Information Technology Reference
In-Depth Information
Definition 3
M
,w
|
=(
a, v
)
, if and only if v
∈
F
(
w,a
)
.
M
,w
|
=
ʱ, if and only if w
∈
V
(
ʱ
)
, for ʱ
∈
PV.
Ind
B
,
1
B
ʱ, if and only if for all w
∈
W with
(
w,w
)
,w
|
M
,w
|
=
∈
M
=
ʱ.
2
B
ʱ, if and only if for all w
∈
W with
(
w,w
)
Sim
B
,
,w
|
M
,w
|
=
∈
M
=
ʱ.
In
B
,
3
B
ʱ, if and only if for all w
∈
W with
(
w,w
)
,w
|
M
,w
|
=
∈
M
=
ʱ.
Note that, unlike the logic in [9], the semantics of LNIS is directly based on
information systems. The extension of a wff
ʱ
relative to a model
M
, denoted
as [[
ʱ
]]
M
, is given by the set
{
w
:
M
,w
|
=
ʱ
}
.Awff
ʱ
is said to be valid in a
model
M
with domain
U
if [[
ʱ
]]
M
=
U
.Awff
ʱ
is called valid, denoted as
|
=
ʱ
,
if
ʱ
is valid in all models.
The following proposition shows that the operators
3
B
capture
the lower approximations relative to the attribute set
B
with respect to indis-
cernibility, similarity and inclusion relations respectively, while
1
B
,
2
B
and
♦
1
B
,
♦
2
B
and
♦
3
B
respectively capture the corresponding upper approximations.
,
a∈A
V
a
,F
)
.
Proposition 1.
Let
M
:= (
S
,V
)
be a model, where
S
:= (
U,
A
Then the following hold.
1
B
ʱ
]]
M
=[
ʱ
]]
M
Ind
B
,
1
B
ʱ
]]
M
= [[
ʱ
]]
M
Ind
B
.
1.
[[
[[
♦
2.
[[
2
B
ʱ
]]
M
=[
ʱ
]]
M
Sim
B
,
♦
2
B
ʱ
]]
M
= [[
ʱ
]]
M
Sim
B
.
[[
3
3
3.
[[
B
ʱ
]]
M
=[
ʱ
]]
M
In
B
,
[[
♦
B
ʱ
]]
M
= [[
ʱ
]]
M
In
B
.
The presence of descriptors in the language helps LNIS to talk about the at-
tributes, attribute-values of the objects, and its effect on the approximation
operators. For instance, the wff (
a, v
)
∧
u∈V
a
\{v}
¬
2
B
p
represents a
decision rule according to which if an object
x
takes the value
v
for the at-
tribute
a
,then
x
is in the lower approximation of the set represented by
p
with respect to similarity relation corresponding to the attribute set
B
.The
valid wff (
b,v
)
(
a, u
)
→
2
B∪{b}
2
∧
ʱ
→
B
((
b,v
)
→
ʱ
) corresponds to the fact that
Y
∩
X
Sim
B∪{b}
ↆ
X
∪
Y
c
Sim
B
,where
Y
=
{
y
:
v
∈
F
(
y,b
)
}
,and
Y
c
is the
set-theoretic complement of
Y
.
3 Axiomatic System
We now present an axiomatic system for LNIS, and prove the corresponding
soundness and completeness theorems.
We recall that
D
denotes the set of all descriptors. Consider the wffs which are
a conjunction of literals from the set
ʱ∈D
{
ʱ,
¬
ʱ
}
, and which contain precisely
one of (
a, v
), or
¬
(
a, v
)foreach(
a, v
)
∈D
.Let
ʘ
be the set of all such wffs.
,
a∈A
V
a
,F
). Then it is not
Let
M
:= (
S
,V
) be a model, where
S
:= (
W,
A
dicult to obtain the following.
