DOCSLIB.ORG

Deciding Properties of Automatic Sequences

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Deciding Properties of Automatic Sequences Deciding Properties of Automatic Sequences by Luke Schaeffer A thesis presented to the University of Waterloo in fulfillment of the thesis requirement for the degree of Master of Mathematics in Computer Science Waterloo, Ontario, Canada, 2013 c Luke Schaeffer 2013 I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii Abstract In this thesis, we show that several natural questions about automatic sequences can be expressed as logical predicates and then decided mechanically. We extend known re- sults in this area to broader classes of sequences (e.g., paperfolding words), introduce new operations that extend the space of possible queries, and show how to process the results. We begin with the fundamental concepts and problems related to automatic sequences, and the corresponding numeration systems. Building on that foundation, we discuss the general logical framework that formalizes the questions we can mechanically answer. We start with a first-order logical theory, and then extend it with additional predicates and operations. Then we explain a slightly different technique that works on a monadic second- order theory, but show that it is ultimately subsumed by an extension of the first-order theory. Next, we give two applications: critical exponent and paperfolding words. In the critical exponent example, we mechanically construct an automaton that describes a set of rational numbers related to a given automatic sequence. Then we give a polynomial- time algorithm to compute the supremum of this rational set, allowing us to compute the critical exponent and many similar quantities. In the paperfolding example, we extend our mechanical procedure to the paperfolding words, an uncountably infinite collection of infinite words. In the following chapter, we address abelian and additive problems on automatic se- quences. We give an example of a natural predicate which is provably inexpressible in our first-order theory, and discuss alternate methods for solving abelian and additive problems on automatic sequences. We close with a chapter of open problems, drawn from the earlier chapters. iii Acknowledgements I would like to thank Jeffrey Shallit for supervising my research, and for his assistance preparing this thesis. I would also like to thank the readers, Shai Ben-David and Jason Bell, for agreeing to read this thesis. Finally, I would like to thank my friends and family for their encouragement. iv Table of Contents List of Figures viii 1 Introduction1 1.1 Introduction...................................1 1.2 Words and Languages.............................2 1.3 Monoids.....................................3 1.4 Finite Automata................................5 1.4.1 ω-Languages and ω-Automata.....................9 1.4.2 Automata with Multiple Inputs.................... 11 1.5 Morphic Words................................. 13 1.6 Properties of Infinite Words.......................... 15 1.6.1 Subword Complexity.......................... 16 1.6.2 Recurrence, Appearance and Condensation.............. 16 2 Automatic Sequences 18 2.1 Numeration Systems.............................. 18 2.2 Properties of Numeration Systems....................... 20 2.2.1 ω-Numeration Systems......................... 23 2.2.2 Morphic Numeration Systems..................... 24 2.3 N -Automatic Sequences............................ 28 v 2.4 k-Regular and k-Synchronized Functions................... 31 2.4.1 k-Regular Functions.......................... 31 2.4.2 k-Synchronized Sequences....................... 32 3 Decidability in Automatic Sequences 34 3.1 Introduction................................... 34 3.2 Deciding First-Order Sentences........................ 35 3.2.1 Additional Operations......................... 39 3.2.2 Decidability using DFAs and DFAOs................. 45 3.2.3 Complexity............................... 47 3.3 Deciding Monadic Second-Order Sentences.................. 48 3.4 DFAO Application and σT ........................... 51 3.4.1 Implementation of σT .......................... 54 3.4.2 Applications of σT ........................... 62 4 Applications 67 4.1 Critical Exponent and k-Automatic Sets of Rational Numbers....... 67 4.1.1 Basic Operations on k-Automatic Rational Sets........... 71 4.1.2 Limit Points and Special Points.................... 72 4.1.3 Computing the Supremum and Largest Special Point........ 73 4.2 Decidability for Paperfolding Words...................... 81 4.2.1 First-Order Theory for Paperfolding Words.............. 83 4.2.2 Paperfolding with DFAs and DFAOs................. 88 5 Abelian and Additive Powers 91 5.1 Definition and Notation............................ 91 5.2 Inexpressibility................................. 91 5.3 Counting Symbols with Automata....................... 95 vi 5.3.1 Linear Algebra and Abelian Properties................ 98 5.3.2 First Example.............................. 