Extremal words in morphic subshifts

Given an infinite word X over an alphabet A a letter b occurring in X, and a total order \sigma on A, we call the smallest word with respect to \sigma starting with b in the shift orbit closure of X an extremal word of X. In this paper we consider the extremal words of morphic words. If X = g(f^{\omega}(a)) for some morphisms f and g, we give two simple conditions on f and g that guarantees that all extremal words are morphic. This happens, in particular, when X is a primitive morphic or a binary pure morphic word. Our techniques provide characterizations of the extremal words of the Period-doubling word and the Chacon word and give a new proof of the form of the lexicographically least word in the shift orbit closure of the Rudin-Shapiro word.Comment: Replaces a previous version entitled "Extremal words in the shift orbit closure of a morphic sequence" with an added result on primitive morphic sequences. Submitte

Periodicity, repetitions, and orbits of an automatic sequence

We revisit a technique of S. Lehr on automata and use it to prove old and new results in a simple way. We give a very simple proof of the 1986 theorem of Honkala that it is decidable whether a given k-automatic sequence is ultimately periodic. We prove that it is decidable whether a given k-automatic sequence is overlap-free (or squareefree, or cubefree, etc.) We prove that the lexicographically least sequence in the orbit closure of a k-automatic sequence is k-automatic, and use this last result to show that several related quantities, such as the critical exponent, irrationality measure, and recurrence quotient for Sturmian words with slope alpha, have automatic continued fraction expansions if alpha does.Comment: preliminary versio

Enumeration and Decidable Properties of Automatic Sequences

We show that various aspects of k-automatic sequences -- such as having an unbordered factor of length n -- are both decidable and effectively enumerable. As a consequence it follows that many related sequences are either k-automatic or k-regular. These include many sequences previously studied in the literature, such as the recurrence function, the appearance function, and the repetitivity index. We also give some new characterizations of the class of k-regular sequences. Many results extend to other sequences defined in terms of Pisot numeration systems

Deciding Properties of Automatic Sequences

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 results 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 sequences. 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

Continuous Spectra For Substitution-Based Sequences

This thesis is chiefly concerned with the continuous spectra of substitution-based sequences. First, motivated by a question of Lafrance, Yee and Rampersad [34], we establish a connection between the ‘root-N’ property and the corresponding sequences that satisfy it having absolutely continuous spectrum. Then we use the recent advances in Bartlett [10, 11] to show that the Rudin–Shapiro-like sequence has singular continuous spectrum, hence does not satisfy the root-N property. This gives a negative answer to the question raised by the authors in [34]. Secondly, we use the connection we establish between the root-N property and absolute continuity to create more substitution-based sequences that have absolutely continuous/Lebesgue spectrum. This is done by modifying Rudin’s original construction [44]. We show that the binary sequences (±1 sequences) from our modification also satisfy the root-N property and they are mutually locally derivable to the corresponding substitution sequences. This shows that the spectral properties of the substitution-based sequences are inherited from their binary counterpart. Finally, we generalise our construction using Fourier matrices. This leads to extending Rudin’s construction to sequences with complex coefficients. This approach allows us to generate substitution sequences of any constant length greater than or equal to two. We show explicitly in the length 3 and 4 cases that these systems exhibit Lebesgue spectrum, employing Bartlett’s algorithm from Chapter 3 and mutual local derivability

Automatic winning shifts

To each one-dimensional subshift $X$, we may associate a winning shift $W(X)$ which arises from a combinatorial game played on the language of $X$. Previously it has been studied what properties of $X$ does $W(X)$ inherit. For example, $X$ and $W(X)$ have the same factor complexity and if $X$ is a sofic subshift, then $W(X)$ is also sofic. In this paper, we develop a notion of automaticity for $W(X)$, that is, we propose what it means that a vector representation of $W(X)$ is accepted by a finite automaton. Let $S$ be an abstract numeration system such that addition with respect to $S$ is a rational relation. Let $X$ be a subshift generated by an $S$-automatic word. We prove that as long as there is a bound on the number of nonzero symbols in configurations of $W(X)$ (which follows from $X$ having sublinear factor complexity), then $W(X)$ is accepted by a finite automaton, which can be effectively constructed from the description of $X$. We provide an explicit automaton when $X$ is generated by certain automatic words such as the Thue-Morse word.Comment: 28 pages, 5 figures, 1 tabl

Automatic Sequences and Decidable Properties: Implementation and Applications

In 1912 Axel Thue sparked the study of combinatorics on words when he showed that the Thue-Morse sequence contains no overlaps, that is, factors of the form ayaya. Since then many interesting properties of sequences began to be discovered and studied. In this thesis, we consider a class of infinite sequences generated by automata, called the k-automatic sequences. In particular, we present a logical theory in which many properties of k-automatic sequences can be expressed as predicates and we show that such predicates are decidable. Our main contribution is the implementation of a theorem prover capable of practically characterizing many commonly sought-after properties of k-automatic sequences. We showcase a panoply of results achieved using our method. We give new explicit descriptions of the recurrence and appearance functions of a list of well-known k-automatic sequences. We define a related function, called the condensation function, and give explicit descriptions for it as well. We re-affirm known results on the critical exponent of some sequences and determine it for others where it was previously unknown. On the more theoretical side, we show that the subword complexity p(n) of k-automatic sequences is k-synchronized, i.e., the language of pairs (n, p(n)) (expressed in base k) is accepted by an automaton. Furthermore, we prove that the Lyndon factorization of k-automatic sequences is also k-automatic and explicitly compute the factorization for several sequences. Finally, we show that while the number of unbordered factors of length n is not k-synchronized, it is k-regular