Thick Sets, Multiple-Valued Mappings, and Possibility Theory

Abstract

Carrying uncertain information via a multivalued function can be found in different settings, ranging from the computation of the image of a set by an inverse function to the Dempsterian transfer of a probabilistic space by a multivalued function. We then get upper and lower images. In each case one handles so-called thick sets in the sense of Jaulin, i.e., lower and upper bounded ill-known sets. Such ill-known sets can be found under different names in the literature, e.g., interval sets after Y. Y. Yao, twofold fuzzy sets in the sense of Dubois and Prade, or interval-valued fuzzy sets, ... Various operations can then be defined on these sets, then understood in a disjunctive manner (epistemic uncertainty), rather than conjunctively. The intended purpose of this note is to propose a unified view of these formalisms in the setting of possibility theory, which should enable us to provide graded extensions to some of the considered calculi.