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The paper approaches the classical combinatorial problem of free-ness of words, in the more general case of partial words. First, we propose an algorithm that tests efficiently whether a partial word is k-free or not. Then, we show that there exist arbitrarily many cube-free infinite partial words containing an infinite number of holes, over binary… (More)

Pseudo-repetitions are a natural generalization of the classical notion of repetitions in sequences. We solve fundamental algorithmic questions on pseudo-repetitions by application of insightful combinatorial results on words. More precisely, we efficiently decide whether a word is a pseudo-repetition and find all the pseudo-repetitive factors of a word. 1… (More)

This paper approaches the combinatorial problem of Thue freeness for partial words. Partial words are sequences over a finite alphabet that may contain a number of ―holes‖. First, we give an infinite word over a three-letter alphabet which avoids squares of length greater than two even after we replace an infinite number of positions with holes. Then, we… (More)

A well known result of Fraenkel and Simpson states that the number of distinct squares in a word of length n is bounded by 2n since at each position there are at most two distinct squares whose last occurrence start. In this paper, we investigate the problem of counting distinct squares in partial words, or sequences over a finite alphabet that may have… (More)

A k-abelian cube is a word uvw, where u, v, w have the same factors of length at most k with the same multiplicities. Previously it has been known that k-abelian cubes are avoidable over a binary alphabet for k ≥ 5. Here it is proved that this holds for k ≥ 3.

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. In this paper, we investigate the problem of avoiding abelian squares in partial… (More)

Erdös raised the question whether there exist infinite abelian square-free words over a given alphabet, that is, words in which no two adjacent subwords are permutations of each other. It can easily be checked that no such word exists over a three-letter alphabet. However, infinite abelian square-free words have been constructed over alphabets of sizes as… (More)

The problem of classifying all the avoidable binary patterns in (full) words has been completely solved (see Chapter 3 of M. Lothaire, Algebraic Combinatorics on Words, Cambridge University Press, 2002). In this paper, we classify all the avoidable binary patterns in partial words, or sequences that may have some undefined positions called holes. In… (More)

We propose an algorithm that given as input a full word w of length n, and positive integers p and d, outputs (if any exists) a maximal p-periodic partial word contained in w with the property that no two holes are within distance d. Our algorithm runs in O(nd) time and is used for the study of freeness of partial words. Furthermore, we construct an… (More)

We initiate a study of languages of partial words related to regular languages of full words. First, we study the possibility of expressing a regular language of full words as the image of a partial-words-language through a substitution that only replaces the hole symbols of the partial words with a finite set of letters. Results regarding the structure ,… (More)