Related papers: Enumeration and Decidable Properties of Automatic …
In this paper we consider a number of natural decision problems involving k-regular sequences. Specifically, they arise from - lower and upper bounds on growth rate; in particular boundedness, - images, - regularity (recognizability by a…
The notion of a k-automatic set of integers is well-studied. We develop a new notion - the k-automatic set of rational numbers - and prove basic properties of these sets, including closure properties and decidability.
We investigate questions related to the presence of primitive words and Lyndon words in automatic and linearly recurrent sequences. We show that the Lyndon factorization of a k-automatic sequence is itself k-automatic. We also show that the…
We illustrate a general technique for enumerating factors of k-automatic sequences by proving a conjecture on the number f(n) of unbordered factors of the Thue-Morse sequence. We show that f(n) <= n for n >= 4 and that f(n) = n infinitely…
We prove that the property of being closed (resp., palindromic, rich, privileged trapezoidal, balanced) is expressible in first-order logic for automatic (and some related) sequences. It therefore follows that the characteristic function of…
Using a novel rewriting problem, we show that several natural decision problems about finite automata are undecidable (i.e., recursively unsolvable). In contrast, we also prove three related problems are decidable. We apply one result to…
We study automatic sequences and automatic systems generated by general constant length (nonprimitive) substitutions. While an automatic system is typically uncountable, the set of automatic sequences is countable, implying that most…
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…
We provide a complete characterisation of automaticity of uniformly recurrent substitutive sequences in terms of the incidence matrix of the return substitution of the underlying purely substitutive sequence. This resolves a recent question…
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…
Let $k\ge 2$. We prove that the characteristic sequence of a regular language over a $k$-letter alphabet is $k$-automatic. More generally, if $t\ge 2$ and $t,k$ are multiplicatively dependent, we show that the characteristic sequence of a…
We prove decidability results on the existence of constant subsequences of uniformly recurrent morphic sequences along arithmetic progressions. We use spectral properties of the subshifts they generate to give a first algorithm deciding…
We study the notion of an asymptotically automatic sequence, which generalises the notion of an automatic sequence. While $k$-automatic sequences are characterised by finiteness of $k$-kernels, the $k$-kernels of asymptotically…
Abstract numeration systems encode natural numbers using radix ordered words of an infinite regular language and linear recurrence sequences play a key role in their valuation. Sequence automata, which are deterministic finite automata with…
Morphic sequences form a natural class of infinite sequences, extending the well-studied class of automatic sequences. Where automatic sequences are known to have several equivalent characterizations and the class of automatic sequences is…
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…
The critical exponent of an infinite word is defined to be the supremum of the exponent of each of its factors. For k-automatic sequences, we show that this critical exponent is always either a rational number or infinite, and its value is…
Automatic sequences have many properties that other sequences (in particular, non-uniformly morphic sequences) do not necessarily share. In this paper we survey a number of different methods that can be used to prove that a given 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…
It is well-known that there exist infinite sequences that are the fixed point of non-uniform morphisms, but not $k$-automatic for any $k$. In this note we show that every $k$-automatic sequence is the image of a fixed point of a {\it…