Ad hoc reasoning in chained fuzzy systems realized with Diens-Rescher implication

Traditional approaches to fuzzy reasoning usually employ Generalized Modus Ponens to generate fuzzy inferences from a given set of fuzzy rules and observations. These observations act as instantiations for the fuzzy propositions present in the antecedent of the rule to infer the fuzzy proposition present in the consequent of the same rule. Unfortunately, in many real world problems, such as system diagnosis, one excluding all fuzzy propositions present in both antecedent and consequent of a rule are instantiated with fuzzy observations, but the rule cannot fire as there is no formalism to handle the situation. This paper proposes a logical basis to handle the above problem both in propositional and fuzzy reasoning systems. It has been shown that two rules are equivalent, where the second is constructed by transposition of one or more propositions of the first rule to the other side of the implication with changed sign. This property has been used here to develop the basis of (ad hoc) reasoning. In case of a fuzzy system comprising chain rules, selection of the firing order of rules using the above property is not easy. An algorithm for ad hoc reasoning for chained fuzzy system is proposed to take care of the complexity in testing the firing condition of chained rules and thereby determining their order of firing. Lastly, a Petri like net has been used as the basic structure of the reasoning system, which takes away the complexity in rule firing and firing order selection for chained rules.