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operator less (<) is called the lower diagonal partition (Figure
1).
Using this scheme partitions corresponding to the fuzzy
modifications of the operators equal, greater, and less , may be
assigned in the search space as shown in Figure 2. This is
possible because, by common sense reasoning, the fuzzy
expression about modifies the expression equal to by
generating a narrow band around the value x. The width of the
band or size of the partition generated also depends on the
magnitude of the crisp value x. By common sense the partition
generated by "about x", "more or less x" , and "roughly equal
to x" are radial partitions stretching over both sides of the line of
equality (Figure 2). Naturally the partition induced by "roughly
equal to x" must be somewhat wider than those generated by
"more or less x", or "about x".
When the fuzzy modifier much is applied to the operators great
and /ess it induces partitions which exclude all values close to
the line of equality. It is logical, therefore, to place the lower
boundary of the partition of much greater than x as far away as
possible in the upper diagonal space. This will create a radial
partition enclosing a "very wide" angle with the line of equality.
Similarly the partition for much less than x encloses a "very
wide" angle with the line of equality in the lower diagonal
space. Note that the partitions generated by greater than x and
less than x are supersets of the partitions induced by much
greater than x and much less than x respectively.
The fuzzy expressions slightly greater than x and slightly less
than x give rise to asymmetric narrow, radial partitions "very
close" to the line of equality. Naturally, the partition induced by
slightly less than x lies in the lower diagonal space. The
partition induced by slightly more than x lies in the upper
diagonal space, very close to the line of equality.
The common sense interpretation of the fuzzy expressions
introduced above, must now be defined mathematically, in
order to form the basis for the proposed fuzzy geometric
partitions based representation of fuzzy objects and comparison
of fuzzy database objects for retrieval purposes.
2.1 Specification of the Fuzzy Partitions Induced
by Fuzzy Restrictions.
Fuzzy expressions, such as about x and more or less x where
x is a number, are said to constitute elastic constraints on the
set of admissible real numbers (Dubois and Prade, 1980,
Kandel, 1986). An arbitrary real number x which satisfies the
elastic constraint is said to be a generic value of the fuzzy
expression (Kandel, 1986; Dubois and Prade, 1980).
The term generic value is used, in this study, to characterize
crisp values lying within a vague interval or the fuzzy partition
associated with a fuzzy number or fuzzy restriction. The basic
idea upon which the concept of fuzzy geometric partitions is
founded, is the simple, intuitive, idea that in common sense
reasoning a vague expression, such as about 5, invokes a
mental band of uncertain but narrow width around the crisp
number 5 as explained in the previous section. The main
assumption is that the human mind realizes this vague interval
by a process in which values picked out from the domain of real
numbers are subconsciously compared with the crisp value 5
and rejected if they differ "too much" from it. In this respect the
vague expression, about 5, is equivalent to the generic binary
relation, about(x, 5), where x is an arbitrary number which
may or may not be an acceptable member of the vague set about
5, depending on its "distance" from the crisp value 5 (see
Figures 2 and 6).
Let the collection of the fuzzy restrictions: equal to, greater than,
less than, much greater than, much less than, slightly greater
than, slightly less than, about, more or less, roughly equal to,
be denoted by PRED.
Using the idea of the diagonal subset generated by the equality
operator in Dowsing et al. (1986), the definition of the equality
operator can be extended and generalized to a general
177
><
line of equality
>>,
e greater than
C mo less than o. greater than
Figure 1: Basic partitions of the search space.
Y
A
line of equality
€ roughly
slightly less than
about
e» more or less
Figure 2: Partitions induced by some common
fuzzy expressions.
slightly greater than
comparison operator R e PRED. Let the partition induced by
the general fuzzy comparison operator, R, in a two dimensional
search space be denoted by Pg. The generalized extension of
the definition of equality operator (Dowsings et al., 1986) is as
follows:
For an interpretation I, with the universe X, the set Ry on
which R is to be true must be a radial or sectoral subset:
{xy)lxe X,ye X,xRy}of XxX
Based on this generalized definition, individual members of the
PRED set can now be defined. If we let R = much greater than
we have:
much greater than, = {(x,y)|x € X, y € X, x >> y} for all
x,y) € X x X. (1)