Information Technology Reference
In-Depth Information
on the existence of extensions. Therefore, it is important to study and discuss the
conditions about the existence of extension.
Theorem 2.3
Let E be a set of closed formulas, and let
∆
=<D, W> be a closed
default theory. Define E0 =W and for i>0 it is
E
i+1
=Th(E
i
)∪{
w
|
(α
M
β
1
,…,
M
β
m
→
w
)
∈
D,
α∈E
i
,
¬β
1
,…,¬β
m
∉E },
∞
=
∪
E
=
E
Then E is an extension for ∆ iff
.
i
i
0
With this theorem, the three extensions of Example 2.8.can be examined to be
right.
:
M
¬
A
There is a special default rule
. A natural question about it is that
¬
A
whether the extension of a default theory determined by this default rule is the
same of the corresponding CWA-augmented theory. Answer for this question is
negative. For example, suppose W={P∨Q} and D={
:
M
:
M
¬
Q
¬
P
,
}. Then is
¬
Q
¬
P
obvious that CWA(∆) is inconsitent, but the set { P∨Q, ¬P} and { P∨Q, ¬Q }
are all consistent extensions for ∆.
Example 2.9
Suppose D={
:
MA
A
}W={A, ¬A}. Then the extension for ∆=<D,
¬
W> is E = Th(W).
This example is surprising since the extension for ∆ is inconsistent. In fact,
some conclusions on the inconsistency of extensions have been summed up:
(1) A closed default theory <D, W> has an inconsistent extension if and only if
the formula set W is inconsistent.
