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1.1. KNOWLEDGE BASE

A linguistic rule "IF X isr A THEN Y isr B" of a fuzzy knowledge base may be interpreted as an information granule, depending on our knowledge of a system characteristics in at least three different ways as follows:

  1. Interpolative Reasoning Perspective: It defines a Cartesian product A ¥ B. This is a standard interpolation model (Zadeh, 1973; 1975; 1996) that is used most often in fuzzy control systems. At times, this is known as Mamdani's (1974) heuristic.
  2. Generalized Modus Ponens Perspective: It defines an implication between A and B, i.e., A —> B, depending on whether it is a strong implication S or a residual implication R (Zadeh, 1973; Demirli and Türks¸en, 1994).

    The interpretations (1) and (2) are a "myopic" view of fuzzy theory known as Type I fuzzy theory. It is based on the reductionist view that the combination of two first-order fuzzy sets produces a fuzzy set of the first order; i.e., µA: X —> [0,1], µB: Y —> [0,1] such that for each interpretation stated above we have:

    µA x B: X ¥ Y—>[0,1] for case (1) above and, and µA—>B: X ¥ Y—>[0,1] for case (2) above

  3. Type II Fuzzy Set Perspective: It is based on the realization that the combination of two first-order, i.e., Type I fuzzy sets, produce a fuzzy set of the second order, known as Type II (Türks¸en,1986; 1995; 1996) fuzzy sets; i.e., µA: X —> [0,1], µB: Y —> [0,1] such that µA—>B: X ¥ Y —> P[0,1] where P[0,1] is the power set of [0,1].

1.2 INFERENCE ENGINE

Now, from the perspective of approximate reasoning, each of the three interpretations stated above provides a different solution when faced with a new system observation A’(x). When a new sytem state is observed, say A’(x), with these three interpretations of knowledge representation, three different consequences may be obtained when we are interested in providing a decision support for system diagnosis or prediction or control. These are:

where, (y), y [element of] Y is a consequence of an inference result that may be obtained for each of the three cases of approximate reasoning for a given rule: "IF x[element of] X isr A, THEN y [element of] Y isr B" and a given observation A’(x). The rule is interpreted in accordance with one of the three perspectives stated above. In the three inference schemes stated above, V is the maximum operator and T is a t-norm to be selected from an infinitely many possibilities of t-norms introduced by Schweitzer and Sklar (1983) where (x), x [element of] X, is the observed membership value µA(x), µB(y) [element of] [0,1], x[contains] X, y [element of] Y are membership values of input, and output variables, respectively. In addition, µA(x)T µB(y) is the interpolation heuristic-based approximate knowledge representation, whereas µCNF(A—>B)(x,y) is the membership value of A —> B, which is a direct and common use of the Boolean Conjunctive normal form, CNF, of implication. Hence, approximate reasoning with cases (1) and (2) are dependent on the myopic view of knowledge representation in fuzzy set and logic theory. (Türks¸en,1986; 1995).

But, approximate reasoning with case (3) is based on nonreductionist interpretation, as stated above. It is worthwhile to emphasize that µFDNF(A—>B)(x,y) and µFCNF(A—>B)(x,y) are the boundaries of Type II "Interval-Valued" fuzzy set representation of our knowledge. It should be recalled that Fuzzy Disjunctive Normal Form, FDNF, of A —> B and Fuzzy Conjunctive Normal Form, FCNF, of A —> B represent bounds on the representation of membership values of A —> B. This is because FDNF(·) [not equal to] FCNF(·) for all the combined concepts in fuzzy theory, whereas DNF(·) = CNF(·) in two-valued theory (Türks¸en,1986; 1995).

Since most of the current fuzzy expert systems are constructed either with case (1) or (2) representation and inference, in Section 3 of this chapter, we will present a unique knowledge representation and inference approach that unifies (1) and (2) interpretations of fuzzy knowledge representation and inference. Type II fuzzy knowledge representation and inference is not discussed in this chapter.


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