127 research outputs found
Languages invariant under more symmetries: overlapping factors versus palindromic richness
Factor complexity and palindromic complexity of
infinite words with language closed under reversal are known to be related by
the inequality for any \,. Word for which
the equality is attained for any is usually called rich in palindromes. In
this article we study words whose languages are invariant under a finite group
of symmetries. For such words we prove a stronger version of the above
inequality. We introduce notion of -palindromic richness and give several
examples of -rich words, including the Thue-Morse sequence as well.Comment: 22 pages, 1 figur
Closed, Palindromic, Rich, Privileged, Trapezoidal, and Balanced Words in Automatic Sequences
We prove that the property of being closed (resp., palindromic, rich,
privileged trapezoidal, balanced) is expressible in first-order logic for
automatic (and some related) sequences. It therefore follows that the
characteristic function of those n for which an automatic sequence x has a
closed (resp., palindromic, privileged, rich, trape- zoidal, balanced) factor
of length n is automatic. For privileged words this requires a new
characterization of the privileged property. We compute the corresponding
characteristic functions for various famous sequences, such as the Thue-Morse
sequence, the Rudin-Shapiro sequence, the ordinary paperfolding sequence, the
period-doubling sequence, and the Fibonacci sequence. Finally, we also show
that the function counting the total number of palindromic factors in a prefix
of length n of a k-automatic sequence is not k-synchronized
String attractors and combinatorics on words
The notion of string attractor has recently been introduced in [Prezza, 2017] and studied in [Kempa and Prezza, 2018] to provide a unifying framework for known dictionary-based compressors. A string attractor for a word w = w[1]w[2] · · · w[n] is a subset Γ of the positions 1, . . ., n, such that all distinct factors of w have an occurrence crossing at least one of the elements of Γ. While finding the smallest string attractor for a word is a NP-complete problem, it has been proved in [Kempa and Prezza, 2018] that dictionary compressors can be interpreted as algorithms approximating the smallest string attractor for a given word. In this paper we explore the notion of string attractor from a combinatorial point of view, by focusing on several families of finite words. The results presented in the paper suggest that the notion of string attractor can be used to define new tools to investigate combinatorial properties of the words
On prefix palindromic length of automatic words
The prefix palindromic length of an infinite
word is the minimal number of concatenated palindromes needed to
express the prefix of length of . Since 2013, it is still
unknown if is unbounded for every aperiodic
infinite word , 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 has been precisely
computed is the Thue-Morse word . This word is -automatic and,
predictably, its function is -regular, but is
this the case for all automatic words?
In this paper, we prove that this function is -regular for every
-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 period-doubling word, the prefix palindromic length
does not look -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.Comment: revised version, to appear in Theoret. Comput. Sc
Generalized Thue-Morse words and palindromic richness
We prove that the generalized Thue-Morse word defined for
and as , where denotes the sum of digits in the base-
representation of the integer , has its language closed under all elements
of a group isomorphic to the dihedral group of order consisting of
morphisms and antimorphisms. Considering simultaneously antimorphisms , we show that is saturated by -palindromes
up to the highest possible level. Using the terminology generalizing the notion
of palindromic richness for more antimorphisms recently introduced by the
author and E. Pelantov\'a, we show that is -rich. We
also calculate the factor complexity of .Comment: 11 page
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