Infinite words without palindrome (original) (raw)
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On the least number of palindromes contained in an infinite word
Theoretical Computer Science, 2013
We investigate the least number of palindromic factors in an infinite word. We first consider general alphabets, and give answers to this problem for periodic and non-periodic words, closed or not under reversal of factors. We then investigate the same problem when the alphabet has size two.
Factor versus palindromic complexity of uniformly recurrent infinite words
Theoretical Computer Science, 2007
We study the relation between the palindromic and factor complexity of infinite words. We show that for uniformly recurrent words one has P(n) + P(n + 1) ≤ ∆C(n) + 2, for all n ∈ N. For a large class of words it is a better estimate of the palindromic complexity in terms of the factor complexity then the one presented in [2]. We provide several examples of infinite words for which our estimate reaches its upper bound. In particular, we derive an explicit prescription for the palindromic complexity of infinite words coding r-interval exchange transformations. If the permutation π connected with the transformation is given by π(k) = r + 1 − k for all k, then there is exactly one palindrome of every even length, and exactly r palindromes of every odd length.
On some problems related to palindrome closure
Theoretical Informatics and Applications, 2008
In this paper we solve some open problems related to (pseudo)palindrome closure operators and to the infinite words generated by their iteration, that is, standard episturmian and pseudostandard words. We show that if ϑ is an involutory antimorphism of A * , then both ϑ-palindromic closures of any factor of a ϑ-standard word are also factors of some ϑ-standard word. We also introduce the class of pseudostandard words with "seed", obtained by iterated pseudopalindrome closure starting from a nonempty word. We prove that pseudostandard words with seed are morphic images of standard episturmian words. Moreover, for any given pseudostandard word s with seed, there exists an integer N such that for any n ≥ N , s has at most one right (resp. left) special factor of length n. * The work for this paper has been supported by the Italian Ministry of Education under Project COFIN 2005 -Automi e Linguaggi Formali: aspetti matematici e applicativi.
Properties of palindromes in finite words
Computing Research Repository - CORR, 2010
We present a method which displays all palindromes of a given length from De Bruijn words of a certain order, and also a recursive one which constructs all palindromes of length n+1n+1n+1 from the set of palindromes of length nnn. We show that the palindrome complexity function, which counts the number of palindromes of each length contained in a given word, has a different shape compared with the usual (subword) complexity function. We give upper bounds for the average number of palindromes contained in all words of length nnn, and obtain exact formulae for the number of palindromes of length 1 and 2 contained in all words of length nnn.
On generating binary words palindromically
Journal of Combinatorial Theory, Series A, 2015
We regard a finite word u = u 1 u 2 · · · u n up to word isomorphism as an equivalence relation on {1, 2, . . . , n} where i is equivalent to j if and only if u i = u j . Some finite words (in particular all binary words) are generated by palindromic relations of the form k ∼ j +i−k for some choice of 1 ≤ i ≤ j ≤ n and k ∈ {i, i + 1, . . . , j}. That is to say, some finite words u are uniquely determined up to word isomorphism by the position and length of some of its palindromic factors. In this paper we study the function µ(u) defined as the least number of palindromic relations required to generate u. We show that if x is an infinite word such that µ(u) ≤ 2 for each factor u of x, then x is ultimately periodic. On the other hand we establish the existence of non-ultimately periodic words for which µ(u) ≤ 3 for each factor u of x, and obtain a complete classification of such words on a binary alphabet (which includes the well known class of Sturmian words). In contrast, for the Thue-Morse word, we show that the function µ is unbounded.
On prefix palindromic length of automatic words
Theoretical Computer Science, 2021
The prefix palindromic length PPL u (n) of an infinite word u is the minimal number of concatenated palindromes needed to express the prefix of length n of u. Since 2013, it is still unknown if PPL u (n) is unbounded for every aperiodic infinite word u, even though this has been proven for almost all aperiodic words. At the same time, the only well-known nontrivial infinite word for which the function PPL u (n) has been precisely computed is the Thue-Morse word t. This word is 2-automatic and, predictably, its function PPL t (n) is 2-regular, but is this the case for all automatic words? In this paper, we prove that this function is k-regular for every k-automatic word containing only a finite number of palindromes. For two such words, namely the paperfolding word and the Rudin-Shapiro word, we derive a formula for this function. Our computational experiments suggest that generally this is not true: for the perioddoubling word, the prefix palindromic length does not look 2-regular, and for the Fibonacci word, it does not look Fibonacci-regular. If proven, these results would give rare (if not first) examples of a natural function of an automatic word which is not regular.
2006
We present a method which displays all palindromes of a given length from De Bruijn words of a certain order, and also a recursive one which constructs all palindromes of length n + 1 from the set of palindromes of length n. We show that the palindrome complexity function, which counts the number of palindromes of each length contained in a given word, has a different shape compared with the usual (subword) complexity function. We give upper bounds for the average number of palindromes contained in all words of length n, and obtain exact formulae for the number of palindromes of length 1 and 2 contained in all words of length n.
Total palindrome complexity of finite words
Discrete Mathematics, 2010
The palindrome complexity function pal w of a word w attaches to each n ∈ N the number of palindromes (factors equal to their mirror images) of length n contained in w. The number of all the nonempty palindromes in a finite word is called the total palindrome complexity of that word. We present exact bounds for the total palindrome complexity and construct words which have any palindrome complexity between these bounds, for binary alphabets as well as for alphabets with the cardinal greater than 2. Denoting by M q (n) the average number of palindromes in all words of length n over an alphabet with q letters, we present an upper bound for M q (n) and prove that the limit of M q (n)/n is 0. A more elaborate estimation leads to M q (n) = O( √ n).
A defect theorem for bi-infinite words
Theoretical Computer Science, 2003
We formulate and prove a defect theorem for bi-inÿnite words. Let X be a ÿnite set of words over a ÿnite alphabet. If a nonperiodic bi-inÿnite word w has two X -factorizations, then the combinatorial rank of X is at most card(X ) − 1, i.e., there exists a set F such that X ⊆ F + with card(F) ¡ card(X ). Moreover, in the case when the combinatorial rank of X equals card(X ), the number of periodic bi-inÿnite words which have two di erent X -factorizations is ÿnite.