99 5.3.3 Second Example............................ 103 5.3.4 Decidability for Abelian Powers.................... 109 6 Open Problems 111 6.1 Complexity Problems.............................. 111 6.2 Abelian Power Decidability.......................... 112 6.3 Automatic Rational Sets............................ 112 6.4 Shift Orbit Closure............................... 114 References 116 vii List of Figures 1.1 B¨uchi automaton accepting words with infinitely many 1s.......... 10 1.2 B¨uchi automaton accepting words with finitely many 1s........... 10 1.3 Automaton deciding if one input is a prefix of the other........... 12 2.1 Addition in base k with LSD first........................ 23 2.2 Automaton for the Thue-Morse word..................... 28 3.1 The Fibonacci word as a cutting sequence................... 53 4.1 DFAO for all paperfolding sequences...................... 83 5.1 Automaton for computing ∆t(n), given n in binary.............. 95 5.2 DFA for L.................................... 104 viii Chapter 1 Introduction 1.1 Introduction The goal of this thesis is to explain and develop a procedure for mechanically deciding questions about automatic sequences, which we state as predicates in first-order logic. After explaining the necessary background for automatic sequences, we present the basic decidability result in Theorem 3.1. This is not a new result, but we will apply it and extend it many different ways. First, we extend our logical theory with a new operation in Section 3.4. This allows us to apply a finite automaton to compare two subwords of a sequence, solving an open problem. It is also general enough to apply to other extensions in later chapters. Second, we give an algorithm (Theorem 4.20) that efficiently processes the output of our mechanical procedure, improving on an earlier result [49]. This algorithm allows us to answer questions about ratios and limits that cannot be expressed directly as predicates. The third major result (Theorem 4.24) extends our decision procedure (and the first- order logical theory) to an uncountably infinite family of sequences, called paperfolding words. This answers an open problem about paperfolding words, and gives proves a number of other interesting new properties. Finally, we address abelian and additive problems in automatic sequences, which are difficult to state as predicates. In fact, we show that a specific, natural query about abelian powers is inexpressible in Theorem 5.5. On the other hand, we show the existence of an additive cube-free word in Theorem 5.12, based on the author’s work in [15]. 1 Let us first introduce a number of basic concepts and definitions. We begin with finite words, languages, monoids, morphisms, automata and ω-automata. The remainder of this chapter covers notation and definitions which make up the foundations of the theory of automatic sequences. 1.2 Words and Languages Let Σ∗ denote the set of (finite) words over a finite alphabet Σ. Unless stated otherwise, we will use the symbols Σ, ∆ and Γ to represent finite alphabets. We write ε ∈ Σ∗ for ∗ the empty word. Let |x| denote the length of a word x ∈ Σ , and |x|a denote the number of occurrences of a symbol a ∈ Σ in the word x. Let Σω denote the set of one-sided right-infinite words over Σ. Let w ∈ Σ∗ be a word and suppose i ≥ 0 is an integer. We write w[i] to denote the symbol in w at position i. We take the first symbol in w to be at position 0, so w = w[0]w[1]w[2] ··· w[n − 1] where n = |w|.A subword of w is a contiguous block of symbols in w. We write w[i..j] for the subword w[i]w[i + 1] ··· w[j − 1]w[j]. In the case of infinite words, we allow ∞ as a position so that we may write w[i..∞] for a suffix of w. A language over Σ is a subset of Σ∗. The traditional set operations of union (A ∪ B), intersection (A ∩ B), set difference (A\B), symmetric difference (A ⊕ B) and complement (Σ∗\A) are common operations on languages A, B ⊆ Σ∗. In addition, we have the following language operations for languages A, B, C ⊆ Σ∗. • The concatenation of two languages, denoted AB, is defined to be AB := {ab : a ∈ A, b ∈ B}. Note that A(BC) = (AB)C, so concatenation is associative, and A{ε} = A = {ε}A. We define powers of a language, An, where {ε}, if n = 0; An := An−1A, if n > 0. 2 • Let A∗ be the Kleene star of A, defined by A∗ := [ An. n≥0 + S n The Kleene plus is similarly defined as A := n≥1 A . • The quotient of A by B is the language A/B := {x ∈ Σ∗ : there exists y ∈ B such that xy ∈ A}. Note that (AB)/B ⊇ A, but in general (AB)/B 6= A. Definition 1.1. We give a recursive definition for the collection of regular languages over an alphabet Σ. A language L ⊆ Σ∗ is regular if 1. L is finite, or 2. there exist regular languages A, B ⊆ Σ∗ such that L = AB, or 3. there exist regular languages A, B ⊆ Σ∗ such that L = A ∪ B, or 4. there exists a regular language A ⊆ Σ∗ such that L = A∗. In other words, close the set of finite languages under concatenation, union and Kleene star to obtain the set of regular languages.
