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On Infinite Prefix Normal Words [article]

Ferdinando Cicalese, Zsuzsanna Lipták, Massimiliano Rossi
2018 arXiv   pre-print
In this paper, we study infinite prefix normal words and explore their relationship to some known classes of infinite binary words.  ...  Prefix normal words are binary words that have no factor with more 1s than the prefix of the same length. Finite prefix normal words were introduced in [Fici and Lipták, DLT 2011].  ...  We wish to thank the participants of the Workshop on Words and Complexity (Lyon, February 2018), for interesting discussions and pointers, and to Péter Burcsi, who first got us interested in Sturmian words  ... 
arXiv:1811.06273v1 fatcat:2ojyohhi2raaljuymutx3fkotu

Bubble-Flip—A new generation algorithm for prefix normal words

Ferdinando Cicalese, Zsuzsanna Lipták, Massimiliano Rossi
2018 Theoretical Computer Science  
We introduce infinite prefix normal words and show that one of the operations used by the algorithm, if applied repeatedly to extend the string, produces an ultimately periodic infinite word, which is  ...  The new algorithm uses two operations on binary strings, which exploit certain properties of prefix normal words in a smart way.  ...  On finite and infinite prefix normal words In this section, we study infinite prefix normal words.  ... 
doi:10.1016/j.tcs.2018.06.021 fatcat:3cpshypcwraklozijkfx4w24j4

Quasiperiodic and Lyndon episturmian words

Amy Glen, Florence Levé, Gwénaël Richomme
2008 Theoretical Computer Science  
Our key tools are morphisms and directive words, in particular "normalized" directive words, which we introduced in an earlier paper.  ...  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.  ...  Krieger who suggested to the third author the notion of ultimate quasiperiodicity and observed the fact that all Sturmian words are ultimately quasiperiodic.  ... 
doi:10.1016/j.tcs.2008.09.056 fatcat:zqhu5um45fbkbgvyfkfwkrir2m

On periodicity of generalized pseudostandard words [article]

Josef Florian, Lubomira Dvorakova
2015 arXiv   pre-print
More precisely, we describe how the directive bi-sequence of a generalized pseudostandard word has to look like in order to correspond to a periodic word.  ...  Generalized pseudostandard words were introduced by de Luca and De Luca in 2006.  ...  Introduction This paper focuses on a recent topic of combinatorics on words: generalized pseudostandard words.  ... 
arXiv:1508.02020v1 fatcat:fv6wcghlabazxarisgi6d2f3ny

On Periodicity of Generalized Pseudostandard Words

Josef Florian, L'ubomíra Dvořáková
2016 Electronic Journal of Combinatorics  
More precisely, we describe how the directive bi-sequence of a generalized pseudostandard word has to look like in order to correspond to a periodic word.  ...  Generalized pseudostandard words were introduced by de Luca and De Luca in 2006.  ...  Introduction This paper focuses on a recent topic of combinatorics on words: generalized pseudostandard words.  ... 
doi:10.37236/4771 fatcat:lnpix5vjpfbaleftsr5uwktaky

Abelian complexity and abelian co-decomposition

Ondřej Turek
2013 Theoretical Computer Science  
We propose a technique for exploring the abelian complexity of recurrent infinite words, focusing particularly on infinite words associated with Parry numbers.  ...  Using that technique, we give the affirmative answer to the open question posed by Richomme, Saari and Zamboni, whether the abelian complexity of the Tribonacci word attains each value in {4,5,6} infinitely  ...  Acknowledgements The author thanks Karel Břinda for a numerical experiment and Edita Pelantová for many valuable comments on the manuscript.  ... 
doi:10.1016/j.tcs.2012.10.034 fatcat:xk6mqhtwkbbfzcqcujsgkbr7xa

Abelian complexity of infinite words

Karel Břinda, Edita Pelantová, Ondřej Turek
2018 Zenodo  
In addition, a new toolkit TigrWords, which facilitates study of infinite words approximated by their prefixes and may serve to discover of new hypotheses, is described.  ...  This thesis deals with Abelian complexity of infinite words, i.e., function describing complexity of a given infinite word using sets of Parikh vectors of finite factors of the word.  ...  Acknowledgments I thank all members of TIGR (Theoretical Informatics Group) at FNSPE for bringing me into the amazing world of combinatorics on words.  ... 
doi:10.5281/zenodo.2112128 fatcat:j4gxptknm5dkbg5d65ccjnvmfu

Fixed points of morphisms among binary generalized pseudostandard words [article]

