Database Reference
In-Depth Information
a
negative
preference: the tuples which do not satisfy it are definitely not matching the whole query
while the latter condition
P
, on the other hand, expresses a
positive
preference. These conditions will be
referred to as a positive and negative condition, for short, and the whole query will be denoted as (
C
,
P
).
We will identify the negative and positive condition of a bipolar query with the predicates that represent
them and denote them as
C
and
P
, respectively. Let us denote the set of all tuples under consideration
with
T.
For a tuple
t
∧
T
,
C
(
t
) and
P
(
t
) will denote that the tuple
t
satisfies the respective condition. The
bipolar query in this approach may be expressed in natural language as follows:
Find tuples t satisfying C and possibly satisfying P
exemplified by: “Find a house cheaper than USD 250,000
and possibly
located not more than two blocks
from a railway station”, and such a query may be formally written as
C
and possibly
P
(20)
The key problem, which we consider here, is a proper modeling of the aggregation of both types of
conditions which is expressed here with the use of the “and possibly” operator. Thus, we are mainly
concerned with how to combine both negative and positive evaluations (assessments) in order to come
up with a “standard” evaluation on a unipolar univariate scale which provides for an obvious ordering
of the tuples in an answer to the query. An alternative way is to not aggregate and order the tuples with
respect to their matching of required and preferred conditions taken separately - this way is adopted, e.g.,
by Dubois and Prade (2002). However, it seems that the interpretation of the “and possibly” operator is
quite intuitive and possesses some interesting properties [cf. also (Bordogna & Pasi, 1995)].
According to the original (crisp) approach by Lacroix & Lavency (1987) such an operator has an
important property: the aggregation result depends not only on the explicit arguments, i.e.,
C
(
t
) and
P
(
t
),
but also on the content of the database. If there are no tuples meeting both conditions then the result of
the aggregation is determined by the negative condition
C
alone. Otherwise the aggregation becomes
a regular conjunction of both conditions. This dependence is best expressed by the following logical
formula (Lacroix & Lavency, 1987):
C
(
t
) and possibly
P
(
t
)
∧
C
(
t
)
∧
∧
s
(
C
(
s
)
∧
P
(
s
))
∧
P
(
t
)
(21)
If conditions
C
and
P
are crisp, then this characteristic property is preserved if the “first select us-
ing
C
and then order using
P
” interpretation of (20) is adopted, i.e., when first tuples satisfying
C
are
selected and then ordered according to
P
. However if both conditions
C
and
P
are fuzzy then it is no
longer clear what it should mean that a tuple satisfies the condition
C
as the satisfaction of this condi-
tion is now a matter of the degree.
In our approach we start with the formula (24) and, using standard fuzzy counterparts of the classical
logical connectives, interpret it in terms of fuzzy logic obtaining the membership function of the fuzzy
answer set to a bipolar query (
C
,
P
), with respect to a set of tuples
T
, ans(
C
,
P
,
T
), as:
μ
ans
(
C
,
P
,
T
)
(
t
) = min (
C
(
t
), max (1-max
s
∧
T
min(
C
(
s
),
P
(
s
)),
P
(
t
)))
(22)
