Information Technology Reference
In-Depth Information
Here, a theory T(KB) is complete if either every ground atom in the language or
its negation is in the theory. The basic idea of the CWA is that everything about
the world is known (i.e., the world is closed); therefore, if a ground atom P can
not be proved according to the theory, then P will be considered to be negative.
The CWA completes the theory by including the negation of a ground atom in
the completed theory whenever that ground atom does not logically follows from
KB.
One of the important applications of the CWA is to complete the database
system. For example, let KB be the following database which contains
information about contiguities of countries:
Neighbor(China, Russia).
Neighbor(China, Mongolia).
∀
x
∀
y
(Neighbor(
x, y
)↔ Neighbor(
y, x
))
Then, it is obvious that T(KB) is incomplete since neither Neighbor(Russia,
Mongolia) nor ¬Neighbor(Russia, Mongolia) can be logically inferred from KB.
According to the CWA, the database KB can be completed by adding the
assertion ¬Neighbor(Russia, Mongolia) into it. It is obvious that the CWA is
nonmonotonic because the set of augmented beliefs would shrink if we added a
new positive ground literal to KB.
Let KBasm be the set of all of the assertions added into KB during the
completing process. According to the CWA, it is obvious that for any ground
atom P:
¬P∈KB
asm
if and only if P∉T(KB)
For example, with respect to the database KB presented in the previous example,
we have KBasm = {¬neighbor(Russia, Mongolia)}.
Let CWA(KB)be the CWA-augmented theory,i.e., CWA(KB)=T(KB∪KBasm).
It is obvious that CWA(KB) is more powerful compared with T(KB), since many
results that can not be deduced from KB can now be derived from KB∪ KBasm.
The augmented theory CWA(KB) might be inconsistent. For example, let
KB={P(A)∨P(B)}, then it is KBasm={¬P(A)¬P(B)} since neither P(A) nor
P(B) can be derived from KB, therefore the set KB∪KBasm is inconsistent.
Inconsistency of the CWA-augmented theory is an important problem that needs
to be solved.
Theorem
2.1
CWA(KB)
is
consistent
if
and
only
if,
for
every
