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Marie-Pierre Béal

Publications and source records attributed to Marie-Pierre Béal.

15 recordsLinked to original sources
arXiv2026

Sofic measures

Sofic measures, also known as hidden Markov measures, have been extensively studied. In this paper, we survey some equivalent definitions of this notion and improve a bound for deciding whether a sofic measure is~$k$-step Markov. We prove that if an invariant sofic measure with a linear representation of dimension~$n$ is a~$k$-step Markov chain, then~$k$ can be chosen at most equal to~$\binom{n}{2}$ and that this bound is tight for sofic measures over an alphabet of size $3$.

math.PR↗
arXiv2026

Checking and producing word attractors

The article focuses on word (or string) attractors, which are sets of positions related to the text compression efficiency of the underlying word. The article presents two combinatorial algorithms based on Suffix automata or Directed Acyclic Word Graphs. The first algorithm decides in linear time whether a set of positions on the word is an attractor of the word. The second algorithm generates an attractor for a given word in a greedy manner. Although the problem of generating a smallest attractor is NP-hard, {the algorithm is efficient and produces very small attractors and even the smallest ones for several well-known infinite families of words.

cs.DS↗
arXiv2025

One-sided Hom shifts

We prove that it is decidable whether a one-sided shift of finite type is conjugate to a one-sided Hom-shift, and whether a tree-shift of finite type is conjugate to a Hom tree-shift. The proof uses Williams's theory for one-sided shifts

cs.FL↗
arXiv2024

Decidable problems in substitution shifts

In this paper, we investigate the structure of the most general kind of substitution shifts, including non-minimal ones, and allowing erasing morphisms. We prove the decidability of many properties of these morphisms with respect to the shift space generated by iteration, such as aperiodicity, recognizability and (under an additional assumption) irreducibility, or minimality.

math.DS↗
arXiv2023

Recognizability in S-adic shifts

We investigate questions related to the notion of recognizability of sequences of morphisms, a generalization of Moss{é}'s Theorem. We consider the most general class of morphisms including ones with erasable letters. The main result states that a sequence of morphisms with finite alphabet rank is eventually recognizable for aperiodic points, improving and simplifying a result of Berth{é} et al. (2019). This also provides a new simple proof for the recognizability of a single morphism on its shift space. The main ingredient of the proof are elementary morphisms.

cs.FL↗
arXiv2023

Fast detection of specific fragments against a set of sequences

We design alignment-free techniques for comparing a sequence or word, called a target, against a set of words, called a reference. A target-specific factor of a target $T$ against a reference $R$ is a factor $w$ of a word in $T$ which is not a factor of a word of $R$ and such that any proper factor of $w$ is a factor of a word of $R$. We first address the computation of the set of target-specific factors of a target $T$ against a reference $R$, where $T$ and $R$ are finite sets of sequences. The result is the construction of an automaton accepting the set of all considered target-specific factors. The construction algorithm runs in linear time according to the size of $T\cup R$. The second result consists of the design of an algorithm to compute all the occurrences in a single sequence $T$ of its target-specific factors against a reference $R$. The algorithm runs in real-time on the target sequence, independently of the number of occurrences of target-specific factors.

cs.DS↗
arXiv2022

Recognizability of morphisms

We investigate several questions related to the notion of recognizable morphism. The main result is a new proof of Mossé's theorem and actually of a generalization to non primitive morphisms due to Berthé et al. We actually prove the result of Berthé et al. for the most general class of morphisms, including ones with erasable letters. It is derived from a result concerning elementary morphisms for which we also provide a new proof. We also show how to decide whether an injective morphism is recognizable on the full shift for aperiodic points.

math.DS↗
arXiv2022

Checking whether a word is Hamming-isometric in linear time

A finite word $f$ is Hamming-isometric if for any two word $u$ and $v$ of same length avoiding $f$, $u$ can be transformed into $v$ by changing one by one all the letters on which $u$ differs from $v$, in such a way that all of the new words obtained in this process also avoid~$f$. Words which are not Hamming-isometric have been characterized as words having a border with two mismatches. We derive from this characterization a linear-time algorithm to check whether a word is Hamming-isometric. It is based on pattern matching algorithms with $k$ mismatches. Lee-isometric words over a four-letter alphabet have been characterized as words having a border with two Lee-errors. We derive from this characterization a linear-time algorithm to check whether a word over an alphabet of size four is Lee-isometric.

cs.DS↗
arXiv2022

Unambiguously coded shifts

We study the coded systems introduced by Blanchard and Hansel. We give several constructions which allow one to represent a coded system as a strongly unambiguous one.

math.DS↗
arXiv2017

$\mathbb{N}$-algebraicity of zeta functions of sofic-Dyck shifts

We prove that the multivariate zeta function of a sofic-Dyck shift is the commutative series of some visibly pushdown language. As a consequence the zeta function of a sofic-Dyck shift is the generating function of a visibly pushdown language and is thus an $\mathbb{N}$-algebraic series.

cs.FL↗
arXiv2015

Sofic-Dyck shifts

We define the class of sofic-Dyck shifts which extends the class of Markov-Dyck shifts introduced by Inoue, Krieger and Matsumoto. Sofic-Dyck shifts are shifts of sequences whose finite factors form unambiguous context-free languages. We show that they correspond exactly to the class of shifts of sequences whose sets of factors are visibly pushdown languages. We give an expression of the zeta function of a sofic-Dyck shift.

cs.FL↗
arXiv2013

Finite-type-Dyck shift spaces

We study some basic properties of sofic-Dyck shifts and finite-type-Dyck shifts. We prove that the class of sofic-Dyck shifts is stable under proper conjugacies. We prove a Decomposition Theorem of a proper conjugacy between edge-Dyck shifts into a sequence of Dyck splittings and amalgamations.

cs.FL↗
arXiv2013

A quadratic algorithm for road coloring

The Road Coloring Theorem states that every aperiodic directed graph with constant out-degree has a synchronized coloring. This theorem had been conjectured during many years as the Road Coloring Problem before being settled by A. Trahtman. Trahtman's proof leads to an algorithm that finds a synchronized labeling with a cubic worst-case time complexity. We show a variant of his construction with a worst-case complexity which is quadratic in time and linear in space. We also extend the Road Coloring Theorem to the periodic case.

cs.DS↗
arXiv2011

Symbolic dynamics

This chapter presents some of the links between automata theory and symbolic dynamics. The emphasis is on two particular points. The first one is the interplay between some particular classes of automata, such as local automata and results on embeddings of shifts of finite type. The second one is the connection between syntactic semigroups and the classification of sofic shifts up to conjugacy.

cs.FL↗