24 research outputs found
Algebraic Characterization of FO for Scattered Linear Orderings
We prove that for the class of sets of words indexed by countable scattered linear orderings, there is an equivalence between definability in first-order logic, star-free expressions with marked product, and recognizability by finite aperiodic semigroups which satisfy some additional equation
Eilenberg Theorems for Free
Eilenberg-type correspondences, relating varieties of languages (e.g. of
finite words, infinite words, or trees) to pseudovarieties of finite algebras,
form the backbone of algebraic language theory. Numerous such correspondences
are known in the literature. We demonstrate that they all arise from the same
recipe: one models languages and the algebras recognizing them by monads on an
algebraic category, and applies a Stone-type duality. Our main contribution is
a variety theorem that covers e.g. Wilke's and Pin's work on
-languages, the variety theorem for cost functions of Daviaud,
Kuperberg, and Pin, and unifies the two previous categorical approaches of
Boja\'nczyk and of Ad\'amek et al. In addition we derive a number of new
results, including an extension of the local variety theorem of Gehrke,
Grigorieff, and Pin from finite to infinite words
Games, equations and dot-depth two monoids
Given any finite alphabet A and positive integers m1, …, mk, congruences on A*, denoted by ~(m1, …, mk) and related to a version of the Ehrenfeucht-Fraisse game, are defined. Level k of the Straubing hierarchy of aperiodic monoids can be characterized in terms of the monoids A*/~(m1, … mk). A natural subhierarchy of level 2 and equation systems satisfied in the corresponding varieties of monoids are defined. For A = 2, a necessary and sufficient condition is given for A*/~(m1, … , mk) to be of dot-depth exactly 2. Upper and lower bounds on the dot-depth of the A*/~(m1, … mk) are discussed
Foundations of Software Science and Computation Structures
This open access book constitutes the proceedings of the 25th International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2022, which was held during April 4-6, 2022, in Munich, Germany, as part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2022. The 23 regular papers presented in this volume were carefully reviewed and selected from 77 submissions. They deal with research on theories and methods to support the analysis, integration, synthesis, transformation, and verification of programs and software systems
Automates à contraintes semilinéaires = Automata with a semilinear constraint
Cette thèse présente une étude dans divers domaines de l'informatique
théorique de modèles de calculs combinant automates finis et contraintes
arithmétiques. Nous nous intéressons aux questions de décidabilité,
d'expressivité et de clôture, tout en ouvrant l'étude à la complexité, la
logique, l'algèbre et aux applications. Cette étude est présentée au travers
de quatre articles de recherche.
Le premier article, Affine Parikh Automata, poursuit l'étude de Klaedtke et Ruess
des automates de Parikh et en définit des généralisations et restrictions.
L'automate de Parikh est un point de départ de cette thèse; nous montrons que
ce modèle de calcul est équivalent à l'automate contraint que nous
définissons comme un automate qui n'accepte un mot que si le nombre de fois
que chaque transition est empruntée répond à une contrainte arithmétique.
Ce modèle est naturellement étendu à l'automate de Parikh affine qui
effectue une opération affine sur un ensemble de registres lors du
franchissement d'une transition. Nous étudions aussi l'automate de
Parikh sur lettres: un automate qui n'accepte un mot que si le nombre de
fois que chaque lettre y apparaît répond à une contrainte arithmétique.
Le deuxième article, Bounded Parikh Automata, étudie les langages
bornés des automates de Parikh. Un langage est borné s'il existe des
mots w_1, w_2, ..., w_k tels que chaque mot du langage peut s'écrire
w_1...w_1w_2...w_2...w_k...w_k. Ces langages sont
importants dans des domaines applicatifs et présentent usuellement de bonnes
propriétés théoriques. Nous montrons que dans le contexte des langages
bornés, le déterminisme n'influence pas l'expressivité des automates de
Parikh.
Le troisième article, Unambiguous Constrained Automata, introduit les
automates contraints non ambigus, c'est-Ã -dire pour lesquels il
n'existe qu'un chemin acceptant par mot reconnu par l'automate. Nous
montrons qu'il s'agit d'un modèle combinant une meilleure expressivité et de
meilleures propriétés de clôture que l'automate contraint déterministe. Le
problème de déterminer si le langage d'un automate contraint non ambigu est
régulier est montré décidable.
Le quatrième article, Algebra and Complexity Meet Contrained Automata,
présente une étude des représentations algébriques qu'admettent les automates
contraints et les automates de Parikh affines. Nous déduisons de ces
caractérisations des résultats d'expressivité et de complexité. Nous
montrons aussi que certaines hypothèses classiques en complexité
computationelle sont reliées à des résultats de séparation et de non clôture
dans les automates de Parikh affines.
La thèse est conclue par une ouverture à un possible approfondissement, au
travers d'un certain nombre de problèmes ouverts.This thesis presents a study from the theoretical computer science
perspective of computing models combining finite automata and arithmetic
constraints. We focus on decidability questions, expressiveness, and closure
properties, while opening the study to complexity, logic, algebra, and
applications. This thesis is presented through four research articles.
The first article, Affine Parikh Automata, continues the study of Klaedtke
and Ruess on Parikh automata and defines generalizations and restrictions of
this model. The Parikh automaton is one of the starting points of this
thesis. We show that this model of computation is equivalent to the
constrained automaton that we define as an automaton which accepts a word
only if the number of times each transition is taken satisfies a given
arithmetic constraint. This model is naturally extended to affine Parikh
automata, in which an affine transformation is applied to a set of registers
on taking a transition. We also study the Parikh automaton on letters, that
is, an automaton which accepts a word only if the number of times each letter
appears in the word verifies an arithmetic constraint.
The second article, Bounded Parikh Automata, focuses on the
bounded languages of Parikh automata. A language is bounded if there
are words w_1, w_2, ..., w_k such that every word in the language can be
written as w_1...w_1w_2...w_2 ... w_k...w_k. These languages
are important in applications and usually display good theoretical
properties. We show that, over the bounded languages, determinism does not
influence the expressiveness of Parikh automata.
The third article, Unambiguous Constrained Automata, introduces the
concept of unambiguity in constrained automata. An automaton is
unambiguous if there is only one accepting path per word of its language. We
show that the unambiguous constrained automaton is an appealing model of
computation which combines a better expressiveness and better closure
properties than the deterministic constrained automaton. We show that it is
decidable whether the language of an unambiguous constrained automaton is
regular.
The fourth article, Algebra and Complexity Meet Constrained Automata,
presents a study of algebraic representations of constrained automata and
affine Parikh automata. We deduce expressiveness and complexity results from
these characterizations. We also study how classical computational
complexity hypotheses help in showing separations and nonclosure properties
in affine Parikh automata.
The thesis is concluded by a presentation of possible future avenues of
research, through several open problems
Foundations of Software Science and Computation Structures
This open access book constitutes the proceedings of the 25th International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2022, which was held during April 4-6, 2022, in Munich, Germany, as part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2022. The 23 regular papers presented in this volume were carefully reviewed and selected from 77 submissions. They deal with research on theories and methods to support the analysis, integration, synthesis, transformation, and verification of programs and software systems