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Top1. Introduction
While designing and implementing knowledge-based systems for decision-making problems, knowledge are often imperfect (Li, Mahmood, et al., 2020). In fact, it is unavoidable to deal with uncertain, imprecise and incomplete knowledge. These different kinds of imperfection are caused by the fact that human usually expresses, thinks, and reasons using imperfect terms. Thus, treated concepts are ill-defined enough so that the development of some heuristics seems crucial to ensure a specific treatment. On the other side, handled information may not be of the same type nor of the same scale. Indeed, they could be quantitative (continuous, numeric) or qualitative (symbolic, ordered symbolic) in nature. Quantitative information are those referring to numerical universe of discourse, such as the car speed or the weigh. Qualitative information are those that are abstract and do not refer to numerical scales such as beauty or degree of satisfaction. Consequently, it is required to establish a unified framework able to handle various types of input information.
Two of the widely used frameworks for modeling and reasoning with imperfect knowledge are fuzzy logic (Zadeh, 1965) and symbolic multi-valued logic (H Akdag et al., 1992). The fuzzy logic, which is based on fuzzy set theory, employs intermediate fuzzy membership degrees between the full and the non-membership. The symbolic multi-valued logic is based on multi-set theory (De Glas, 1989) and generalizes the classical set theory by authorizing intermediate membership to a multi-set. For the latter, degrees are linguistic terms (little, very, perfectly, etc.) and not numbers like in fuzzy logic.
Working under a numerical framework as fuzzy set theory would ensure a rigorous framework and a precise treatment of imperfect knowledge (Yu et al., 2019). Nonetheless, some authors like De Glas (De Glas, 1989), Pacholczyk (Pacholczyk, 1992), Chung and Schwartz (Chung & Schwartz, 1995) point out limits of this logic. According to them, it is more-or-less complex to model qualitative terms by fuzzy logic, such as clever, tired, and sick since in this case, it is required to supply numerical membership functions for each predicate. They consider that this procedure is subjective, artificial, and often difficult to realize. Furthermore, as they reported, the fuzzy logic is characterized by a considerable computational complexity since it requires computing fuzzy sets of multi-dimensional Cartesian products, especially for significant applications. However, the multi-valued logic allows a symbolic representation of qualitative even quantitative terms and then their treatment will be computationally simpler. In addition, they think that working under a symbolic context is more intuitive, and very close to how human naturally thinks and reasons. Actually, the symbolic multi-valued logic proved its performance in many applications such as in Colorimetric (Truck et al., 2001, 2002) and in manipulating qualitative degrees under uncertainty to aggregate words (Truck & Akdag, 2006). In addition, it was used in inference systems to solve approximate reasoning problems (generalized modus tollens (Phuong & Khang, 2014), generalized modus ponens (Phuong & Khang, 2012). More recently, a work proposed a diagnosis aid system for autism based in this logic (Bel Hadj Kacem et al., 2016; Chaoued et al., 2017b), and it has given very satisfactory results. Besides, Chaoued et al. (Chaoued et al., 2017a) employ it in rule-based decision system for the camphor odor recognition, which has also given satisfying results. Therefore, we can adopt the symbolic multi-valued logic as a perfect framework to deal with imprecise knowledge.