Abstract
Abduction is the procedure to derive a set of hypotheses that explains a set of observed events under given knowledge. The derived set of hypotheses is called an explanation. We have already proposed a theory of fuzzy abduction that was an expansion of the abduction with fuzzy theory, and shown the way to derive fuzzy explanations. In the theory, observed events and hypotheses are expressed in fuzzy sets, and a set of implications with a truth value between zero and one is used as the given knowledge. However, it is not guaranteed that fuzzy explanations always exist. In this paper, we specify the necessary and sufficient conditions of existence of fuzzy explanations and propose a method to obtain approximate explanations when the conditions are not satisfied. Fuzzy abduction is a procedure similar to the inverse problem of fuzzy relational equations. However, unlike methods proposed so far in the inverse problem of fuzzy relational equations, our method has a merit that it does not need iterative calculation to obtain approximate solutions.
