i. e., fixed points of uniform morphisms with all the images of...
Arithmetical Complexity of Infinite Words
"... We introduce a new notion of the arithmetical complexity of a word, that is the number of words of a given length which occur in it in arithmetical progressions. The arithmetical complexity is related to a wellknown function of subword complexity and cannot be less than it. However, our main result ..."
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We introduce a new notion of the arithmetical complexity of a word, that is the number of words of a given length which occur in it in arithmetical progressions. The arithmetical complexity is related to a wellknown function of subword complexity and cannot be less than it. However, our main results show that the behavour of the arithmetical complexity is not determined only by the subword complexity growth: if the latter grows linearly, the arithmetical complexity can increase both linearly and exponentially. To prove it, we consider a family of D0L words with high arithmetical complexity and a family of Toeplitz words with low one. In particular, we nd the arithmetical complexity of the ThueMorse word and of the paperfolding word. 1
SEQUENCES OF LOW ARITHMETICAL COMPLEXITY
 THEORETICAL INFORMATICS AND APPLICATIONS
, 2005
"... Arithmetical complexity of a sequence is the number of words of length n that can be extracted from it according to arithmetic progressions. We study uniformly recurrent words of low arithmetical complexity and describe the family of such words having lowest complexity. ..."
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Arithmetical complexity of a sequence is the number of words of length n that can be extracted from it according to arithmetic progressions. We study uniformly recurrent words of low arithmetical complexity and describe the family of such words having lowest complexity.
Language Structure of Pattern Sturmian Words
"... Pattern Sturmian words introduced by Kamae and Zamboni [KZ1, KZ2] are an analogy of Sturmian words for the maximal pattern complexity instead of the block complexity. So far, two kinds of recurrent pattern Sturmian words are known, namely, rotation words and Toeplitz words. But neither a structural ..."
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Pattern Sturmian words introduced by Kamae and Zamboni [KZ1, KZ2] are an analogy of Sturmian words for the maximal pattern complexity instead of the block complexity. So far, two kinds of recurrent pattern Sturmian words are known, namely, rotation words and Toeplitz words. But neither a structural characterization nor a reasonable classi cation of the recurrent pattern Sturmian words is known. In this paper, we introduce a new notion, pattern Sturmian sets, which are used to study the language structure of pattern Sturmian words. We prove that there are exactly two primitive structures for pattern Sturmian words. Consequently, we suggest a classi cation of pattern Sturmian words according to structures of pattern Sturmian sets and prove that there are at most three classes in this classi cation. Rotation words and Toeplitz words fall into two di erent classes, but no examples of words from the third class are known.
Languages Obtained From Infinite Words
, 1996
"... Answering two problems formulated by Marcus and Paun, we prove that it is decidable whether or not a contextfree language can be written as the set of all finite prefixes of an infinite word and that it is decidable whether or not a regular language can be written as the set of all finite subwords ..."
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Answering two problems formulated by Marcus and Paun, we prove that it is decidable whether or not a contextfree language can be written as the set of all finite prefixes of an infinite word and that it is decidable whether or not a regular language can be written as the set of all finite subwords of an infinite word. TUCS Research Group
Language structure of pattern Sturmian words
, 2006
"... www.elsevier.com/locate/disc Pattern Sturmian words introduced by Kamae and Zamboni [Sequence entropy and the maximal pattern complexity of infinite words, Ergodic Theory Dynamical Systems 22 (2002) 1191–1199; Maximal pattern complexity for discrete systems, Ergodic Theory Dynamical Systems 22 (2002 ..."
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www.elsevier.com/locate/disc Pattern Sturmian words introduced by Kamae and Zamboni [Sequence entropy and the maximal pattern complexity of infinite words, Ergodic Theory Dynamical Systems 22 (2002) 1191–1199; Maximal pattern complexity for discrete systems, Ergodic Theory Dynamical Systems 22 (2002) 1201–1214] are an analogy of Sturmian words for the maximal pattern complexity instead of the block complexity. So far, two kinds of recurrent pattern Sturmian words are known, namely, rotation words and Toeplitz words. But neither a structural characterization nor a reasonable classification of the recurrent pattern Sturmian words is known. In this paper, we introduce a new notion, pattern Sturmian sets, which are used to study the language structure of pattern Sturmian words. We prove that there are exactly two primitive structures for pattern Sturmian words. Consequently, we suggest a classification of pattern Sturmian words according to structures of pattern Sturmian sets and prove that there are at most three classes in this classification. Rotation words and Toeplitz words fall into two different classes, but no examples of words from the third class are known. © 2006 Elsevier B.V. All rights reserved.