# 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 s^{2}. If *X* is a random variable which takes values in a finite set *{u _{1},...,u_{n}}* with respective probabilities

*p*then

_{1},...,p_{n},*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 *{u _{i} ,...,u_{n}}* with respective verity (truth) values

*t*then

_{1},...,t_{n},*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 *A _{i}* and

*B*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).

_{i}, i =*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).