Load more

Recommended publications
  • A Short Walk Across Combinatorics of Words
    A short walk across combinatorics of words J.-P. Allouche CNRS, IMJ-PRG, Sorbonne Universit´e 4 Place Jussieu F-75252 Paris Cedex 05 France [email protected] Abstract M. P. Sch¨utzenberger was extremely influential in theoretical com- puter science. We propose a general short survey on combinatorics of words and sequences, where we point to some of his contributions. 1 Introduction Thue is often considered as the father of combinatorics of words. In two papers at the beginning of the 20th century Thue addressed the question of finding an infinite sequence on two symbols that is cube-free, i.e., that does not contain three consecutive identical blocks. May be Thue should be considered as a grandfather of combinatorics of words, since his work was later followed and largely extended by other researchers: among the fathers of combinatorics of words, Sch¨utzenberger certainly had a special and important r^ole. Combinatorics on words deal with finite or infinite sequences taking there values in a finite set. Finite such sequences are called words. Structures are added: concatenation of words (leading to the study of finitely generated free monoids), morphisms of monoids, topology on monoids, infinite sequences viewed as limits of words and their morphisms, periodicity and periodicity-like properties of infinite sequences... We will visit some of the concepts and results which go from Thue's discovery to recent papers. To keep this survey short, we will not give all proofs but either hints and sketches or precise references. We will only give a succinct bibliography.
    [Show full text]
  • Finite Automata and Arithmetic
    Finite automata and arithmetic J.-P. Allouche ∗ 1 Introduction The notion of sequence generated by a finite automaton, (or more precisely a finite automaton with output function, i. e. a \uniform tag system") has been introduced and studied by Cobham in 1972 (see [19]; see also [24]). In 1980, Christol, Kamae, Mend`esFrance and Rauzy, ([18]), proved that a sequence with values in a finite field is automatic if and only if the related formal power series is algebraic over the rational functions with coefficients in this field: this was the starting point of numerous results linking automata theory, combinatorics and number theory. Our aim is to survey some results in this area, especially transcendence results, and to provide the reader with examples of automatic sequences. We will also give a bibliography where more detailed studies can be found. See in particular the survey of Dekking, Mend`esFrance and van der Poorten, [22], or the author's, [2], where many relations between finite automata and number theory (and between finite automata and other mathematical fields) are described. For applications of finite automata to physics see [6]. In the first part of this paper we will recall the basic definitions and give the theorem of Christol, Kamae, Mend`esFrance and Rauzy. We will also give five typical examples of sequences generated by finite automata. In the second part we will discuss transcendence results related to au- tomata theory, giving in particular some results concerning the Carlitz zeta function. We will indicate in the third part of this paper the possible generaliza- tions of these automatic sequences.
    [Show full text]
  • Some Results Around Automatic Sequences Shuo Li
    Some results around automatic sequences Shuo Li To cite this version: Shuo Li. Some results around automatic sequences. Combinatorics [math.CO]. Sorbonne Université UPMC, 2020. English. tel-02530147 HAL Id: tel-02530147 https://tel.archives-ouvertes.fr/tel-02530147 Submitted on 2 Apr 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Sorbonne Université École doctorale de sciences mathématiques de Paris centre Thèse de doctorat Discipline : Mathématiques présentée par Shuo LI Des résultats autour des suites automatiques dirigée par Jean-Paul ALLOUCHE Soutenue le 22 Janvier 2020 devant le jury composé de : M. Jean-Paul Allouche Institut de mathématiques de Jussieu-PRG directeur Mme Anna Frid Aix-Marseille Université membre invitée Mme Elise Janvresse Université de Picardie Jules Verne examinatrice Mme Sophie Morier-Genoud Institut de mathématiques de Jussieu-PRG examinatrice M. Michel Rigo Université de Liège rapporteur 2 Institut de mathématiques de Jussieu- Université Pierre et Marie Curie. Paris Rive gauche. UMR 7586. École doctorale de sciences Boîte courrier 247 mathématiques de Paris centre. 4 place Jussieu Boîte courrier 290 75 252 Paris Cedex 05 4 place Jussieu 75 252 Paris Cedex 05 Remerciements Premièrement je voudrais exprimer mes sincères remerciements à mon directeur, Jean-Paul Allouche, pour son encadrement, ses patiences et ses conseils pendant les quatre dernières années.
