Approximate Reasoning to Unify Norm - Based And

In Gerla [2000] a fuzzy logic in narrow sense is proposed as a theoretical framework for triangular norm based fuzzy control. In this note we show that the resulting theory is also able to express the implicationbased fuzzy control. Introduction The aim of control theory is to define a function f : X → Y whose intended meaning is that f(x) is the correct answer given the input x. Fuzzy approach to control, as devised in Zadeh [1965], [1975]a, [1975]b and in Mamdani [1981], furnishes an approximation of such a (ideal) function f : X → Y on the basis of pieces of fuzzy information (fuzzy granules). This approximation is represented by a system of fuzzy IFTHEN rules like IF x is Ai THEN y is Bi where i = 1,...,n and Ai and Bi are labels for fuzzy subsets ai and bi. We associate the i-rule with the Cartesian product ai×bi and the whole system with the fuzzy function f = »i=1,...,nai×bi. A suitable defuzzification process enable us to define a function f' we consider a suitable approximation of f. Now, as it is well known, the interpretation of such a rule as a logical implication A(x)→B(y) in a formalized logic is rather questionable (see, e.g., Hájek [1998]). Then in Gerla [2000]a, we propose to give a logical meaning to a fuzzy IF-THEN rule by translating the system of rules into the set Ai(x) ∧ Bi(y) → Good(x,y) of first order formulas. The intended meaning of Good(x,y) is that given x the value y gives a correct control (see also Gerla [2000]). Since it is natural to assign suitable weights to these formulas, the information carried on by a system of fuzzy IF-THEN rules is represented by a fuzzy theory in a fuzzy logic. Such a theory is a fuzzy program, i.e. a fuzzy set of Horn clauses. So, the computation of the fuzzy function f is equivalent to the computation of the least fuzzy Herbrand model of this fuzzy program. Now, in literature we have an alternative procedure for fuzzy control