36 research outputs found
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
Numeration systems on a regular language: Arithmetic operations, Recognizability and Formal power series
Generalizations of numeration systems in which N is recognizable by a finite
automaton are obtained by describing a lexicographically ordered infinite
regular language L over a finite alphabet A. For these systems, we obtain a
characterization of recognizable sets of integers in terms of rational formal
series. We also show that, if the complexity of L is Theta (n^q) (resp. if L is
the complement of a polynomial language), then multiplication by an integer k
preserves recognizability only if k=t^{q+1} (resp. if k is not a power of the
cardinality of A) for some integer t. Finally, we obtain sufficient conditions
for the notions of recognizability and U-recognizability to be equivalent,
where U is some positional numeration system related to a sequence of integers.Comment: 34 pages; corrected typos, two sections concerning exponential case
and relation with positional systems adde
Abstract numeration systems on bounded languages and multiplication by a constant
A set of integers is -recognizable in an abstract numeration system if
the language made up of the representations of its elements is accepted by a
finite automaton. For abstract numeration systems built over bounded languages
with at least three letters, we show that multiplication by an integer
does not preserve -recognizability, meaning that there always
exists a -recognizable set such that is not
-recognizable. The main tool is a bijection between the representation of an
integer over a bounded language and its decomposition as a sum of binomial
coefficients with certain properties, the so-called combinatorial numeration
system
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
Structural properties of bounded languages with respect to multiplication by a constant
peer reviewedWe consider the preservation of recognizability of a set of integers after multiplication by a constant for numeration systems built over a bounded language. As a corollary we show that any nonnegative integer can be written as a sum of binomial coefficients with some prescribed properties
Breadth-first serialisation of trees and rational languages
We present here the notion of breadth-first signature and its relationship
with numeration system theory. It is the serialisation into an infinite word of
an ordered infinite tree of finite degree. We study which class of languages
corresponds to which class of words and,more specifically, using a known
construction from numeration system theory, we prove that the signature of
rational languages are substitutive sequences.Comment: 15 page