Lubomira Dvorakova, Tereza Velka
2017 arXiv   pre-print
We conjecture that this class contains all fixed points of primitive morphisms among aperiodic binary generalized pseudostandard words that are not standard Sturmian words.  ...  We introduce a class of fixed points of primitive morphisms among aperiodic binary generalized pseudostandard words.  ...  If a finite word w = pvs, where p, v, s ∈ {0, 1} * , then p is called a prefix of w and s is called a suffix of w. An infinite word u over {0, 1} is any binary infinite sequence.  ... 
arXiv:1701.04472v1 fatcat:uryio7jvhfgynhqdsheyzcazky

Unavoidable Set: Extension and Reduction

Phan Trung Huy, Nguyen Huong Lam
1999 RAIRO - Theoretical Informatics and Applications  
As u,v j£ e, consider the periodic bi-infinite word "vxv? -w u w = "v^, in which the subwords having an occurence of x as prefix and the next one as sufnx are ail the same s.  ...  Since every occurence of x in this word is invariably followed by the infinité word u w we get that t is a prefix of every u^.  ... 
doi:10.1051/ita:1999114 fatcat:w6df7dsiqbachhf4uo3yvbjcme

Abelian complexity of infinite words associated with quadratic Parry numbers

L'ubomíra Balková, Karel Břinda, Ondřej Turek
2011 Theoretical Computer Science  
We derive an explicit formula for the Abelian complexity of infinite words associated with quadratic Parry numbers.  ...  Our method consisted in the construction of infinite words v and w whose prefixes are factors of the infinite word in question and such that the prefixes of v are the richest in A's (among all factors  ...  Moreover, these cases are the only ones among all Parry numbers β (not only the quadratic ones) for which the infinite word u β is Sturmian, see [10] .  ... 
doi:10.1016/j.tcs.2011.08.016 fatcat:b6nw4jvx5fefzcstlmnnaaccsq

Determining Sets of Quasiperiods of Infinite Words

Guilhem Gamard, Gwenaël Richomme, Marc Herbstritt
2016 International Symposium on Mathematical Foundations of Computer Science  
Assume by contradiction that n is the least integer such that ψ n (q) is quasiperiodic, and let Q be one of its quasiperiods. Necessarily n ≥ 1. The word ψ(a)ba is a prefix of ψ n (q).  ...  Based on links between quasiperiods, right special factors and square factors, we introduce a method to determine the set of quasiperiods of a given right infinite word.  ...  This decomposition is called the left normal form (resp. right normal form) of the word. As observed by Marcus and Monteil [19] this result extends naturally to infinite words.  ... 
doi:10.4230/lipics.mfcs.2016.40 dblp:conf/mfcs/GamardR16 fatcat:viporj3zyfg7fisnhywyzedmfy

On Periodicity and Complexity of Generalized Pseudostandard Words [article]

Josef Florian, Lubomira Balkova
2016 arXiv   pre-print
Vuillon, On Generalized Pseudostandard Words over Binary Alphabet (Journal of Int.  ...  We present two new results concerning these words. The first one is a necessary and sufficient condition for their periodicity.  ...  Introduction This paper focuses on an actual topic of combinatorics on words: generalized pseudostandard words.  ... 
arXiv:1408.5210v2 fatcat:vzlfsldlbrevji5ycicye3ehp4

On Selection Functions that Do Not Preserve Normality [chapter]

Wolfgang Merkle, Jan Reimann
2003 Lecture Notes in Computer Science  
In fact, for both types of languages it is possible to select a constant sequence from a normal one.  ...  We show that there are such languages that are only slightly more complicated than regular ones, namely, normality is preserved neither by deterministic one-counter languages nor by linear languages.  ...  There is a normal sequence N and a deterministic one-counter language L such that the sequence selected from N by L is infinite and constant. Proof.  ... 
doi:10.1007/978-3-540-45138-9_54 fatcat:2gvcy2ifqvcfdkmqmbhunjtdr4

Thompson's group F is 1-counter graph automatic

Murray Elder, Jennifer Taback
2016 Groups - Complexity - Cryptology  
We consider two cases, depending on whether wx − is already in infinite normal form, or must be simplified in order to obtain the infinite normal form.  ...  . (3.5) We again consider subcases, depending on whether or not the expression in (3.5) is the infinite normal form for gx − .  ... 
doi:10.1515/gcc-2016-0001 fatcat:hwnibopn5jajdom3iun4kmpp5i

A new estimate on complexity of binary generalized pseudostandard words [article]

Lubomira Dvorakova, Josef Florian
2016 arXiv   pre-print
Generalized pseudostandard words were introduced by de Luca and De Luca in 2006.  ...  In comparison to the palindromic and pseudopalindromic closure, only little is known about the generalized pseudopalindromic closure and the associated generalized pseudostandard words.  ...  We would like to thank Edita Pelantová for her useful comments and advice concerning in particular complexity of infinite words.  ... 
arXiv:1611.05482v1 fatcat:c7ahkjz5ynhk7e4zhx4fzrld6q
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