Remarks on Two Nonstandard Versions of Periodicity in Words (original) (raw)
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Periodic-like words, periodicity, and boxes
Acta Informatica, 2001
We introduce the notion of periodic-like word. It is a word whose longest repeated prefix is not right special. Some different characterizations of this concept are given. In particular, we show that a word w is periodic-like if and only if it has a period not larger than |w| − R w , where R w is the least non-negative integer such that any prefix of w of length ≥ R w is not right special. We derive that if a word w has two periods p, q ≤ |w| − R w , then also the greatest common divisor of p and q is a period of w. This result is, in fact, an improvement of the theorem of Fine and Wilf. We also prove that the minimal period of a word w is equal to the sum of the minimal periods of its components in a suitable canonical decomposition in periodic-like subwords. Moreover, we characterize periodic-like words having the same set of proper boxes, in terms of the important notion of root-conjugacy. Finally, some new uniqueness conditions for words, related to the maximal box theorem are given.
Computers & Mathematics with Applications, 2004
Made available courtesy of Elsevier: http://www.elsevier.com ***Reprinted with permission. No further reproduction is authorized without written permission from Elsevier. This version of the document is not the version of record. Figures and/or pictures may be missing from this format of the document.*** Abstract: A partial word of length n over a finite alphabet A is a partial map from {0, … , n-1} into A. Elements of {0, … , n-1} without image are called holes (a word is just a partial word without holes). A fundamental periodicity result on words due to Fine and Wilf [1] intuitively determines how far two periodic events have to match in order to guarantee a common period. This result was extended to partial words with one hole by Berstel and Boasson [2] and to partial words with two or three holes by Blanchet-Sadri and Hegstrom [3]. In this paper, we give an extension to partial words with an arbitrary number of holes.
Periodicity properties on partial words
Information and Computation, 2008
The concept of periodicity has played over the years a centra1 role in the development of combinatorics on words and has been a highly valuable too1 for the design and analysis of algorithms. Fine and Wilf's famous periodicity result, which is one of the most used and known results on words, has extensions to partia1 words, or sequences that may have a number of "do not know" symbols. These extensions fal1 into two categories: the ones that relate to strong periodicity and the ones that relate to weak periodicity. In this paper, we obtain consequences by generalizing, in particular, the combinatoria1 property that "for any word u over {a, b}, ua or ub is primitive," which proves in some sense that there exist very many primitive partia1 words.
Some Periodicity of Words and Marcus Contextual Grammars
数理解析研究所講究録, 2006
In this paper, first we define aperiodic (semi-periodic, quasi-periodic) word and then we define a primitive (strongly primitive, hyper primitive) word. After we define several Marcus contextual grammars, we show that the set of all primitive (strongly primitive, hyper primitive) words can be generated by some Marcus contextual grammar.
Quasiperiodic and Lyndon episturmian words
Theoretical Computer Science, 2008
Recently the second two authors characterized quasiperiodic Sturmian words, proving that a Sturmian word is non-quasiperiodic if and only if it is an infinite Lyndon word. Here we extend this study to episturmian words (a natural generalization of Sturmian words) by describing all the quasiperiods of an episturmian word, which yields a characterization of quasiperiodic episturmian words in terms of their directive words. Even further, we establish a complete characterization of all episturmian words that are Lyndon words. Our main results show that, unlike the Sturmian case, there is a much wider class of episturmian words that are non-quasiperiodic, besides those that are infinite Lyndon words. Our key tools are morphisms and directive words, in particular normalized directive words, which we introduced in an earlier paper. Also of importance is the use of return words to characterize quasiperiodic episturmian words, since such a method could be useful in other contexts.
Quasiperiodicity and string covering
Theoretical Computer Science, 1999
In this paper, we study word regularities and in particular extensions of the notion of the word period: quasiperiodicity, covers and seeds. We present overviews of algorithms for computing the quasiperiodicity, the covers and the seeds of a given word. We also present an overview of an algorithm that finds maximal word factors with the above regularities. Finally, we show how Fine and Wilf's Theorem fails if we try to extend it directly to quasiperiodicity, as well as a new property on concatenation of periodic words.
Fine and Wilf 's Periodicity on Partial Words and Consequences
The concept of periodicity has played over the years a central role in the development of combinatorics on words and has been a highly valuable tool for the design and analysis of algorithms. There are many fundamental periodicity results on words. Among them is the famous result of Fine and Wilf which intuitively determines how far two periodic events have to match in order to guarantee a common period. This result states that for positive integers p and q, if the word u has periods p and q and the length of u is not less than p + q − gcd(p, q), then u has also period gcd(p, q). Fine and Wilf's result, which is one of the most used and known results on words, has extensions to partial words, or sequences that may have a number of "do not know" symbols. These extensions fall into two categories: The ones that relate to strong periodicity and the ones that relate to weak periodicity. In this paper, we study some consequences of these results.
Lecture Notes in Computer Science, 2009
In this paper we investigate the periodic structure of rich words (i.e., words having the highest possible number of palindromic factors), giving new results relating them with periodic-like words. In particular, some new characterizations of rich words and rich palindromes are given. We also prove that a periodic-like word is rich if and only if the square of its fractional root is also rich.
A periodicity result of partial words with one hole
Computers & Mathematics with Applications, 2003
The study of the combinatorial properties of strings of symbols from a finite alphabet, also referred to as words, is profoundly connected to numerous fields such as biology, computer science, All rights reserved.