    [Show full text]
  • Self-Similar Groups, Automatic Sequences, and Unitriangular Representations
    Self-similar groups, automatic sequences, and unitriangular representations R. Grigorchuk, Y. Leonov, V. Nekrashevych, V. Sushchansky March 14, 2018 Contents 1 Introduction 2 2 Groups acting on rooted trees 3 2.1 Rootedtreesandtheirautomorphisms . 3 2.2 Self-similarity of Aut(X∗)...................... 4 2.3 Self-similarity bimodule . 6 n 2.4 Inductive limit of kX ........................ 10 3 Automata 14 3.1 MealyandMooreautomata . 14 3.2 Diagramsofautomata ........................ 16 3.3 Non-deterministicautomata . 17 4 Automatic matrices 19 4.1 Automaticsequences. .. .. .. .. .. .. .. .. .. .. 19 4.2 Automatic infinite matrices . 21 4.3 Representation of automata groups by automatic matrices . 24 4.4 Creation and annihilation operators . 26 4.5 Cuntz algebrasandHigman-Thompsongroups . 30 arXiv:1409.5027v1 [math.GR] 17 Sep 2014 5 Representations by uni-triangular matrices 32 5.1 Sylow p-subgroup of Aut(X∗).................... 32 ω 5.2 Polynomial bases of C(X , Fp) ................... 34 5.3 Thefirstdiagonal........................... 40 5.4 Generatingseries ........................... 42 5.5 Principalcolumns........................... 46 5.6 Uniseriality . 47 1 1 Introduction Self-similar groups is an active topic of modern group theory. They initially appeared as interesting examples of groups with unusual properties (see [Ale72, Sus79, Gri80, GS83]). The main techniques of the theory were developed for the study of these examples. Later a connection to dynamical systems was discovered (see [Nek03, Nek05]) via the notion of the iterated monodromy group. Many interesting problems were solved using self-similar groups (see [Gri98, GLSZ00,˙ BV05, BN06]). One of the ways to define self-similar groups is to say that they are groups generated by all states of an automaton (of Mealy type, also called a transducer, or sequential machine).
    [Show full text]
  • Automatic Sequences and Generalised Polynomials 3
    AUTOMATIC SEQUENCES AND GENERALISED POLYNOMIALS JAKUB BYSZEWSKI AND JAKUB KONIECZNY Abstract. We conjecture that bounded generalised polynomial functions can- not be generated by finite automata, except for the trivial case when they are ultimately periodic. Using methods from ergodic theory, we are able to partially resolve this conjecture, proving that any hypothetical counterexample is periodic away from a very sparse and structured set. In particular, we show that for a polynomial p(n) with at least one irrational coefficient (except for the constant one) and integer m ≥ 2, the sequence ⌊p(n)⌋ mod m is never automatic. We also prove that the conjecture is equivalent to the claim that the set of powers of an integer k ≥ 2 is not given by a generalised polynomial. Contents Introduction 2 Contents 5 Acknowledgements 5 1. Background 5 Notations and generalities 5 Automatic sequences 6 Dynamical systems 9 2. Density 1 results 10 Polynomial sequences 10 Generalised polynomials 11 3. Combinatorial structure of automatic sets 14 Arid sets 14 IPS sets and automatic sequences 16 4. Examplesandpropertiesofautomaticsets 17 arXiv:1705.08979v2 [math.NT] 18 Dec 2017 -free sets 17 TranslatesB of IP sets 19 Densities of symbols 20 Linear recurrence sequences 21 5. Proof of Theorem D 24 6. Concluding remarks 26 Morphic words 27 Regular sequences 27 References 27 2010 Mathematics Subject Classification. Primary: 11B85, 37A45. Secondary: 37B05, 37B10, 11J71, 11B37, 05C20. Key words and phrases. Generalised polynomials, automatic sequences, IP sets, nilmanifolds, linear recurrence sequences, regular sequences. 1 2 J.BYSZEWSKIANDJ.KONIECZNY Introduction Automatic sequences are sequences whose n-th term is produced by a finite- state machine from the base-k digits of n.
