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# Principal modalities of generalized constraints

(a) Possibilistic (r = blank) with R playing the role of the possibility distribution of X. For example, means that [a, b] is the set of possible values of X. Another example is In this case, the fuzzy set labeled small is the possibility distribution of X (Zadeh, 1978; Dubois and Prade, 1988). If p,small is the membership function of small, then the semantics of "X is small" is defined by where u is a generic value of X.

(b) Probabilistic (r = p) with R playing the role of the probability distribution of X. For example, means that X is a normally distributed random variable with mean m and variance s2. If X is a random variable which takes values in a finite set {u1,...,un} with respective probabilities p1,...,pn, then X may be expressed symbolically as with the semantics What is important to note is that in GTU a probabilistic constraint is viewed as an instance of a generalized constraint. When X is a generalized constraint, the expression is interpreted as a probability qualification of X, with R being the probability of X (Zadeh, 1979a; 1981a; 1981b). For example, where small is a fuzzy subset of the real line, means that probability of the fuzzy event {X is small} is likely. More specifically, if X takes values in the interval [a, b] and g is the probability density function of X, then the probability of the fuzzy even "X is small" may be expressed as (Zadeh, 1968) Hence, This expression for test-score function defines the semantics of probability qualification of a possibilistic constraint.

(c) Veristic (r = v) where R plays the role of a verity (truth) distribution of X. In particular, if X takes values in a finite set {ui ,...,un} with respective verity (truth) values t1,...,tn, then X may be expressed as meaning that Ver (X = щ) = t, i = ,...,n.

For example, if Robert is half-German, quarter-French and quarter- Italian, then

Ethnicity(Robert) isv 0.5|German + 0.25|French + 0.25|Italian. When X is a generalized constraint, the expression is interpreted as verity (truth) qualification of X. For example, should be interpreted as "It is very true that X is small." The semantics of truth qualification is defined by (Zadeh, 1979b) where ms-1 is inverse of the membership function of R, and t is a fuzzy truth value which is a subset of [0, 1], see Figure 6.10.

Note: There are two classes of fuzzy sets: (a) possibilistic, and (b) veristic. In the case of a possibilistic fuzzy set, the grade of membership is the degree of possibility. In the case of a veristic fuzzy set, Figure 6.10 Truth qualification: (X is small) is t

the grade of membership is the degree of verity (truth). Unless stated to the contrary, a fuzzy set is assumed to be possibilistic.

(d) Usuality (r = u) The usuality constraint presupposes that X is a random variable, and that probability of the event {X isu R} is usually, where usually plays the role of a fuzzy probability which is a fuzzy number (Kaufmann and Gupta, 1985). For example, means that "usually X is small" or, equivalently, In this expression, small may be interpreted as the usual value of X. The concept of a usual value has the potential of playing a significant role in decision analysis, since it is more informative than the concept of an expected value.

(e) Random set (r = vs)

In X is a fuzzy-set-valued random variable and R is a fuzzy random set.

(f) Fuzzy graph (r = fq)

In X is a function, f, and R is a fuzzy graph (Zadeh, 1974; 1996) which constrains f (Figure 6.11). A fuzzy graph is a disjunction of Cartesian granules expressed as where the Ai and Bi, i = 1,..., n, are fuzzy subsets of the real line, and X is the Cartesian product. A fuzzy graph is frequently described as a collection of fuzzy if-then rules (Zadeh, 1973; 1996; Pedrycz and Gomide, 1998; Bardossy and Duckstein, 1995).  Figure 6.11 Fuzzy graph

The concept of a fuzzy-graph constraint plays an important role in applications of fuzzy logic (Bardossy and Duckstein, 1995; Di Nola et al., 1989; Filev and Yager, 1994; Jamshidi et al., 1997; Ross, 2004; Yen, Langari and Zadeh, 1995).

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