Modeling Consensus and Competition with improper S-Implications

This paper discusses an approach to modeling rule interaction using a family of improper S-implications derived from an extension of the Schweizer-Sklar family of parameterized improper t-norms defined over -infin < p <+infin. Improper S-implications define a relation whose membership grade can exceed 100%. The advantage of this is that when improper S-implications are used to interpolate between fuzzy points, they provide a reasonable definition of a unique best defuzzified consequent value even when the corresponding proper S-implication leads to multiple consequent values that fully satisfy the rules embodying the fuzzy points. When the output from a fuzzy rule based system using improper S-implications is defuzzified using mode defuzzification, the parameter p determines the degree to which the output represents a consensus between different rules (versus a fuzzy switching between whichever rule is dominant depending on the antecedent variable). After reviewing the foundations for the method, the paper illustrates it using two examples. Similar results to these could be obtained by varying the shapes of the antecedent and consequent membership functions of the rules themselves, but doing so by varying the parameter of the implication operator achieves the effect using a single parameter in a more systematic way