Information Technology Reference
In-Depth Information
Minimal model is nonmonotonic. The following example reflect the property
of minimal model:
p
M
¬q
p
∨
q
M
¬p
∨
¬q
p
、
q
、
p
∨
q
M
p
∧
q
Definition 2.15
Let Z = {z
1
, z
2
, …, z
n
} be all the propositions occurring in a
formula A. Then, a satisfying truth assignment P is called a
X
Z-
minimal
satisfying assignment of A if and only if there is no other satisfying truth
assignment P' of A such that P
X
Z
P'. Where, P
X
Z
P' if and only if P'(z)=l for
any proposition z which holds z
∈
Z and P(z)=l.
Definition 2.16
Let P = {p
1
, p
2
, …, p
n
} be all the propositions occurring in a
formula A. Then, a formula
ϕ
is entailed by the propositional circumscription of
P in A, written as A
P
ϕ
, if and only if
ϕ
is true with respect to any
X
Z-
minimal
satisfying assignment of A.
ż
The propositional circumscription CIRC(A, P) is defined as the following
formual:
∀
P'(A(P')
∧
P'
P))
(P
P')
A(P)
∧
(2.5)
Where A(P') is the result of replacing all occurrence of P in A by P'. If we use
P'
X
P to replace P'
Pthen CIRC(A,P) can also be rewrited as:
A(P)
∧
¬
∃
P'(A(P')
∧
P'
X
P)
(2.6)
Therefore, logical inferences in the propositional circumscription can be
represented as schemas of the form A
P
ϕ
or CIRC(A,P)
ϕ
. The following
theorem on the soundness and completeness has been proved:
Theorem 2.4
A
ũ
ż
p
ϕ
.
In the following we advance the idea of propositional circumscription into
predicate circumscription.
p
ϕ
if and only if A
