Modulated reasoning for Mamdani fuzzy systems: singleton fuzzification

A Modulated-Reasoning fuzzy rule operator is one involving a product of Logical-Reasoning and Mamdani fuzzy rule operators, in which each operator is raised to a power and the sum of the powers equals one. In this paper we establish which of 30 possible Modulated-Reasoning fuzzy rule operators lead to fired-rule output fuzzy sets (or a property of such a set) for a fuzzy logic system (FLS) that can be expressed as closed form mathematical expressions that are (piecewise-) differentiable with respect to membership function (MF) parameters, since such operators could then be used in the back-propagation design of a FLS. We do this for singleton fuzzification and two kinds of defuzzifiers, combined-centroid and property-centroid. We show that Modulated Reasoning is practical for singleton fuzzification, but if Gaussian (or any other exponential) MFs are used there is no difference between it and Mamdani Reasoning.