RL-bags: A conceptual, level-based approach to fuzzy bags

In this paper we claim that, though algebraically well-defined, bags are not well suited for representing and reasoning with real-world information, and we propose suitable alternatives. We extend the same discussion to the fuzzy case, in which membership of elements to bags is gradual, extending the proposed alternatives. There are two main ideas behind our approach. The first is that in general the elements of a bag can be identified and are distinguishable in the real world and, when this is not the case, we are facing a problem of representation of partial knowledge, i.e., we have a lack of information. Under this consideration, we discuss and criticize the usual operations for bags. The second is to manage the fuzziness by levels using a recent level-based representation approach that extends that of fuzzy sets and keeps all the properties of the crisp case. The classical notion of bag can be seen in our approach as a bag summary. We propose a new model that generalizes the existing approach, defining new operations from this new perspective. We also propose how to deal with fuzziness and incompleteness following our approach, doing a review of the existing approaches and applications concerning bags.