Superadditive strong implication in fuzzy rule-based interpolation

The three standard S-implications, Lukasiewicz, Reichenbach, and Kleene-Dienes, are derived from Schweizer-Sklav T-norms with respective parameters of -1, 0, and positive infinity. The focus of this paper is the Quadratic S-implication with parameter -2 I(a,b)=1-√a² +b²-2b if a²-b²>2b, 1 otherwise, and on the Drastic S-implication with parameter minus infinity, 1(a,b)=1-a if b=0, b if a=1, 1 otherwise. As well as on numerical methods for general Schweizer-Sklar parameters, Parameters below -1 yield S-implication values greater than the bounded sum of Lukasiewicz, (1-a)+b, except where they both equal 1; hence the term “superadditive.” An interesting property of the Drastic S-implication with p=-∞; is that it produces a fuzzy interpolation that is a dual of the Kleene-Dienes S-implication with p=+∞; the defuzzified relation between base variables X and Y using a given set of fuzzy rules with one of these implication operators is identical to the defuzzified relation using the converses of those rules with the other operator. The defuzzified relation produced by Lukasiewicz implication is self-dual, since the same relation is produced under Lukasiewicz when the converses of the rules are used