(x) 
IS 
re 
nd 
ns. 
be 
m 
3. COMPUTATION OF FUZZY GENERIC VALUES AND 
CONSTRUCTION OF MEMBERSHIP FUNCTIONS. 
When appropriately specified, fuzzy geometric partitions 
provide a means for performing direct comparison of fuzzy 
objects for data base search purposes. Alternatively fuzzy 
membership functions may be constructed from the assigned 
partitions and used to compare fuzzy objects based on existing 
theories of fuzzy set inclusion, equality and composition of 
fuzzy sets as outlined in Zadeh et al.(1975); Dubois and Prade 
(1980); Kandel (1986); Dubois and Prade (1988); Klir and 
Folger (1988); and Zadeh (1979, 1989);. 
In section 2.1 the vague expression about 5 was said to be 
equivalent to a binary fuzzy relation about(x, 5) such that x is 
a generic value satisfying the fuzzy restriction. Based on this the 
concept of the generic value of a fuzzy number may be defined 
as follows: 
Let R be any fuzzy predicate in PRED, then the unary fuzzy 
expression R(x) where x €X, is said to induce a generic 
value x € X such that the expression, x = R(x), or 
equal(x, R(x)), evaluated over the universe of discourse is 
true. 
Using this definition the fuzzy restriction greater than 5 has a 
corresponding binary fuzzy relation greater than(x, 5) where x 
is a generic value satisfying the fuzzy restriction. 
To characterize generic values in a mathematically meaningful 
way, tentative values for the parameters a; and a; defined in 
Figure 3 are given in Table 2. The angular parameters 0; and oi 
represent the band width of left and right tailed fuzzy 
sets(Dubois and Prade, 1980) respectively. For symmetric 
fuzzy sets, o; and o; represent the left and right half-band 
widths. The term band width is used in the same sense as it is 
used to characterize standard membership functions(Kandel, 
1986). 
Notice that in Table 2 the parameters o; and o; are assigned 
values by logarithmically weighting the fuzzy constants defined 
in Table 1. This is necessary to preserve the fact that perception 
of changes in numerical magnitudes vary as the difference 
between the numbers involved change from very small to very 
large. It is intuitive to use logarithmic weighting since 
logarithmic functions are also used in modelling image 
intensities in natural vision, photography and image processing 
to reflect the human physiological response to increasing light 
stimulus (Land et al, 1989). 
Based on the. values in Table 2, functions for computing 
arbitrary generic values for the fuzzy expressions in the PRED 
set can be derived. These functions are summarized in Table 3. 
3.1 Comparison of Fuzzy Objects Using Generic 
Values. 
The equations required for computing upper and lower 
bounding generic values for all the fuzzy predicates in the 
PRED set are summarized in Table 3. Generic values computed 
by these equations can be used to facilitate direct comparison of 
fuzzy objects for the purposes of database searching. For 
example the query object more or less x can be interpreted as a 
request to retrieve all database objects satisfying the elastic 
constraint more or less(y,x). Valid generic objects y must 
therefore have values lying close to the crisp value x. This 
condition may be expressed as 
XQ Xy x (9) 
where x, and X, are generic values corresponding to the upper 
and lower bounds of the partition induced by the fuzzy 
restriction more or less(x). The values of x, and X, can be 
computed from Eqs. (10) and (11) respectively, where the term 
CLOSE ' is as defined in Table 2. 
Table 1: DEFINITION OF THE FUZZY CONSTANTS DETERMINING THE BAND 
WIDTH OF THE FUZZY PATITIONS. 
LABEL VERYWIDE WIDE 
GENERIC WIDTH T/3 7/6 
CLOSE1 CLOSE VERYCLOSE 
1/12 T/24 1/48 
Table 2: FORMULAE FOR COMPUTING THE GENERIC BAND WIDTH OF THE 
PARTITIONS. 
Fuzzy Predicate Left and Right Tails: Symbolic Value 
much greater than(x) 
much less than(x) 
slightly more. than(x) 
slightly less than(x) 
more or less(x) 
WIDE+VERYCLOSE/(4+log(x)) | VERYWIDE' 
WIDE+VERYCLOSE/(4+log(x))  VERYWIDE' 
3*VERYCLOSE/(1+0.5log(x)) VERYCLOSE' 
3*VERYCLOSE/(1+0.5log(x)) VERYCLOSE' 
5*CLOSE1/(8+l0g(x)) if x<=5 
CLOSE1/(2+log(x)) if x<=10 CLOSE1' 
CLOSE1/(3+10g(x)) if x>10 
about(x) 4*CLOSE/(3+3log(x)) if x<=5 
4*CLOSE/(6+3log(x)) if x<=10  CLOSE' 
4*CLOSE/(9+3log(x)) if x>10 
roughly(x) CLOSE1/(1+log(x)) if x <=5 
CLOSE1/(2+log(x)) if x <=10 CLOSE" 
CLOSE1/(3+log(x)) if x >10 
179