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Selection and correction of weighted rules based on Łukasiewicz’s fuzzy logic with evaluated syntax. (English) Zbl 1424.28028

The paper is based on the application of the, so called, fuzzy logic with evaluated syntax (\(Ev_{\text{Ł}}\)) to expert systems. Let us emphasize that \(Ev_{\text{Ł}}\) is developed on the idea that axioms need not be fully true (they can form a fuzzy set). This leads to a special formal system dealing with evaluated formulas, i.e., formulas that are assigned a lower bound of a truth value that they can attain in a model. Consequently, also inference rules manipulate with evaluated formulas and we can thus form a proof having a value. We naturally arrive at the concept of a fuzzy theory. The completeness theorem then says that the provability degree of a formula \(\psi\) in the fuzzy theory (that is supremum of values of all proofs of \(\psi\)) is equal to its truth (that is infimum of its truth in all models). It seems natural that an expert system which is formed by rules can be understood as a fuzzy theory and search of a conclusion in it can be taken as inference in this fuzzy theory. This paper continues with this idea and presents precise description the method how such an expert system should be formed and how the reasoning in it should be realized. It is interesting that the suggested theory deals with weights of rules taken from the interval \([-1,+1]\). This is necessary because the weights must distinguish positive or negative support for a given knowledge while 0 has the meaning of “neutral”. These weights are encoded in the suggested fuzzy theory by couples of evaluated formulas \([\psi^+\&\neg\psi^-,a]\) and \([\psi^-\&\neg\psi^+,b]\).

MSC:

28E10 Fuzzy measure theory
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