Analogy on Sequences : a Definition and an Algorithm

We present a definition of analogy on sequences which is based on two principles : the definition of an analogy between the letters of an alphabet and the use of the edit distance between sequences. Our definition generalizes the algorithm given by Lepage and is compatible with another definition of analogy in sequences given by Yvon

Analogical proportions and the factorization of information in distributive lattices

International audienceAnalogical proportions are statements involving four enti- ties, of the form 'A is to B as C is to D'. They play an important role in analogical reasoning. Their formalization has received much attention from different researchers in the last decade, in particular in a proposi- tional logic setting. Analogical proportions have also been algebraically defined in terms of factorization, as a generalization of geometric nu- merical proportions (that equate ratios). In this paper, we define and study analogical proportions in the general setting of lattices, and more particularly of distributive lattices. The decomposition of analogical pro- portions in canonical proportions is discussed in details, as well as the resolution of analogical proportion equations, which plays a crucial role in reasoning. The case of Boolean lattices, which reflects the logical mod- eling, and the case corresponding to entities described in terms of gradual properties, are especially considered for illustration purposes

Relation d'analogie et distance sur un alphabet défini par des traits

Cet article rappelle d'abord ce qu'est une relation d'analogie entre quatre ensembles finis (ou alphabets) selon la définition de Lepage. Il étudie le cas où ils sont définis par des traits binaires et présente ensuite une distance sur ces alphabets, compatible avec la relation d'analogie. Il étudie aussi comment construire des alphabets définis par des traits binaires au sein desquels se trouvent des relations d'analogie systématiques et examine le nombre d'analogies possibles dans ces alphabets. Pour finir, il prouve l'équivalence entre la définition de l'analogie entre ensembles de Lepage et celle de Stroppa

La proportion analogique dans les groupes. Application aux permutations et aux matrices.

Ce rapport de recherche étudie la notion de proportion analogique entre quatre éléments d'un groupe, en respectant les axiomes proposés par Lepage [1] et en suivant aussi la notion de factorisation, donnée comme fondamentale par Stroppa et Yvon pour dénir la proportion analogique [2]. On montre d'abord qu'en général, l'équation analogique n'a pas de solution dans un groupe non commutatif. On s'intéresse ensuite aux conditions que doivent respecter trois éléments pour qu'il existe un quatrième en proportion analogique avec eux. Ces conditions sont présentées de diérentes manières, que l'on démontre comme équivalentes. On dénit ensuite la notion de dissemblance analogique à partir d'une distance sur le groupe. On s'intéresse ensuite au cas particulier du groupe des permutations sur un ensemble à n éléments. On caractérise les conditions d'existence et on montre comment construire des proportions analogiques à partir de deux éléments et on en dénombre les cas possibles. On présente une distance sur le groupe et on en déduit une dissimilarité analogique entre quatre permutations. Pour terminer, on étudie le groupe des matrices inversibles. On caractérise les conditions d'existence et on montre comment construire des proportions analogiques à partir de deux matrices. Quelques cas particuliers sont présentés. Enn, on dénit une dissimilarité analogique entre quatre matrices

Les techniques d'appariement entre arbres. Rapport Bibliographique

le problème de comparer deux arbres intervient dans divers domaines comme les documents structurés (XML), la bioinformatique (les structures secondaires d'ARN), etc. Les algorithmes reposent sur le principe de l'appariement, ou édition (editing), d'un arbre en un autre par la composition d'opérations élémentaires, en visant à minimiser leur coût cumulé (la distance d'édition). Dans ce cadre, nous étudions un certain nombre de travaux antérieurs sur l'appariement entre arbres, qui représentent un large éventail des méthodes existantes. Notre but est d'étendre ces méthodes pour définir une analogie entre quatre arbres

Steps in the Representation of Concept Lattices and Median Graphs

International audienceMedian semilattices have been shown to be useful for dealing with phylogenetic classication problems since they subsume median graphs, distributive lattices as well as other tree based classica-tion structures. Median semilattices can be thought of as distributive ∨-semilattices that satisfy the following property (TRI): for every triple x, y, z, if x ∧ y, y ∧ z and x ∧ z exist, then x ∧ y ∧ z also exists. In previous work we provided an algorithm to embed a concept lattice L into a dis-tributive ∨-semilattice, regardless of (TRI). In this paper, we take (TRI) into account and we show that it is an invariant of our algorithmic approach. This leads to an extension of the original algorithm that runs in polynomial time while ensuring that the output is a median semilattice

When nominal analogical proportions do not fail

International audienceAnalogical proportions are statements of the form "a is to b as c is to d", where a, b, c, d are tuples of attribute values describing items. The mechanism of analogical inference, empirically proved to be efficient in classification and reasoning tasks, started to be better understood when the characterization of the class of classification functions with which the analogical inference always agrees was established for Boolean attributes. The purpose of this paper is to study the case of finite attribute domains that are not necessarily two-valued, i.e., when attributes are nominal. In particular, we describe the more stringent class of "hard" analogy preserving (HAP) functions f : X1×· · ·×Xm → X over finite domains X1,. .. , Xm, X for binary classification purposes. This description is obtained in two steps. First we observe that such AP functions are almost affine, that is, their restriction to any S1×· · ·×Sm, where Si ⊆ Xi and |Si| ≤ 2 (1 ≤ i ≤ m), can be turned into an affine function by renaming variable and function values. We then use this result together with some universal algebraic tools to show that they are essentially unary or quasi-linear, which provides a general representation of HAP functions. As a by-product, in the case when X1 = · · · = Xm = X, it follows that this class of HAP functions constitutes a clone on X, thus generalizing several results by some of the authors in the Boolean case

Twelve numerical, symbolic and hybrid supervised classification methods

International audienceSupervised classification has already been the subject of numerous studies in the fields of Statistics, Pattern Recognition and Artificial Intelligence under various appellations which include discriminant analysis, discrimination and concept learning. Many practical applications relating to this field have been developed. New methods have appeared in recent years, due to developments concerning Neural Networks and Machine Learning. These "hybrid" approaches share one common factor in that they combine symbolic and numerical aspects. The former are characterized by the representation of knowledge, the latter by the introduction of frequencies and probabilistic criteria. In the present study, we shall present a certain number of hybrid methods, conceived (or improved) by members of the SYMENU research group. These methods issue mainly from Machine Learning and from research on Classification Trees done in Statistics, and they may also be qualified as "rule-based". They shall be compared with other more classical approaches. This comparison will be based on a detailed description of each of the twelve methods envisaged, and on the results obtained concerning the "Waveform Recognition Problem" proposed by Breiman et al which is difficult for rule based approaches

Analogical proportions in a lattice of sets of alignments built on the common subwords in a finite language

International audienceWe define the locally maximal subwords and locally min- imal superwords common to a finite set of words. We also define the corresponding sets of alignments. We give a partial order relation between such sets of alignments, as well as two operations between them. We show that the constructed family of sets of alignments has the lattice structure. The study of analogical proportion in lattices gives hints to use this structure as a machine learning basis, aiming at inducing a generalization of the set of words