Partially abelian squarefree words (original) (raw)

RAIRO - Theoretical Informatics and Applications

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On dot-depth two

RAIRO - Theoretical Informatics and Applications

Abelian Square-Free Partial Words

Lecture Notes in Computer Science, 2010

Erdös raised the question whether there exist infinite abelian square-free words over a given alphabet (words in which no two adjacent subwords are permutations of each other). Infinite abelian square-free words have been constructed over alphabets of sizes as small as four.

New Abelian Square-Free DT0L-Languages over 4 Letters

In 1906 Axel Thue (34) started the systematic study of structures in words. Consequently, he studied basic objects of theoretical computer science long before the invention of the computer or DNA. In 1961 Paul Erdös (13) raised the question whether abelian squares can be avoided in infinitely long words. In 1992, we presented in (19), see also (20-23), an abelian square-free (a-2-free) endomorphism g85 on the four letter alphabet S4 = {a , b , c , d}. The size of g85 , i.e. |g85 (abcd)|, is equal to 4×85. Until now, all known methods for constructing arbitrarily long a-2-free words on S4 have been based on the structure of this g85 ; see Arturo Carpi (4-6). In this paper, we report of a completely new endomorphism g98 of S4* , the iteration of which produces an infinite abelian square-free word. The size of g98 is 4×98, and the image words for letters are constructed, in part, differently from the case of g85 . For g85 they were directly obtained by permutating letters cyclically. T...

Finite degrees of ambiguity in pattern languages

RAIRO - Theoretical Informatics and Applications, 1994

Combinatorics on Words

The Mathematica journal, 2010

We explain extensive computer-aided searches that have been carried out for many years to find new ways of constructing abelian square-free words over four letters. These structures have turned out to be very rare and hard to find. We also encountered highly nonlinear phenomena that considerably affected our calculations and usually made them hard to accomplish. However, quite recently, we gained new insight into why these structures are so very rare. Consequently, the present work has the potential to make future explorations easier. The rarity of long words that avoid abelian squares can be explained, at least partly, by using the concept of an unfavorable factor. The purpose of this article is to describe the use of Mathematica in searching for and suppressing these factors. In principle, the same programs can be used with slight modifications for other kinds of word patterns as well.

Squares in partial words

Theoretical Computer Science, 2014

We investigate the number of positions that do not start a square, the number of square occurrences, and the number of distinct squares in partial words, i.e., sequences that may have undefined positions called holes. We show that the limit of the ratio of the maximum number of positions not starting a square in a binary partial word with h holes over its length n is 15/31 and the limit of the ratio of the minimum number of square occurrences in a binary partial word with h holes over its length n is 103/187, provided the limit of h/n is 0. Both limits turn out to match with the known limits for binary full words (those without holes). We prove another surprising result that the maximal proportion of defined positions that are square-free to the number of defined positions in a binary partial word with h holes of length n is 1/2, provided the limit of h/n is in the interval [1/11, 1). We also give a 2k h tight bound on the number of rightmost occurrences of squares per position in a k-ary partial word with h holes. In addition, we provide a more detailed analysis than earlier ones for the maximum number of distinct squares in a one-hole partial word of length n over an alphabet of size k, bound that is independent of k.

Maximum number of distinct and nonequivalent nonstandard squares in a word

The combinatorics of squares in a word depends on how the equivalence of halves of the square is defined. We consider Abelian squares, parameterized squares and order-preserving squares. The word uv is an Abelian (parameterized, order-preserving) square if u and v are equivalent in the Abelian (parameterized, order-preserving) sense. The maximum number of ordinary squares is known to be asymptotically linear, but the exact bound is still investigated. We present several results on the maximum number of distinct squares for nonstandard subword equivalence relations. Let SQ Abel (n, k) and SQ Abel (n, k) denote the maximum number of Abelian squares in a word of length n over an alphabet of size k, which are distinct as words and which are nonequivalent in the Abelian sense, respectively. We prove that SQ Abel (n, 2) = Θ(n 2 ) and SQ Abel (n, 2) = Ω(n 1.5 / log n). We also give linear bounds for parameterized and order-preserving squares for small alphabets: SQ param (n, 2) = Θ(n) and SQ op (n, O(1)) = Θ(n). As a side result we construct infinite words over the smallest alphabet which avoid nontrivial order-preserving squares and nontrivial parameterized cubes (nontrivial parameterized squares cannot be avoided in an infinite word).

