Extending the Contraposition Property of Propositional Logic for Fuzzy Abduction

Abduction deals with assumption-based reasoning to explain an observation. In the context of fuzzy reasoning, abduction attempts to determine the membership function of the fuzzy propositions present in the antecedent of a rule when the membership functions for the propositions in the consequent of the rule are given. Currently available models of fuzzy abduction are capable of inferring the membership function of the antecedent clause accurately when the antecedent includes single fuzzy proposition. However, when the antecedent clause of a rule contains multiple fuzzy propositions, these models fail to determine the independent membership function of the individual propositions present in the antecedent. This paper presents a new formulation to handle the above problem by fuzzy extension of the well-known contraposition property of propositional logic. Several interesting properties due to the fuzzy extension of the classical contraposition have been derived. An algorithm for automated abduction using the extended contraposition property has been developed to demonstrate the principle of abduction with rules containing one or more fuzzy propositions in the antecedent/consequent. The time complexity of the proposed fuzzy abduction for a sequence of n-chained rules, where each rule has m fuzzy propositions, is O(mn), considering a uniform cost for composition operation and t-norm computation of the antecedent.