J. Berstel. Axel Thue's work on repetitions in words. Invited Lecture at the 4th Conference on Formal Power Series and Algebraic Combinatorics, Montreal, 1992.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....word is not ultimately periodic, both proven di erently in the literature. Section 4 contains the constructions of in nite sequences of words in four letters with density one for every word, in nite sequences of ternary words which has a limit of their densities at one, using square free words [12, 1, 2], and in nite sequences of binary words which has a limit of their densities at two, using Thue Morse words [11, 10, 1, 2] We also show that these limits are optimal. 2 Preliminaries In this section we x the notations for this paper. We refer to [8, 4] for more basic and general de nitions. ....

.... nite sequences of words in four letters with density one for every word, in nite sequences of ternary words which has a limit of their densities at one, using square free words [12, 1, 2] and in nite sequences of binary words which has a limit of their densities at two, using Thue Morse words [11, 10, 1, 2]. We also show that these limits are optimal. 2 Preliminaries In this section we x the notations for this paper. We refer to [8, 4] for more basic and general de nitions. Let A be a nite nonempty alphabet and A be the monoid of all nite words in A; the empty word is denoted by . Let ....

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Jean Berstel. Axel Thue's work on repetitions in words. In Pierre Leroux and Christophe Reutenauer, editors, Series formelles et combinatoire algebrique, volume 11 of Publications du LaCIM, pages 65-80. Universite du Quebec a Montreal, June 1992.

....in j(0) are distinct (i.e. if t i 6= t j for i 6= j) then the symmetric D0L word w is overlap free, i.e. contains no factor of the form axaxa for any x 2 S and a 2 S. Keywords: overlap free word, D0L word, symmetric morphism 1 Introduction In his classical 1912 paper [15] see also [3]) A. Thue gave the first example of an overlap free infinite word, i.e. of a word which contains no subword of the form axaxa for any symbol a and word x. Thue s example is known now as the Thue Morse word w TM = 01101001100101101001011001101001 : It was rediscovered several times, can be ....

J. Berstel. Axel Thue's work on repetitions in words. In P. Leroux and C. Reutenauer, eds., S eries Formelles et Combinatoire Alg ebrique, no. 11 in Publications du LACIM, Universite du Quebec a Montreal, 1992, 65--80.

.... Uber die gegenseitige Lage gleicher Teile gewisser Zeichenreihen of 1912 constitute a foundation of the subject of combinatorics on words, in particular of morphic in nite words (of which the Thue Morse word is the most prominent example) This work has been discussed in detail by Berstel in [Bers92]. In the paper Probleme uber Ver anderungen von Zeichenreihen nach gegebenen Regeln (1914) which is, in fact, a continuation of the paper to be considered here, Thue introduces word rewriting systems and poses the word problem for them. This problem was shown to be undecidable by Emil Post in ....

J. Berstel, Axel Thue's work on repetitions in words, In: P. Leroux and C. Reutenauer, eds., Series formelles et combinatoire algebrique, Publications du LaCIM, Universite du Quebec a Montreal, Montreal 1992, pp. 65-80.

.... from B to B de ned by : 0 7 01 1 7 10 The xed point (1) 100101100110 : is the characteristic word of the predicate P with n 2 P i the binary expansion of n contains an even number of 1. The xed point (1) is the well known Thue Morse word. We refer the reader to [5] for an interesting survey on that sequence and related works of A. Thue. Recall that a B uchi automaton is an automaton A = Q; E; I ; F ) where Q is a nite set of states, E Q A Q is the set of transitions and I and F are the sets of initial and nal states. A path is successful if it ....

Berstel, J. Axel Thue's work on repetitions in words. In Series formelles et combinatoire algebrique (1990), P. Leroux and C. Reutenauer, Eds., Publications du LaCIM, Universite du Quebec a Montreal, pp. 65-80.

....the morphism from B to B given by : 0 7 01; 1 7 10. The xed point (1) 100101100110 : is the characteristic word of the predicate P with n 2 P i the binary expansion of n contains an even number of 1. The xed point (1) is the well known Thue Morse word (see e.g. [4]) Recall that a B uchi automaton is an automaton A = Q; E; I ; F ) where Q is a nite set of states, E Q A Q is the set of transitions and I and F are the sets of initial and nal states. A path is successful if it starts in an initial state and goes in nitely often through a nal state. ....

