A unified and extended framework for operator selection in generalized modus ponens type fuzzy reasoning

Abstract

This thesis deals with issues related to fuzzy expert systems, in general, and implication functions and the inference known as Generalized Modus Ponens (GMP), in particular. A unified framework to the generation of implications is adopted and based on the properties of the unified framework, two new implication functions are generated. Existing results about the relationship between the conjunctions and implications are reviewed and extended. A general and unified framework is provided for the identification of appropriate operators (implication functions and composition operators) in GMP. It is shown that there are implication functions and composition operators, other than the extreme solutions, which produce conclusions in GMP in accordance with the identified desirable properties of inference. The set of such operators are characterized and a broader set of composition operators are identified in order to be used with a given implication function. Conversely, a broader set of implication functions are identified in order to be used with a given composition operator. This analysis is extended from the implications under the unified framework to general implications. Based on the analysis of the properties of implications, results regarding the rule decomposition principle are extended and operation decomposition principle is proposed. These principles lead to two efficient implementations of GMP with multi-antecedent rules. These are called inference with rule decomposition and inference with operation decomposition. The selection of a t-norm operator in the multi antecedent case is also investigated in conjunction with rule and operation decomposition principles. Inference with rule and operation decomposition procedures are considered and inference with operation decomposition procedure is successfully applied to implement a navigation algorithm for an autonomous mobile robot in real time. A new heuristic method is also suggested to combine the outcome of individual rules. In summary, a broader set and variety of composition operators and implication functions are incorporated into GMP. With the efficient implementations, computational complexity of GMP is reduced from an exponential function to a polynomial function of the number of antecedent variables. Finally, some of the theoretical findings of this thesis are successfully applied in the solution of a practical problem.