Information Technology Reference
In-Depth Information
Weak Negative Similarity Relation
-
B
ʱ
→
C
ʱ
for
∅
=
C
ↆ
B
ↆA
.
→
B
b∈B
v∈V
b
(
b,v
))
,
B
1
∅
-
i
¬
(
b,v
)
∧
(
i
→¬
=
∅
.
-
∅
↥
.
Weak Complementarity Relation
-
B
ʱ
→
C
ʱ
for
∅
=
C
ↆ
B
ↆA
.
→
B
b∈B
v∈V
b
(
b,v
))
,
B
-
i
¬
(
b,v
)
ₔ
1
∅
(
i
→
∅
=
.
-
∅
↥
.
We sketch briefly how the three axioms of negative similarity relation given above
will give the corresponding completeness theorem. One can proceed similarly
for the other relations. Let
R
B
be the canonical relation corresponding to the
modal operator
B
for the negative similarity relation (cf. (1) of page 125). Then
corresponding to the Proposition 4, we will obtain the following:
1.
R
B
ↆ
R
C
for
C
ↆ
B
(using first axiom).
2.
ʓ
a
∩
ʔ
a
∅
if and only if (
ʓ,ʔ
)
∈
R
a
(using second axiom and third axiom
=
for
B
=
∅
).
3. If (
ʓ,ʔ
)
∈
R
B
and
ʓ
a
∩
ʔ
a
=
∅
,then(
ʓ,ʔ
)
∈
R
B∪{a}
(using third axiom).
As a consequence of these facts, we will obtain
R
B
to be the negative similarity
relation induced from the canonical information system (cf. Definition 4) and
consequently we get the Truth Lemma, and hence completeness theorem.
6 Conclusions
In NISs, object-attribute pair is mapped to a set of attribute values. This repre-
sents uncertainty, in the sense that we know some possible attribute values that
an object can take for an attribute, but we do not know exactly which ones. In
this kind of situation, one could get information which reduces or removes this
uncertainty. In the line of work in [10,11], a natural question would be about the
proposal of an update logic for NISs, which can capture such an information and
its effect on the approximations of concepts. The current work has completed the
first step in this direction and given the static part of such an update logic. It
remains to extend the language of LNIS to obtain the update logic for NISs.
