Logical Interpretation of Fuzzy Control
In spite of the appearance there are several difficulties to give a rigorous logical interpretation of the IF-THEN rules. As an example, interpret a rule as IF (temperature is "cold") THEN (heater is "high") by the first order formula Cold(x)→High(y) and assume that r is an input such that Cold(r) is false. Then the formula Cold(r)→High(t) is true for any t and therefore any t gives a correct control given r. A rigorous logical justification of fuzzy control is given in Hájek's book (see Chapter 7) where fuzzy control is represented as a theory of Hájek's basic logic. Also in Gerla 2005 a logical approach to fuzzy control is proposed based on fuzzy logic programming. Indeed, denote by f the fuzzy function arising of a IF-THEN systems of rules. Then we can translate this system into fuzzy program in such a way that f is the interpretation of a vague predicate Good(x,y) in the least fuzzy Herbrand model of this program. This gives further useful tools to fuzzy control.
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