Information Technology Reference
In-Depth Information
Proof.
Recalling the definition, W,w
O
if and only if
(1) W,w
B
, and
(2) w'∈W for any w' which satisfys W,w'
.
Now suppose W,w
O
. Then we can construct a set
| W,w
B
ϕ
j
,1
∧
···
∧
B
ϕ
j,mj
∧
¬
B
ϕ
j
,1
∧
···
∧
¬
B
ϕ
j,mj
}. It is easy to demonstrate that
J
={
j
J
has
the
O
-property; furthermore, according to (1) we will get W ⊆
∪
j∈J
◇
jj
, and
according to (2) we will get
∪
j∈J
◇
jj
⊆W. Therefore we have W =
∪
j∈J
◇
jj
.
The other direction can be similarly demonstrated.
Corollary 2.3
The number of stable expansions of the basic formula
ɂ
is
equivalent with the number of sets which have the
O
-property and are subsets of
{1,
, k}.
Corollary 2.4
···
has exactly one stable expansion if and only
if there is only one subset of {1, …, k} that has the
The basic formula
ɂ
O
-property.
In the following are some examples.
Example
Suppose
Then
2.11
is
B
p
.
can
be
transformed
as
B
). Therefore, there is no stable expansion for
, since
◇
(q
∨
¬q), i.e. 2
L
, is the only maximal set for examine.
p
∧
¬
B
(
r
∧
¬
r
)
∧
(
q
∨
¬
q
Example
2.12
Suppose
is
p.
Then
can
be
transformed
as
B
(
q
∨
¬
q
)
∧
¬
B
(
r
∧
¬
r
)
∧
p. Therefore, there is just one stable expansion since
)=2
L
and
◇
p
⊄
◇
(r
∧
¬r) =Ø.
◇
p
⊆
◇
(
q
∨
¬
q
