Centre of Excellence IT4Innovations
Division University of Ostrava
Institute for Research and Applications of Fuzzy Modeling 30. dubna 22, 70103 Ostrava
Abstract: Fuzzy rule based systems were standing at the origin of the whole fuzzy mod- elling area and undoubtedly, they still keep one of the key roles in this field. Briefly speaking, these systems employ a certain inference mechanism deducing conclusions based on an appropriate model of a fuzzy rule base (a finite set of fuzzy IF-THEN rules). They became very popular mainly due the first success- ful industrial experiment by E. H. Mamdani and S. Assilian. It is worth noticing that the authors used so called ‘conjunctive’ fuzzy model of a fuzzy rule base, i.e., a fuzzy relation that does not employ any kind of implication that would be naturally expected in the model by mathematicians and logicians. Success of the implementation elicited numerous followers and the conjunctive models became very popular among the community of practitioners. The popularity of these models nearly eliminated any interest of practitioners in the models that directly employ fuzzy implications – the ‘implicative’ fuzzy models. This becomes more interesting having in mind that even L. A. Zadeh already in 1973 (two years before the publication of E. H. Mamdani and S. Assilian) recalled the truth table of classical implication and recalled the fact that in classical logic the following holds
a ⇒ b ≡ ¬a ∨ b.
Moreover, only three years after the successful experiment, i.e., in 1978, Lotfi ZadehrepresentedasinglefuzzyIF-THENrulewithhelpoftheL ukasiewicz fuzzy implication.
Logicians who originally did not share the enthusiasm for conjunctive fuzzy models, later on showed that both implicative and conjunctive models are logi- cally correct. However, it is important to stress that they have to be understood and treated in a different way.
In this tutorial, we do not intend to contribute too much to the logical, philo- sophical and interpretation aspects of the discussion on these two basic families of models of fuzzy rule bases. The goal of this contribution is totally different – in particular, to provide an insight into the most fundamental mathematical properties of such systems in order to find an appropriate combination of the “bricks” we use to built a fuzzy rule-based system in order to preserve some of the desirable properties. Therefore, we will for example present how to combine a fuzzy-rule based model (either implicative or conjunctive) with an appropriate inference mechanism and an appropriate fuzzy partition of the input space in order to preserve the modus ponens property or, how to combine the fuzzy rule based model with an appropriate defuzzification and appropriate antecedent and consequent fuzzy sets in order to preserve the monotonicity property. We will also address the question of mutually exclusivity of such properties or their possible co-existence.