Detecting Local Inconsistency and Incompleteness in Fuzzy Rule Bases

Fuzzy rule bases are built of linguistic, qualitative knowledge. By using fuzzy rules we are able to specify simple models of complex systems. But, we have to pay a price for this simpliication. In general, fuzzy knowledge is gradually incomplete and gradually inconsistent. This paper deals with the detection of such partial gaps of knowledge or local contradictions. In order to do so we introduce the notion of {completeness and of {consistency. I Introduction and Goals In general, a fuzzy rule base will neither be totally complete nor absolutely consistent. This is, due to the use of fuzzy predicates with overlapping membership functions there will always be gradual gaps of knowledge as well as local contradictions. With the theories of possibilistic and evidential reasoning 2, 1, 8, 4, 5] there are two complementary theoretical frameworks to deal with fuzzy rule bases. But whereas the possibilistic approach is very appropriate for dealing with incomplete knowledge, it is very sensitive in case of inconsistent knowledge. The new evidential approach to fuzzy reasoning corresponds to Mamdani's inference mechanism and shows complementary properties. This method is well suited for inconsistent rule bases, but inappropriate for incomplete knowledge. Therefore, in the possibilistic or the evidential setting it is important to have a feeling for the overall degree of consistency or for the overall degree of completeness, respectively. Consequently, this paper addresses the detection of inconsistency in the context of possilistic reasoning and the detection of incompleteness in the evidential framework. We propose a mechanism that is able to evaluate both degrees in the corresponding context. Furthermore, by pointing at the part of the rule base that is responsible for incompleteness or inconsistency our mechanism helps the designer to adjust the rule base in an appropriate way. We are convinced that the lack of success with complex, highly structured rule bases so far is mainly due to the fact, that small errors caused by gradually imperfect knowledge are accumulated when inference is extended over several layers of rules. From this point of view this proposal may be a rst step in order to manage complex fuzzy rule bases. This paper is organized as follows: In section II, we brieey discuss a theoretically sound framework for fuzzy reasoning, namely possibilistic and evidential reasoning. We analyze the advantages and drawbacks of each mechanism and show their complementary behaviour with respect to incomplete and inconsistent knowledge. In section III, …