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In-Depth Information
Definition 2.24
A default theory is a pair <∆, Σ>, where ∆ is a set of default rules,
and Σ is a sentence of first-order predicate calculus.
Σ encapsulates certain knowledge of the domain, from which we can draw
deductively valid inferences, while ∆ represents the default knowledge, from
which we draw defeasible conclusions.
The default theory <∆, Σ> considerated here is:
Ê
Ú
:
¬
Ab
(
a
,
f
,
s
)
∆
=
Ë
Û
¬
Ab
(
a
,
f
,
s
)
and Σ is the conjunction of (Y
1
) to (Y
4
) with (F
2
).
Definition 2.25
A set of well-formed formulas is an extension of a default theory
<
∆
,
Σ
> if it is a fixed point of theoperator
Γ
. Here the operator is defined as
follows. If S is a set of well-formed formulas with no free variables, then
Γ
(S) is
the smallest set such that:
(1) Σ ⊆ Γ(S);
(2) if φ is a logical consequence of Γ(S), then φ∈Γ(S); and
(3) if ∆ include the default rule
φ
(
x
φ
)
:
φ
(
x
)
1
2
(
x
)
3
and φ
1
(τ
1
…τ
n
) ∈ Γ(S) and ¬φ
2
(τ
1
…τ
n
)∉ Γ(S), then φ
3
(τ
1
…τ
n
) ∈ Γ(S), where
each τ
i
is a non-variable term.
Each extension of a default theory will represent an acceptable set of beliefs.
The way to construct extensions is natural: start with a set containing just Σ and
its logical consequences, then repeatedly choose an applicable default, add its
consequence, and form the corresponding deductive closure, until nothing more
can be added.
