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

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).

Partially abelian squarefree words (original) (raw)

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