Study of the Systems with Uncertainty in Presence of a Fuzzy Similarity TIC2003-04564

In real problems, there are basically two kind of uncertainty: uncertainty due to the lack of knowledge with respect to well defined objects, but for which there are different hypothetical situations, and uncertainty due to the intrinsic vagueness of the objects. Traditionally, the first type of uncertainty has been studied with the help of Probability Theory while the second one has used Fuzzy Set Theory and Possibility Theory. In situations where the two kind of uncertainty are present, theories capable of handling the probability and the possibility of fuzzy events together are needed. On the other hand, many times the vagueness of the objects is due to the existence of a fuzzy equality that makes indistinguishable different objects of the universe. This can happen, for example, because of the imprecision of the measurement tools. Indistinghishability operators or similarities model this kind of equalities in a satisfactory way. The main target of this project is to formulate a general framework that allows us to deal with different types of uncertainty in universes where a similarity between their objects is defined in a coherent way. This general target can be divided in the following three partial ones: a) Study of probability distributions compatible with a given similarity relation. b) Generation of similarity relations on a universe starting from a similarity defined on a set of prototypes of this universe. c) Extending the way normalized fuzzy sets generate possibility distributions to the non-normalized case.