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.