A Combined Approach to Fuzzy Reasoning

Due to the enormous success of fuzzy control, there has been a great interest in fuzzy reasoning, i.e. the formalization and application of human rule based knowledge. But when we take a closer look at the practical success of fuzzy reasoning, we nd that there are only simple rule bases involved. So far, we are not able to cope with complex fuzzy rule bases. Despite lots of theoretical investigations and practical eeorts, today fuzzy reasoning seems to be restricted to knowledge bases with one or at most two inference steps. From our point of view, this lack of success is due to two properties that are characteristic for fuzzy knowledge: In general, a fuzzy rule base will be neither totally complete nor absolutely consistent. By building models based on fuzzy predicates we get the opportunity to specify complex situations by simple relations. But this simpliication carries a cost: A fuzzy model will never be a perfect model, but a coarse grained one on an appropriate level of abstraction. Speaking in terms of a rule base, our knowledge matches some input situations better than others (gradual incompleteness) and often several rules with diierent consequents partially match a given input (gradual inconsistency). When dealing with simple fuzzy models this is no problem at all. But when working with large knowledge bases all these little inconsistencies and gradual gaps of knowledge tend to accumulate and, without any further precautions, we will receive a completely useless answer in the end. From a mathematical point of view, there are two basic fuzzy inference mechanisms with a sound theoretical background. The rst one is the well known possibilistic approach 12, 1, 2]. The second one is Mamdani's heuristic approach to fuzzy reasoning that has been successfully applied in fuzzy controllers for years and has found a theoretical framework recently 8, 10, 11]. The rst mechanism is based on possibility distributions and we will call it {reasoning, for short. The second inference mechanism is based on support distributions and is therefore referred to as {reasoning. With respect to incomplete and inconsistent information, {reasoning and {reasoning behave in a complementary way. {reasoning is very appropriate for dealing with incomplete knowledge, but at the same time very sensitive when faced with inconsistent information. {reasoning, on the other hand, is the appropriate mechanism to process partially inconsistent, but mainly complete knowledge. Considering this result we propose to combine both mechanisms …