A new approach to fuzzy reasoning

Over the last years fuzzy control has become a very popular and successful control paradigm. The basic idea of fuzzy control is to incorporate human expert knowledge. This expert knowledge is speciied in a rule based manner on a high and granular level of abstraction. By using vague predicates a fuzzy rule base neglects useless details and concentrates on important relations. Following L.A. Zadeh's famous principle of incompatibility, this technique is most promising when applied to large and complex problems. Nevertheless, nowadays most fuzzy rule bases are small and represent simple knowledge. From our point of view this surprising and somewhat disappointing observation is due to a major lack of understanding how to handle a fuzzy rule base. In this paper we present a new theory for fuzzy reasoning. This theory is twofold. In general, a fuzzy rule base is both partially inconsistent and partially incomplete. This is the price to pay for abstraction and granularization. We show, that if a fuzzy rule base maximizes consistency at the cost of completenes, the well known possibilistic approach to fuzzy inference is the right choice. For a fuzzy rule base, that maximizes completeness at the cost of consistency, we derive a new type of inference called {reasoning. Together, both mechanisms form an embracing theory for fuzzy reasoning in general. We propose a combined approach to be applied in order to manage complex rule bases. I Introduction Fuzzy reasoning deals with the formalization and activation of rule based knowledge that is based on vague precidates. In general, a fuzzy rulebase is neither complete nor consistent. Due to the imprecise nature of fuzzy knowledge, there is always some detailed information missing and we have to cope with gradual contradictions, as well. By modeling a complex system in terms of a fuzzy rule base, we may hope for a simple and treatable solution. To describe the system on a granular and qualitative level may be the only treatable way to handle it (Zadeh 1973, cf. principle of incompatibility). This is, we have to live with gradual incompleteness and gradual inconsistency to a certain extent and we know a priori, that they do exist. From an intuitive point of view this situation may be characterized as follows. If we ask experts about their knowledge, there may be two extreme types of informants: The reliable expert is very cautious and tells us only things he is sure …