621 research outputs found
Decomposable Theories
We present in this paper a general algorithm for solving first-order formulas
in particular theories called "decomposable theories". First of all, using
special quantifiers, we give a formal characterization of decomposable theories
and show some of their properties. Then, we present a general algorithm for
solving first-order formulas in any decomposable theory "T". The algorithm is
given in the form of five rewriting rules. It transforms a first-order formula
"P", which can possibly contain free variables, into a conjunction "Q" of
solved formulas easily transformable into a Boolean combination of
existentially quantified conjunctions of atomic formulas. In particular, if "P"
has no free variables then "Q" is either the formula "true" or "false". The
correctness of our algorithm proves the completeness of the decomposable
theories.
Finally, we show that the theory "Tr" of finite or infinite trees is a
decomposable theory and give some benchmarks realized by an implementation of
our algorithm, solving formulas on two-partner games in "Tr" with more than 160
nested alternated quantifiers
The Bottom-Up Position Tree Automaton, the Father Automaton and their Compact Versions
The conversion of a given regular tree expression into a tree automaton has
been widely studied. However, classical interpretations are based upon a
Top-Down interpretation of tree automata. In this paper, we propose new
constructions based on the Gluskov's one and on the one of Ilie and Yu one
using a Bottom-Up interpretation. One of the main goals of this technique is to
consider as a next step the links with deterministic recognizers, consideration
that cannot be performed with classical Top-Down approaches. Furthermore, we
exhibit a method to factorize transitions of tree automata and show that this
technique is particularly interesting for these constructions, by considering
natural factorizations due to the structure of regular expression.Comment: extended version of a paper accepted at CIAA 201
Root-Weighted Tree Automata and their Applications to Tree Kernels
In this paper, we define a new kind of weighted tree automata where the
weights are only supported by final states. We show that these automata are
sequentializable and we study their closures under classical regular and
algebraic operations. We then use these automata to compute the subtree kernel
of two finite tree languages in an efficient way. Finally, we present some
perspectives involving the root-weighted tree automata
Construction of rational expression from tree automata using a generalization of Arden's Lemma
Arden's Lemma is a classical result in language theory allowing the
computation of a rational expression denoting the language recognized by a
finite string automaton. In this paper we generalize this important lemma to
the rational tree languages. Moreover, we propose also a construction of a
rational tree expression which denotes the accepted tree language of a finite
tree automaton
L'indépendant faiblement connexe : études algorithmiques et polyédrales
In this work, we focus on a topology for Wireless Sensor Networks (WSN). A wireless sensor network can be modeled as an undirected graph G = (V,E). Each vertex of V represents a sensor and an edge e = {u, v} in E implies a direct transmission between the two sensors u and v. Unlike wired devices, wireless sensors are not a priori arranged in a network. Topology should be made by selecting some sensor as dominators nodes who manage transmissions. Architectures that have been studied in the literature are mainly based on connected dominating sets and weakly connected dominating sets.This study is devoted to weakly connected independent sets. An independent set S ⊂ V is said Weakly Connected if the graph GS = (V, [S, V \S]) is connected, where [S, V \S] is the set of edges with exactly one end in S. A sensor network topology based on weakly connected sets is partition into three groups, slaves, masters and bridges. The first performs the measurements, the second gathers the collected data and the later provides the inter-group communications. We first give some properties of this combinatorial structure when the undirected graph G is connected. Then we provide complexity results for the problem of finding the minimum weakly connected independent set problem (MWCISP). We also describe an exact enumeration algorithm of complexity O∗(1.4655|V |) (for the (MWCISP)). Numerical tests of this exact procedure are also presented. We then present an integer programming formulation for the minimum weakly connected independent set problem and discuss its associated polytope. Some classical graph operations are also used for defining new polyhedra from pieces. We give valid inequalities and describe heuristical separation algorithms for them. Finally, we develop a branch-and-cut algorithm and test it on two classes of graphs.Dans ce travail, nous nous intéressons à une topologie pour les réseaux de capteurs sans fil. Un réseau de capteurs sans fil peut être modélisé comme un graphe non orienté G = (V,E). Chaque sommet de V représente un capteur et une arête e = {u, v} dans E indique une transmission directe possible entre deux capteurs u et v. Contrairement aux dispositifs filaires, les capteurs sans fil ne sont pas a priori agencé en réseau. Une topologie doit être créée en sélectionnant des noeuds "dominants" qui vont gérer les transmissions. Les architectures qui ont été examinées dans la littérature reposent essentiellement sur les ensembles dominants connexes et les ensembles dominants faiblement connexes. Cette étude est consacrée aux ensembles indépendants faiblement connexes. Un indépendant S ⊂ V est dit faiblement connexe si le graphe GS = (V, [S, V \S]) est connexe, où [S, V \S] est l’ensemble des arêtes e = {u, v} de E avec u ∈ S et v ∈ V \S. Une topologie basée sur les ensembles faiblement connexes permet de partitionner l’ensemble des capteurs en trois groupes, les esclaves, les maîtres et les intermédiaires. Les premiers effectuent les mesures, les seconds rassemblent les données collectées et les troisièmes assurent les communications inter-groupes. Nous donnons d’abord quelques propriétés de cette structure combinatoire lorsque le graphe non orienté G est connexe. Puis nous proposons des résultats de complexité pour le problème de la recherche de l’indépendant faiblement connexe de cardinalité minimale (MWCISP). Nous décrivons également un algorithme d’énumération exact de complexité O∗(1.4655|V |) pour le MWCISP. Des tests numériques de cette procédure exacte sont présentés. Nous formulons ensuite le MWCISP comme un programme linéaire en nombres entiers. Le polytope associé aux solutions de ce problème est complètement caractérisé lorsque G est un cycle impair. Nous étudions des opérations de composition de graphes et leurs conséquences polyédrales. Nous introduisons des inégalités valides notamment les contraintes dites de multibord. Par la suite, nous développons un algorithme de coupes et branchement sous CPLEX pour résoudre ce problème en utilisant des heuristiques pour la séparation de nos familles de contraintes. Des résultats expérimentaux de ce programme sont exposés
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