Jean Berstel. Axel Thue's work on repetitions in words. In P. Leroux and C. Reutenauer, editors, Series formelles et combinatoire algebrique, pages 65-80. Publications du LaCIM, Universite du Quebec a Montreal, 1990.

.... from B to B de ned by : 0 7 01 1 7 10 The xed point (1) 100101100110 : is the characteristic word of the predicate P with n 2 P i the binary expansion of n contains an even number of 1. The xed point (1) is the well known Thue Morse word. We refer the reader to [4] for an interesting survey on that sequence and related works of A. Thue. Now we can state the main result of this section. Theorem 4 Let x = a) be a xed morphic word where : A A and : A B are morphisms. For any B uchi automaton A, it can be decided whether x ....

Berstel, J. Axel Thue's work on repetitions in words. In Series formelles et combinatoire algebrique (1990), P. Leroux and C. Reutenauer, Eds., Publications du LaCIM, Universite du Quebec a Montreal, pp. 65-80.

.... such as algebra, number theory, game theory (see [15, 21] The oldest results of this kind, dating back to the beginning of the century, are Thue s famous constructions of in nite square free and (strongly) cube free words over alphabets of three and two letters respectively [22, 23] see also [4]) A word is square free (respectively cube free, strongly cube free) if it does not contain a subword uu (respectively uuu, uua) where u is a non empty word and a is the rst letter of u. During the last two decades, di erent generalizations of Thue s results have been studied. A natural ....

....is another formulation of this result: There exists an in nite word over three letters that not only avoids repetitions (subwords uu) but does not admit subwords uv, where v is a pre x of u of length more than 3juj=4. Generalizations of this result for bigger alphabets have been obtained (see [4] for more references; see also [8] for a related result) In this paper, that ts into this general research direction, we address the following general problem. Assume that each letter has some weight, and we try to minimize the total weight of a word of given length avoiding the pattern. For ....

J. Berstel. Axel thue's work on repetitions in words. Invited Lecture at the 4th Conference on Formal Power Series and Algebraic Combinatorics, Montreal, 1992, June 1992. accessible at http://www-igm.univ-mlv.fr/~berstel/index.html.

....G. Andrews [2, Theorem 10.3] The purpose of this note is to prove (2) Before doing this, however, we will show that c has a number of other interesting properties. Chief among these is the fact that c is closely related to the famous Thue Morse sequence, t. See the survey article of J. Berstel [3] for more information about t. First we need to have a characterization of the integers in the sequence c. 1 Proposition 1 If n is any positive integer then n 2 c if and only if n = 2 2i (2j 1) for some nonnegative integers i and j. Proof. Every positive integer n can be uniquely written ....

J. Berstel, Axel Thue's work on repetitions in words, in "S'eries Formelles et Combinatoire Alg'ebrique," P. Leroux and C. Reutenauer eds., Publications du LACIM, Vol. 11, Universit'e du Qu'ebec `a Montr'eal, 1992, 65--80.

....A are the identity Id, the morphism E which exchanges a and b, and the morphism such that (a) ab and (b) ba. Clearly, the composition of two overlap free morphisms is overlap free. Thus for any integer k, k and E ffi k are overlap free. The converse was proved by Thue [4] see also [1, 5]) Theorem 1 A morphism h on fa; bg is overlap free iff h = k or h = E ffi k for some integer k 0. In this note, we will characterize all the test sets for the overlap freeness of morphism on fa; bg i.e. each set X such that for any morphism h, h is overlap free iff for all x in X, h(x) ....

J. Berstel, Axel Thue's work on repetitions in words, (4th Conf. on Formal Power Series an Algebraic Combinatorics, Montreal 06/1992), Rapport LITP 92.70

.... 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 square free and (strongly) cube free words over alphabets of three and two letters respectively [17, 18] see also [5]) A word is square free (respectively cube free, strongly cube free) if it does not contain a subword uu (respectively uuu, uua) where u is a non empty word and a is the first letter of u. During the last two decades, different generalizations of Thue s results have been studied. A natural ....

