Pure Refined Variable Inclusion Logics

In this article, we explore the semantic characterization of the (right) pure refined variable inclusion companion of all logics, which is a further refinement of the nowadays well-studied pure right variable inclusion logics. In particular, we will focus on giving a characterization of these fragments via a single logical matrix, when possible, and via a class of finite matrices, otherwise. In order to achieve this, we will rely on extending the semantics of the logics whose companions we will be discussing with infectious values in direct and in more subtle ways. This further establishes the connection between infectious logics and variable inclusion logics

Some Quantitative Characterizations of Certain Symplectic Groups

Given a finite group $G$, denote by ${\rm D}(G)$ the degree pattern of $G$ and by ${\rm OC}(G)$ the set of all order components of $G$. Denote by $h_{{\rm OD}}(G)$ (resp. $h_{{\rm OC}}(G)$) the number of isomorphism classes of finite groups $H$ satisfying conditions $|H|=|G|$ and ${\rm D}(H)={\rm D}(G)$ (resp. ${\rm OC}(H)={\rm OC}(G)$). A finite group $G$ is called OD-characterizable (resp. OC-characterizable) if $h_{\rm OD}(G)=1$ (resp. $h_{\rm OC}(G)=1$). Let $C=C_p(2)$ be a symplectic group over binary field, for which $2^p-1>7$ is a Mersenne prime. The aim of this article is to prove that $h_{\rm OD}(C)=1=h_{\rm OC}(C)$

: Méthodes d'Inférence Symbolique pour les Bases de Données

This dissertation is a summary of a line of research, that I wasactively involved in, on learning in databases from examples. Thisresearch focused on traditional as well as novel database models andlanguages for querying, transforming, and describing the schema of adatabase. In case of schemas our contributions involve proposing anoriginal languages for the emerging data models of Unordered XML andRDF. We have studied learning from examples of schemas for UnorderedXML, schemas for RDF, twig queries for XML, join queries forrelational databases, and XML transformations defined with a novelmodel of tree-to-word transducers.Investigating learnability of the proposed languages required us toexamine closely a number of their fundamental properties, often ofindependent interest, including normal forms, minimization,containment and equivalence, consistency of a set of examples, andfinite characterizability. Good understanding of these propertiesallowed us to devise learning algorithms that explore a possibly largesearch space with the help of a diligently designed set ofgeneralization operations in search of an appropriate solution.Learning (or inference) is a problem that has two parameters: theprecise class of languages we wish to infer and the type of input thatthe user can provide. We focused on the setting where the user inputconsists of positive examples i.e., elements that belong to the goallanguage, and negative examples i.e., elements that do not belong tothe goal language. In general using both negative and positiveexamples allows to learn richer classes of goal languages than usingpositive examples alone. However, using negative examples is oftendifficult because together with positive examples they may cause thesearch space to take a very complex shape and its exploration may turnout to be computationally challenging.Ce mémoire est une courte présentation d’une direction de recherche, à laquelle j’ai activementparticipé, sur l’apprentissage pour les bases de données à partir d’exemples. Cette recherches’est concentrée sur les modèles et les langages, aussi bien traditionnels qu’émergents, pourl’interrogation, la transformation et la description du schéma d’une base de données. Concernantles schémas, nos contributions consistent en plusieurs langages de schémas pour les nouveaumodèles de bases de données que sont XML non-ordonné et RDF. Nous avons ainsi étudiél’apprentissage à partir d’exemples des schémas pour XML non-ordonné, des schémas pour RDF,des requêtes twig pour XML, les requêtes de jointure pour bases de données relationnelles et lestransformations XML définies par un nouveau modèle de transducteurs arbre-à-mot.Pour explorer si les langages proposés peuvent être appris, nous avons été obligés d’examinerde près un certain nombre de leurs propriétés fondamentales, souvent souvent intéressantespar elles-mêmes, y compris les formes normales, la minimisation, l’inclusion et l’équivalence, lacohérence d’un ensemble d’exemples et la caractérisation finie. Une bonne compréhension de cespropriétés nous a permis de concevoir des algorithmes d’apprentissage qui explorent un espace derecherche potentiellement très vaste grâce à un ensemble d’opérations de généralisation adapté àla recherche d’une solution appropriée.L’apprentissage (ou l’inférence) est un problème à deux paramètres : la classe précise delangage que nous souhaitons inférer et le type d’informations que l’utilisateur peut fournir. Nousnous sommes placés dans le cas où l’utilisateur fournit des exemples positifs, c’est-à-dire deséléments qui appartiennent au langage cible, ainsi que des exemples négatifs, c’est-à-dire qui n’enfont pas partie. En général l’utilisation à la fois d’exemples positifs et négatifs permet d’apprendredes classes de langages plus riches que l’utilisation uniquement d’exemples positifs. Toutefois,l’utilisation des exemples négatifs est souvent difficile parce que les exemples positifs et négatifspeuvent rendre la forme de l’espace de recherche très complexe, et par conséquent, son explorationinfaisable

On Group-Characterizability of Homomorphic Secret Sharing Schemes

A group-characterizable (GC) random variable is induced by a finite group, called main group, and a collection of its subgroups [Chan and Yeung 2002]. The notion extends directly to secret sharing schemes (SSS). It is known that multi-linear SSSs can be equivalently described in terms of GC ones. The proof extends to abelian SSSs, a more powerful generalization of multi-linear schemes, in a straightforward way. Both proofs are fairly easy considering the notion of dual for vector spaces and Pontryagin dual for abelian groups. However, group-characterizability of homomorphic SSSs (HSSSs), which are generalizations of abelian schemes, is non-trivial, and thus the main focus of this paper. We present a necessary and sufficient condition for a SSS to be equivalent to a GC one. Then, we use this result to show that HSSSs satisfy the sufficient condition, and consequently they are GC. Then, we strengthen this result by showing that a group-characterization can be found in which the subgroups are all normal in the main group. On the other hand, GC SSSs whose subgroups are normal in the main group can easily be shown to be homomorphic. Therefore, we essentially provide an equivalent characterization of HSSSs in terms of GC schemes. We also present two applications of our equivalent definition for HSSSs. One concerns lower bounding the information ratio of access structures for the class of HSSSs, and the other is about the coincidence between statistical, almost-perfect and perfect security notions for the same class