Qualitative Capacities as Imprecise Possibilities.

National audienceThis paper studies the structure of qualitative capacities, that is, monotonic set-functions, when they range on a finite totally ordered scale equipped with an order-reversing map. These set-functions correspond to general representations of uncertainty, as well as importance levels of groups of criteria in multicriteria decision-making. More specifically, we investigate the question whether these qualitative set-functions can be viewed as classes of simpler set-functions, typically possibility measures, paralleling the situation of quantitative capacities with respect to imprecise probability theory. We show that any capacity is characterized by a non-empty class of possibility measures having the structure of an upper semi-lattice. The lower bounds of this class are enough to reconstruct the capacity, and their number is characteristic of its complexity. We introduce a sequence of axioms generalizing the maxitivity property of possibility measures, and related to the number of possibility measures needed for this reconstruction. In the Boolean case, capacities are closely related to non-regular multi-source modal logics and their neighborhood semantics can be described in terms of qualitative Moebius transforms

Towards Contingent World Descriptions in Description Logics

The philosophical, logical, and terminological junctions between Description Logics (DLs) and Modal Logic (ML) are important because they can support the formal analysis of modal notions of ‘possibility’ and ‘necessity’ through the lens of DLs. This paper introduces functional contingents in order to (i) structurally and terminologically analyse ‘functional possibility’ and ‘functional necessity’ in DL world descriptions and (ii) logically and terminologically annotate DL world descriptions based on functional contingents. The most significant contributions of this research are the logical characterisation and terminological analysis of functional contingents in DL world descriptions. The ultimate goal is to investigate how modal operators can – logically and terminologically – be expressed within DL world descriptions

Capacités qualitatives et information incomplète

International audienceCet article étudie les capacités qualitatives, qui sont des fonctions d'ensemble monotones croissantes à valeurs sur un ensemble totalement ordonné muni d'une fonction de renversement de l'ordre. En nous inspirant du rôle joué par les probabilités pour les capacités quantitatives, nous cherchons à savoir si les capacités qualitatives peuvent être considérées comme des ensembles de mesures de possibilité. Plus précisément nous montrons que toute capacité qualitative est caracterisée par une classe de mesures de possibilité. De plus, les bornes inférieures de cette classe sont suffisantes pour reconstruire la capacité et leur nombre caractérise sa complexité. Nous présentons aussi un axiome généralisant la maxitivité des mesures de possibilité qui revient à préciser le nombre de mesures de possibilité nécessaires à la reconstruction de la capacité. Cet axiome nous permet aussi d'établir un lien entre capacité qualitative et logique modale non regulière. Enfin nous donnons quelques résultats pour caractériser la quantité d'information contenue dans une capacité

Fuzzy measures on finite scales as families of possibility measures

International audienceWe show that any capacity or fuzzy measure ranging on a qualitative scale can be viewed both as the lower bound of a set of possibility measures, and the upper bound of a set of necessity measures. An algorithm is provided to compute the minimal set of possibility measures dominating a given capacity. This algorithm relies on the representation of the capacity by means of its qualitative Moebius transform, and the use of selection functions of the corresponding focal sets. We also introduce the counterpart of a contour function, that turns out to be the union of all most specific possibility distributions dominating the capacity. Finally we show the connection between Sugeno integrals and lower possibility measures