# The concept of granular value

The concept of a generalized constraint provides a basis for an important conceptâ€”the concept of a granular value. Let *X* be a variable taking values in a universe of discourse U, *U = {u}.* If *a* is an element of U, and it is known that the value of *X* is *a,* then *a *is referred to as a singular value of *X.* If there is some uncertainty about the value of *X,* the available information induces a restriction on the possible values of *X* which may be represented as a generalized constraint GC(X), *X* isr *R.* Thus a generalized constraint defines a granule which is referred to as a granular value of X, Gr(X ) (see Figure 6.14). For example, if *X* is known to lie in the interval [*a, h*], then [*a, h*] is a granular value of *X*. Similarly, if *X* isp *N (m,* s^{2}), then *N (m,* s^{2}) is a granular value of X. What is important to note is that defining a granular value in terms of a generalized constraint makes a granular value *mm*-precise. It is this characteristic of granular values that underlies the concept of a linguistic variable

*Figure 6.14* A granule defined as a generalized constraint

*Figure 6.15* Singular and granular values

(Zadeh, 1973). Symbolically, representing a granular value as a generalized constraint may be expressed as Gr(X) = GC(X). It should be noted that, in general, perception-based information is granular (see Figure 6.15).

The importance of the concept of a granular value derives from the fact that it plays a central role in computation with information described in natural language. More specifically, when a proposition expressed in a natural language is represented as a system of generalized constraints, it is, in effect, a system of granular values. Thus, computation with information described in natural language ultimately reduces to computation with granular values. Such computation is the province of Granular Computing. (Zadeh, 1979a; 1979b; 1997; 1998; Lin, 1998; Bargiela and Pedrycz, 2002; Lawry, 2001; Lawry, Shanahan and Ralescu, 2003; Mares, 1994; Yager, 2006).