Database Reference
In-Depth Information
where
p
,
q
are conditions. The minimum and maximum may be replaced by, e.g.,
t
-norm and
t
-conorm
(Klement, Mesiar & Pap, 2000) to model the conjunction and disjunction connectives, respectively.
Among earlier contributions, using the
relational calculus
instead of the relational algebra, is Taka-
hashi (1995) where he proposes the FQL (Fuzzy Query Language), meant as a fuzzy extension of the
domain relational calculus
(DRC). A more complete approach has been proposed Buckles, Petry &
Sachar (1989) in a more general context of
fuzzy databases
, which is however also applicable for the
crisp relational databases considered here. Zadrożny & Kacprzyk (2002) proposed to interpret elements
of DRC in terms of a variant of fuzzy logic. This approach makes it also possible to account for prefer-
ences between query conditions in an uniform way.
Fuzzy Preferences Between Query Conditions
A query usually comprises several conditions and they may differ in their importance. For instance, a
user may look for a cheap car with a low mileage but the price may be much more important to him or
her. Thus, it may be wortwhile to offer him or her the possibility to assign different
importance weights
to various parts of a query condition. A weight
w
i
is usually assumed to be represented by a real number,
w
i
∧
[0,1], and
w
i
=0 models 'not important at all' and
w
i
=1 represents 'fully important'. A weight
w
i
is
associated with each part of a (fuzzy) condition
p
i
. The matching degree of a condition
p
i
with an im-
portance weight
w
i
is denoted by
g
p t
( )
.
In order to be meaningful, weights should satisfy some natural conditions [cf. (Dubois & Prade, 1997;
Dubois, Fargier & Prade, 1997)]. An interesting disctinction is between static and dynamic weights.
Basically, for the static weights which are used in most approaches, Dubois and Prade (1997) propose
the following framework. Assume that a query condition
p
is a conjunction (or disjunction) of weighted
elementary query conditions
p
i
, and denote by
g
p t
*
i
i
( )
the matching degree for a tuple
t
of
p
i
without
any importance weight assigned. Then, the matching degree,
g
p t
i
(
)
, of an elementary condition
p
i
*
,
with an importance weight
w
i
∧
[0,1] assigned is:
(
)
= ⇒
(
(
)
)
g
p t
*
,
w
g
p t
,
(7)
i
i
i
where
∧
is fuzzy implication connective. The overall matching degree of the whole query composed
of the conjunction of conditions
p
i
is calculated using the standard min-operator.
Depending on the type of the fuzzy implication operator used we get various interpretations of im-
portance weights. For example, using the Dienes implication we obtain from (7):
(
)
=
(
)
(
)
−
g
p t
*
,
max
g
p t
,
,
1
w
(8)
i
i
i
for the Gödel implication:
(
)
≥
1
if
g
p t
,
w
(
)
=
*
i
i
g
p t
,
(9)
(
)
i
g
p t
,
otherwise
i
