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Thick sets, multiple-valued mappings, and possibility theory

Didier Dubois 1 Luc Jaulin 2 Henri Prade 1
1 IRIT-ADRIA - Argumentation, Décision, Raisonnement, Incertitude et Apprentissage
IRIT - Institut de recherche en informatique de Toulouse
2 Lab-STICC_ENSTAB_CID_PRASYS
Lab-STICC - Laboratoire des sciences et techniques de l'information, de la communication et de la connaissance
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.
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Didier Dubois, Luc Jaulin, Henri Prade. Thick sets, multiple-valued mappings, and possibility theory. Vladik Kreinovich. Statistical and Fuzzy Approaches to Data Processing, with Applications to Econometrics and Other Areas, 892, Springer, pp.101-109, 2020, Studies in Computational Intelligence, 978-3-030-45618-4. ⟨10.1007/978-3-030-45619-1_8⟩. ⟨hal-02924428⟩

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