Databases Reference
In-Depth Information
To represent incomplete information or multi-valued data, we can use
fuzzy
data tables
(FDT). An FDT is a pair
S
=(
U,A
), where
U
is a finite set of
objects,
A
=
˜
,and
˜
{
f
i
:
U
→
P
(
V
i
)
|
1
≤
i
≤
m
}
P
(
V
i
) denotes the class of
fuzzy sets of domain
V
i
.
3 Rule Representation
In [12], decision logic (DL) is proposed as a means to represent knowledge
discovered from data tables. This logic is called decision logic because it is
particularly useful for a
decision table,
which is a data table
S
=(
U,A
),
where
A
can be partitioned into two sets,
C
(condition attributes) and
D
(de-
cision attributes). Through data analysis, decision rules relating condition and
decision attributes can be derived from the table. A rule is then represented
as an implication between two formulas of DL.
Since DL can represent knowledge discovered from precise data tables, we
generalize it to fuzzy decision logic (FDL) for rule representation in FDT.
The basic alphabet of FDL consists of a finite set of attribute symbols
A
=
{
a
1
,a
2
,...,a
m
}
and, for 1
≤
i
≤
m
, a finite set of linguistic terms
L
i
.The
atomic formula of an FDL is a descriptor (
a
i
,l
i
), where
a
i
∈A
and
l
i
∈L
i
.
The set of well-formed formulas (wff) of FDL is the smallest set containing
the atomic formulas and closed under the Boolean connectives
¬
,
∧
,and
∨
.
If
ϕ
and
ψ
are wffs of FDL, then
ϕ
ψ
is a rule in FDL, where
ϕ
is the
antecedent of the rule and
ψ
is the consequent.
Each element in the universe of an FDT corresponds to an object and
an atomic formula (i.e., an attribute-value pair) describes the value of an
individual attribute of an object. Thus, atomic formulas (and wffs) can be
verified or falsified in each object. This gives rise to a satisfaction relation
between the universe and the set of wffs.
Many natural language terms are highly context-dependent. For example,
the word “tall” in “a tall basketball player” has a quite different meaning than
it has in “a tall child”. To model context-dependency, we associate a context
with each FDL. The context determines the domain of values of each attribute
and assigns an appropriate meaning to each linguistic term. Formally, given
an FDT (
U,A
), a context associated with an FDL is a function,
ct
,thatmaps
each linguistic term
l
i
∈L
i
to
ct
(
l
i
)
−→
˜
m
,where
V
i
is the
domain of values of attribute
f
i
. We assume each FDT has a fixed context.
Each linguistic term is interpreted as a fuzzy subset of attribute values, so
an object may satisfy an atomic formula in FDL to some degree. Thus, the
satisfaction between data records and wffs is a quantitative relation.
The semantics of FDL depend on how the fuzzy sets in the FDT and
FDL contexts are interpreted. A fuzzy set can be interpreted disjunctively or
conjunctively, and the difference between disjunctive and conjunctive inter-
pretations corresponds to the bipolar representation of possibilistic logic [1].
∈
P
(
V
i
)for1
≤
i
≤