    [Show full text]
  • Hankel Determinants of the Thue-Morse Sequence Annales De L’Institut Fourier, Tome 48, No 1 (1998), P
    ANNALES DE L’INSTITUT FOURIER JEAN-PAUL ALLOUCHE JACQUES PEYRIÈRE ZHI-XIONG WEN ZHI-YING WEN Hankel determinants of the Thue-Morse sequence Annales de l’institut Fourier, tome 48, no 1 (1998), p. 1-27 <http://www.numdam.org/item?id=AIF_1998__48_1_1_0> © Annales de l’institut Fourier, 1998, tous droits réservés. L’accès aux archives de la revue « Annales de l’institut Fourier » (http://annalif.ujf-grenoble.fr/) implique l’accord avec les conditions gé- nérales d’utilisation (http://www.numdam.org/conditions). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une in- fraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Ann. Inst. Fourier, Grenoble 48, 1 (1998), 1-27 HANKEL DETERMINANTS OF THE THUE-MORSE SEQUENCE by J.-P. ALLOUCHE, J. PEYRIERE, Z.-X. WEN (*), Z.-Y. WEN (**) 0. Introduction. Let S = {0,6} be a two-letter alphabet and 5* the free monoid generated by S. Consider the endomorphism 0 defined on 5'* by 0: a i—^ a6, b ^—> ba. Since the word 0n(a) is the left half of the word ^"^(a), it has a limit as n goes to infinity: the infinite sequence c = eoci " -€n"' € {a, b}^ which is called the Thue-Morse (or sometimes the Prouhet-Thue-Morse) sequence. In this article, except in Section 4, we take a = 1, 6=0. Then the sequence e satisfies the following relations: CQ = 1, e^n = ^n, C2n+i = 1 —€n.
    [Show full text]
  • AUTOMATIC SEQUENCES and GENERALISED POLYNOMIALS Introduction Automatic Sequences Are Sequences Whose N-Th Term Is Produced by A
    AUTOMATIC SEQUENCES AND GENERALISED POLYNOMIALS JAKUB BYSZEWSKI AND JAKUB KONIECZNY Abstract. We conjecture that bounded generalised polynomial functions can- not be generated by finite automata, except for the trivial case when they are ultimately periodic. Using methods from ergodic theory, we are able to partially resolve this conjecture, proving that any hypothetical counterexample is periodic away from a very sparse and structured set. In particular, we show that for a polynomial p(n) with at least one irrational coefficient (except for the constant one) and integer m ≥ 2, the sequence bp(n)c mod m is never automatic. We also prove that the conjecture is equivalent to the claim that the set of powers of an integer k ≥ 2 is not given by a generalised polynomial. Introduction Automatic sequences are sequences whose n-th term is produced by a finite- state machine from the base-k digits of n. (A precise definition is given below.) By definition, automatic sequences can take only finitely many values. Allouche and Shallit [AS92, AS03b] have generalised the notion of automatic sequences to a wider class of regular sequences and demonstrated its ubiquity and links with multiple branches of mathematics and computer science. In a broader context, automatic sequences are closely tied to regular languages (a sequence is automatic precisely when its level sets are regular), which are at the lowest level in Chomsky's hierarchy of formal languages [Cho59]. The problem of demonstrating that a certain sequence is or is not automatic or regular has been widely studied, particularly for sequences of arithmetic origin (see, e.g., [AS92, AS03b, Bel07, SY11, MR15, SP11, Mos08, Row10]).