....There is another formulation of this result: There is an infinite word over three letters that not only avoids repetitions (subwords uu) but does not admit subwords uv, where v is a prefix of u of length more than 3juj=4. Generalizations of this result for bigger alphabets have been obtained (see [5] for more references; see also [8] for a related result) In this paper that fits into this general research direction, we address the following general problem which, to the best of our knowledge, has not been studied so far. Assume that each letter has some weight, and we try to minimize the ....

J. Berstel. Axel thue's work on repetitions in words. Invited Lecture at the 4th Conference on Formal Power Series and Algebraic Combinatorics, Montreal, 1992, June 1992. accessible at http://www-litp.ibp.fr:80/berstel/.

....words that don t contain such subwords are called n th power free. Back in the beginning of the century, Thue proved that there exist infinite 2 nd power free (square free) words over the three letter alphabet, and 3 rd power free (cube free) words over the two letter alphabet [15,16] see also [4]) To recast it in terms of pattern avoidability, Thue showed that pattern xx is avoidable on the three letter alphabet, and pattern xxx is avoidable on the two letter alphabet. Note that xx is unavoidable on the 2 letter alphabet, and this illustrates the On leave from French Russian ....

J. Berstel. Axel thue's work on repetitions in words. Invited Lecture at the 4th Conference on Formal Power Series and Algebraic Combinatorics, Montreal, 1992, June 1992. accessible at http://www-litp.ibp.fr:80/berstel/.

....were til 1970 s only scattered samples in the literature, during the last quarter of the century the research on words has been systematic, extensive, and we believe, also successful. Actually, it was already at the beginning of this century when A. Thue initiated a systematic study on words, cf. [Be6] for a survey of Thue s work. However, his fundamental results, cf. T1] T2] and also [Be8] remained quite unnoticed for decades, mainly due to the unknown journals he used. Later many of his results were discovered several times in different connections. The modern systematic research on ....

....to many other fragments of formal language theory, results of this theory depend on the size of the alphabet. 8.1 Prelude The theory of avoidable words is among the oldest in formal language theory. A systematic study was carried out by A. Thue at the beginning of this century, see [T1] T2] [Be6] and [Be8] for a survey of Thue s work. Later these problems have been encountered several times in different connections, and manyimportant results, including most of Thue s original ones, have been discovered or rediscovered, cf. Chapter 3 in [Lo] The topic has been under a very active research ....

J. Berstel, Axel Thue's work on repetitions in words, Proc. 4th FPSAC, Montreal, 1992; also LITP Report 70, 1992.

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J. Berstel, Axel Thue's work on repetitions in words, in "Series Formelles et Combinatoire Algebrique," P. Leroux and C. Reutenauer eds., Publications du LACIM, Vol. 11, Universite du Quebec a Montreal, 1992, 65--80.

No context found.

J. Berstel. Axel Thue's work on repetitions in words. Invited Lecture at the 4th Conference on Formal Power Series and Algebraic Combinatorics, Montreal, 1992.

No context found.

J. Berstel, Axel Thue's work on repetitions in words, Series formelles et combinatoire algebrique (P. Leroux, C. Reutenauer ed.), Publications du LaCIM, Universite du Quebec a Montreal, 1992; pp. 65-80.

No context found.

J. Berstel. Axel Thue's work on repetitions in words. Invited Lecture at the 4th Conference on Formal Power Series and Algebraic Combinatorics, Montreal, 1992.

No context found.

J. Berstel [1992], Axel Thue's work on repetitions in words, in: S'eries Formelles et Combinatoire Alg'ebrique (P. Leroux and C. Reutenauer, eds.), Publ. du LaCIM, Vol. 11, Universit'e de Qu'ebec `a Montr'eal, Canada, pp. 65-80.

No context found.

J. Berstel [1992], Axel Thue's work on repetitions in words, in: S'eries Formelles et Combinatoire Alg'ebrique (P. Leroux and C. Reutenauer, eds.), Publ. du LACIM, Vol. 11, Universit'e de Qu'ebec, `a Montr'eal, pp. 65-80.

No context found.

J. Berstel [1992], Axel Thue's work on repetitions in words, in: S'eries Formelles et Combinatoire Alg'ebrique (P. Leroux and C. Reutenauer, eds.), Publ. du LaCIM, Vol. 11, Universit'e de Qu'ebec `a Montr'eal, Canada, pp. 65-80.

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