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Example 2.4 Suppose W = {
bird(tweety)
,
x(ostrich(x) ã ¬ flies(x))
}, and
bird
(
x
)
:
M
flies
(
x
)
D = {
}.
flies
(
x
)
Then the formula flies(tweety) can be deduced from the default theory. However,
if we add the knowledge ostrich(tweety) into W, then flies(tweety) can not be
deduced any more.
Example 2.5 Suppose W = {
feathers(tweety)
}, and
feathers
(
x
)
:
M
bird
(
x
)
bird
(
x
)
:
M
flies
(
x
)
D = {
,
}.
bird
(
x
)
flies
(
x
)
Then the formula flies(tweety) can be deduced from the default theory. However,
it can not be deduced any more if we add the following knowledge into W:
ostrich(tweety)
,
x(ostrich(x) ã ¬
flies(x))
x(ostrich(x) ã feathers(x)).
Definition 2.8
Let =<D, W> be a closed default theory. Γ is an operator
defined w.r.t. such that, for any set S of closed formulas, Γ (S) is the smallest set
satisfying the following three properties:
(1) W⊆ Γ(S);
(2) Γ(S) is deductively closed, i.e., Th(Γ(S))= Γ(S);
α
:
M
β
,...,
M
β
w
(3) For any default rule
contained in D: if
α ∈Γ(S)
1
m
¬
¬
and
, …,
∉S, then it must be w∈Γ(S).
1
m
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