    [Show full text]
  • Thue, Combinatorics on Words, and Conjectures Inspired by the Thue-Morse Sequence Tome 27, No 2 (2015), P
    Jean-Paul ALLOUCHE Thue, Combinatorics on words, and conjectures inspired by the Thue-Morse sequence Tome 27, no 2 (2015), p. 375-388. <http://jtnb.cedram.org/item?id=JTNB_2015__27_2_375_0> © Société Arithmétique de Bordeaux, 2015, tous droits réservés. L’accès aux articles de la revue « Journal de Théorie des Nom- bres de Bordeaux » (http://jtnb.cedram.org/), implique l’accord avec les conditions générales d’utilisation (http://jtnb.cedram. org/legal/). Toute reproduction en tout ou partie de cet article sous quelque forme que ce soit pour tout usage autre que l’utilisation à fin strictement personnelle du copiste est constitutive d’une infrac- tion pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.cedram.org/ Journal de Théorie des Nombres de Bordeaux 27 (2015), 375–388 Thue, Combinatorics on words, and conjectures inspired by the Thue-Morse sequence par Jean-Paul ALLOUCHE On the occasion of Axel Thue’s 150th birthday Résumé. Nous décrivons quelques résultats récents sur la suite de Thue-Morse, ainsi que des questions ou conjectures, dont l’une, due à Shevelev, est résolue dans cet article. Abstract. We describe some recent results on the Thue-Morse sequence. We also list open questions and conjectures, one of which is due to Shevelev and proved in this paper. 1. Introduction The birth of combinatorics on words, i.e., the study of finite sequences (called words) taking their values in a finite set (called alphabet), aka the study of the free monoid generated by a finite set, can be dated to 1906, when the first of the two seminal papers [60, 61] of Axel Thue appeared.
    [Show full text]
  • Automatic Sequences of Rank
    Automatic Sequences of Rank Two Jason Bell Jeffrey Shallit Department of Pure Mathematics School of Computer Science University of Waterloo University of Waterloo Waterloo, ON N2L 3G1 Waterloo, ON N2L 3G1 Canada Canada [email protected] [email protected] August 13, 2021 Abstract Given a right-infinite word x over a finite alphabet A, the rank of x is the size of the smallest set S of words over A such that x can be realized as an infinite concatenation of words in S. We show that the property of having rank two is decidable for the class of k-automatic words for each integer k ≥ 2. 1 Introduction Let k ≥ 2 be an integer. In this paper we study k-automatic sequences, which are those sequences (or infinite words) (an)n≥0 over a finite alphabet, generated by a deterministic finite automaton with output (DFAO) taking, as input, the base-k representation of n and outputting an. Many interesting classical examples of sequences, including the Thue-Morse sequence, the Rudin-Shapiro sequence, and the paper-folding sequence are in this class. For more information about this well-studied class of sequences, see, for example, [1]. We arXiv:2108.05434v1 [cs.FL] 11 Aug 2021 mention that there is another well-known characterization of the k-automatic sequences, as the image, under a coding, of the fixed point of a k-uniform morphism [8]. Here a morphism is called k-uniform if the length of the image of every letter is k, and a coding is a 1-uniform morphism.
    [Show full text]
  • On Factors of Synchronized Sequences
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Theoretical Computer Science 411 (2010) 3932–3937 Contents lists available at ScienceDirect Theoretical Computer Science journal homepage: www.elsevier.com/locate/tcs On factors of synchronized sequences Arturo Carpi a,∗, Valerio D'Alonzo b a Dip. di Matematica e Informatica, Università di Perugia, Italy b Dip. di Matematica e Applicazioni, Università di Napoli ``Federico II'', Italy article info a b s t r a c t Article history: Let k ≥ 2 be an integer. A sequence of natural numbers is k-synchronized if its graph is Received 15 January 2010 represented, in base k, by a right-synchronized rational relation. We show that the factor Received in revised form 22 July 2010 complexity and the palindromic complexity of a k-synchronized sequence are k-regular Accepted 3 August 2010 sequences. We derive that the palindromic complexity of a k-automatic sequence is k- Communicated by D. Perrin automatic. ' 2010 Elsevier B.V. All rights reserved. Keywords: Synchronized sequence Automatic sequence Factor complexity Palindromic complexity 1. Introduction In the study of the combinatorial properties of infinite words, a fundamental role is played by factor complexity (see, e.g., [2] and references therein). For instance, Sturmian words are characterized as non-eventually periodic words with minimal subword complexity. Some further information on the structure of an infinite word can be obtained by counting the distinct factors of each length satisfying some given property. For instance, Allouche, Baake, Cassaigne and Damanik studied the notion of palindromic complexity [1].
    [Show full text]