Automatic sequences based on Parry or Bertrand numeration systems

We study the factor complexity and closure properties of automatic sequences based on Parry or Bertrand numeration systems. These automatic sequences can be viewed as generalizations of the more typical k-automatic sequences and Pisot-automatic sequences. We show that, like k-automatic sequences, Parry-automatic sequences have sublinear factor complexity while there exist Bertrand-automatic sequences with superlinear factor complexity. We prove that the set of Parry-automatic sequences with respect to a fixed Parry numeration system is not closed under taking images by uniform substitutions or periodic deletion of letters. These closure properties hold for k-automatic sequences and Pisot-automatic sequences, so our result shows that these properties are lost when generalizing to Parry numeration systems and beyond. Moreover, we show that a multidimensional sequence is U -automatic with respect to a positional numeration system U with regular language of numeration if and only if its U -kernel is finite

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

Automatic sequences: from rational bases to trees

The $n$th term of an automatic sequence is the output of a deterministic finite automaton fed with the representation of $n$ in a suitable numeration system. In this paper, instead of considering automatic sequences built on a numeration system with a regular numeration language, we consider these built on languages associated with trees having periodic labeled signatures and, in particular, rational base numeration systems. We obtain two main characterizations of these sequences. The first one is concerned with $r$-block substitutions where $r$ morphisms are applied periodically. In particular, we provide examples of such sequences that are not morphic. The second characterization involves the factors, or subtrees of finite height, of the tree associated with the numeration system and decorated by the terms of the sequence.Comment: 25 pages, 15 figure

Automatic sequences based on Parry or Bertrand numeration systems

We study the factor complexity and closure properties of automatic sequences based on Parry or Bertrand numeration systems. These automatic sequences can be viewed as generalizations of the more typical $k$-automatic sequences and Pisot-automatic sequences. We show that, like $k$-automatic sequences, Parry-automatic sequences have sublinear factor complexity while there exist Bertrand-automatic sequences with superlinear factor complexity. We prove that the set of Parry-automatic sequences with respect to a fixed Parry numeration system is not closed under taking images by uniform substitutions or periodic deletion of letters. These closure properties hold for $k$-automatic sequences and Pisot-automatic sequences, so our result shows that these properties are lost when generalizing to Parry numeration systems and beyond. Moreover, we show that a multidimensional sequence is $U$-automatic with respect to a positional numeration system $U$ with regular language of numeration if and only if its $U$-kernel is finite.</p

Generalization of automatic sequences for numeration systems on a regular language

Let L be an infinite regular language on a totally ordered alphabet (A,<). Feeding a finite deterministic automaton (with output) with the words of L enumerated lexicographically with respect to < leads to an infinite sequence over the output alphabet of the automaton. This process generalizes the concept of k-automatic sequence for abstract numeration systems on a regular language (instead of systems in base k). Here, I study the first properties of these sequences and their relations with numeration systems.Comment: 10 pages, 3 figure

Automatic sequences in rational base numeration systems (and even more)

The nth term of an automatic sequence is the output of a deterministic finite automaton fed with the representation of n in a suitable numeration system. Here, instead of considering automatic sequences built on a numeration system with a regular numeration language, we consider these built on languages associated with trees having periodic labeled signatures and, in particular, rational base numeration systems. We obtain two main characterizations of these sequences. The first one is concerned with r-block substitutions where r morphisms are applied periodically. In particular, we provide examples of such sequences that are not morphic. The second characterization involves the factors, or subtrees of finite height, of the tree associated with the numeration system and decorated by the terms of the sequence

Positional Numeration Systems: Ultimate Periodicity, Complexity and Automatic Sequences

This dissertation thesis is made up of three distinct parts, connected especially by complexity notion, factorial complexity as well as state complexity. We study positional numeration systems and recognizable sets through decision problems and automatic sequences. The first part is devoted to the following problem: given a numeration system U and a finite automaton accepting U-representations of a set X ⊆ N, can we decide whether the set X is ultimately periodic (i.e. a finite union of arithmetic progressions)? We prove that this problem is decidable for a large class of numeration systems based on linear recurrent sequences. Thanks to the given automaton, we bound the possible periods of X via an arithmetical study of the linear recurrent sequence, as well as p-adic methods. The second part is dealing with the set of non-negative integers whose base-2 representation contains an even number of 1, called the Thue-Morse set and denoted by T. We study of the minimal automaton of the base-2^p expansions of sets of the form mT+r, where m and p are positive integers and r a remainder between 0 and m−1. In particular, we give the state complexity of such sets. The proposed method is constructive and general for any b-recognizable set of integers. As an application, we get a procedure to decide whether a 2^p-recognizable set given via an automaton is a set of the form mT+r. Finally, in the third part, we study properties of automatic sequences based on Parry and Bertrand numeration systems. We show that Parry-automatic sequences, like Pisot-automatic sequences (and thus in particular like b-automatic sequences) have a sublinear factor complexity. Furthermore, we exhibit a Bertrand-automatic sequence whose factor complexity is quadratic. We also prove that, contrarily to Pisot-automatic sequences, the image of a Parry-automatic sequence under a uniform morphism is not always a Parry-automatic sequence. The same happens for periodic deletion of letters. Last, we give the generalization to multidimensional sequences of a well-known result: a sequence is U-automatic if and only if its U-kernel is finite, U being such that the numeration language is regular

Multidimensional Generalized Automatic Sequences and Shape-symmetric Morphic Words

An infinite word is S-automatic if, for all n>=0, its (n + 1)st letter is the output of a deterministic automaton fed with the representation of n in the considered numeration system S. In this extended abstract, we consider an analogous definition in a multidimensional setting and present the connection to the shape-symmetric infinite words introduced by Arnaud Maes. More precisely, for d>=2, we state that a multidimensional infinite word x : N^d \to \Sigma over a finite alphabet \Sigma is S-automatic for some abstract numeration system S built on a regular language containing the empty word if and only if x is the image by a coding of a shape-symmetric infinite word