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3.2. CRISP CONNECTIVES OF FUZZY THEORY

In order to cover various types of set operators used in fuzzy set theory, a parameterized family of t-norms (T) and t-conorms (S) is chosen among the parametric Schweizer and Skalar's operators (1983):

The main features of this parameterized family of t-norms and co-norms are that they are analytically simple and symmetric, which are basic advantages in implementations. Furthermore, when p changes continuously as a positive real number, this parametric form covers all t-norms and t-conorms.

Analogous to the classical set theory, De Morgan laws establish a link between union and intersection via complementation. The conjoint t-norms and t-conorms introduced above are De Morgan duals when considered with the standard negation c(a) = 1-a (Ruan and Kerre, 1993).

Although t-norm and t-conorm functions are defined as binary operators on [0,1], their associativity property allows them to be extended to n-ary operations as :

where, T(...) and S(...) are the binary operators. It is proved that the n-ary operators Tn and Sn satisfy similar properties as the original binary T and S (Ruan and Kerre, 1993; Emami,Türks¸en, and Goldenberg, 1996).

The extension of these t-conorm operators to n arguments can be computed with a fast algorithm (Emami, Türks¸en,and Goldenberg, 1996). That is, the original formula:

can be transformed into:

where for both T(·) formula is computed as:

T(a1, a2, ..., an) = 1 - S((1 - a1), (1 - a2), ..., (1 - an))

It should be noted that the computational complexity of the original formula is O(2n), whereas for the transformed formula, it is O(n).

3.3. IMPLICATION AND AGGREGATION

Within the theory of approximate reasoning (Zadeh, 1973), each fuzzy rule is specified as:

Ri - IF X1 isr Ai1 AND ... AND Xr isr Air THEN Y isr Bi

where R is a fuzzy relation defined on the Cartesian product universe of X1 ¥ X2 ¥ ... ¥ Xr ¥ Y.

In agreement with the discussion presented above, t-norm operators are used to define conjunctions in the antecedent of the multi-input rule. Furthermore, modeling of implication relations is not unique in fuzzy logics. Two extreme paradigms for forming the implication relation are the conjunctive and disjunctive methods, which are identified as cases (1) and (2) in the introduction section. Under the conjunctive method, the fuzzy relation R is simply the conjunction of antecedent and consequent spaces. Therefore:

Rci(x1, x2, ..., xr, y) = T(T’(Ai1(x1), Ai2(x2), ..., Air(xr)), Bi(y))

where Aij(xj) and Bi(y) are the membership grades used in the computations of Rci, T is t-norm operator (with parameter p) for the rule implication, and T’ is t-norm operator (with parameter q) for the antecedent aggregation. On the other hand, the disjunctive method is obtained directly by generalizing the material implication defined in classical set theory as A --> B [identically equal to] [bar A] B, where [bar A] is the standard negation of A. Therefore, we have:

in which, S and S’ are t-conorm operators with parameters p and q, respectively.

Selection of the rule aggregation function depends on the selection of implication function for individual rules (Türks¸enand Tian, 1993). For the conjunctive method, the aggretation of the rules should be with a disjunction (union) operator. In other words, the ALSO connective should be an OR operator, i.e., a t-conorm, in particular, it is the Max operator.

Let x = (x1, x2, ..., xr), then more compactly, we have:

This method of aggregation was introduced originally by Zadeh (1993) as the interpolative reasoning, and at times, it is known as the Mamdani (1974) method, which is a myopic inference method.

On the other hand, if each basic proposition is regarded as "[X isr [bar A]ij] [Y isr Bi]," which is the disjunctive method, then the knowledge "(X1, X2, ..., Xr, Y) isr F’ should be aggregated with the conjunction (intersection) of the rules. In other words, the ALSO connective is an AND opeartor; i.e., t-norm, in particular with Min operator.

More compactly, we have:

This method is the formal logical myopic approximate reasoning, as explained in the introduction.


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