Firing Fuzzy Rules With Measure Type Inputs

Our concern here is with the issue of the determination of the satisfaction (firing level) of the antecedent condition associated with a variable in a fuzzy systems rule base where this antecedent condition is expressed in terms of a normal fuzzy set, linguistic value. We first consider the case where the input information about the variable is also expressed in terms of a normal fuzzy set. After looking at some approaches for determining this firing level we provide the requirements needed by any formulation for this operation when our input information is a fuzzy set. We next introduce the idea of a measure and show how it can be used to more generally express our knowledge about an uncertain value associated with a variable. We then generalize the requirements for any formulation that can be used to determine the satisfaction (firing level) of the antecedent fuzzy set when the input information about the variable is expressed using a measure. We further provide some examples of formulations. Since a probability distribution is a special case of a measure we are able to determine the firing level of fuzzy rules with probabilistic inputs.