Abelian powers and repetitions in Sturmian words

Theoretical Computer Science, 2016

) proved that at every position of a Sturmian word starts an abelian power of exponent k for every k > 0. We improve on this result by studying the maximum exponents of abelian powers and abelian repetitions (an abelian repetition is an analogue of a fractional power) in Sturmian words. We give a formula for computing the maximum exponent of an abelian power of abelian period m starting at a given position in any Sturmian word of rotation angle α. By considering all possible abelian periods m, we recover the result of Richomme, Saari and Zamboni. As an analogue of the critical exponent, we introduce the abelian critical exponent A(s α) of a Sturmian word sα of angle α as the quantity A(sα) = lim sup km/m = lim sup k m /m, where km (resp. k m ) denotes the maximum exponent of an abelian power (resp. of an abelian repetition) of abelian period m (the superior limits coincide for Sturmian words). We show that A(s α) equals the Lagrange constant of the number α. This yields a formula for computing A(sα) in terms of the partial quotients of the continued fraction expansion of α. Using this formula, we prove that A(sα) ≥ √ 5 and that the equality holds for the Fibonacci word. We further prove that A(sα) is finite if and only if α has bounded partial quotients, that is, if and only if sα is β-power-free for some real number β. Concerning the infinite Fibonacci word, we prove that: i) The longest prefix that is an abelian repetition of period F j , j > 1, has length Fj(Fj+1 + Fj-1 + 1) -2 if j is even or Fj(Fj+1 + Fj-1) -2 if j is odd, where Fj is the jth Fibonacci number; ii) The minimum abelian period of any factor is a Fibonacci number. Further, we derive a formula for the minimum abelian periods of the finite Fibonacci words: we prove that for j ≥ 3 the Fibonacci word f j , of length Fj, has minimum abelian period equal to F j/2 if j = 0, 1, 2 mod 4 or to F 1+ j/2 if j = 3 mod 4.

Abelian antipowers in infinite words

Advances in Applied Mathematics, 2019

An abelian antipower of order k (or simply an abelian k-antipower) is a concatenation of k consecutive words of the same length having pairwise distinct Parikh vectors. This definition generalizes to the abelian setting the notion of a k-antipower, as introduced in [G. Fici et al., antipowers in infinite words, J. Comb. Theory, Ser. A, 2018], that is a concatenation of k pairwise distinct words of the same length. We aim to study whether a word contains abelian k-antipowers for arbitrarily large k.Š. Holub proved that all paperfolding words contain abelian powers of every order [Abelian powers in paperfolding words. J. Comb. Theory, Ser. A, 2013]. We show that they also contain abelian antipowers of every order.

Autonomous Posets and Quantales

Theoretical Informatics and Applications, 1993

On the hypergroups with four proper pairs and without scalars

Annales mathématiques Blaise Pascal, 1996

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When Size Matters. Legal Perspective(s) on N-grams

Linköping Electronic Conference Proceedings

N-grams are of utmost importance for modern linguistics and language theory. The legal status of n-grams, however, raises many practical questions. Traditionally, text snippets are considered copyrightable if they meet the originality criterion, but no clear indicators as to the minimum length of original snippets exist; moreover, the solutions adopted in some EU Member States (the paper cites German and French law as examples) are considerably different. Furthermore, recent developments in EU law (the CJEU's Pelham decision and the new right of newspaper publishers) also provide interesting arguments in this debate. The proposed paper presents the existing approaches to the legal protection of n-grams and tries to formulate some clear guidelines as to the length of n-grams that can be freely used and shared.

On the discriminant of the Artin component

Compositio Mathematica

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Constructing Words with High Distinct Square Densities

Electronic Proceedings in Theoretical Computer Science

Fraenkel and Simpson showed that the number of distinct squares in a word of length n is bounded from above by 2n, since at most two distinct squares have their rightmost, or last, occurrence begin at each position. Improvements by Ilie to 2n − Θ(logn) and by Deza et al. to ⌊11n/6⌋ rely on the study of combinatorics of FS-double-squares, when the maximum number of two last occurrences of squares begin. In this paper, we first study how to maximize runs of FS-double-squares in the prefix of a word. We show that for a given positive integer m, the minimum length of a word beginning with m FS-double-squares, whose lengths are equal, is 7m + 3. We construct such a word and analyze its distinct-square-sequence as well as its distinct-square-density. We then generalize our construction. We also construct words with high distinct-square-densities that approach 5/6.

On a class of infinitary codes

RAIRO - Theoretical Informatics and Applications, 1990

On a class of infinitary codes Informatique théorique et applications, tome 24, n o 5 (1990), p. 441-458. http://www.numdam.org/item?id=ITA\_1990\_\_24\_5\_441\_0 © AFCET, 1990, tous droits réservés. L'accès aux archives de la revue « Informatique théorique et applications » implique l'accord avec les conditions générales d'utilisation (http://www.numdam. org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/