A new approach to fuzzy reasoning

Abstract

 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 specified 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 completeness, 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 σ-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.