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In-Depth Information
1.
W
(
O;D
)
;
2. either
Initially
(
:f
ab
)or
:Initially
(
:f
ab
), for each abnormality flu-
ent
f
ab
.
Then
E
=
Th
(
F
) is called a potential extension of
(
O;D
)
. Notice that po-
tential extensions may be inconsistent, e.g., if
Initially
(
:f
ab
)
2 F
although
W
(
O;D
)
necessitates that
f
ab
be true in any
s
which satises
Result
([ ]
;s
).
Given a potential extension
E
=
Th
(
F
), we call
induced by E
any strict
ordering
D
that extends
<
D
such that
f
ab
D
f
ab
0
whenever
Initially
(
:f
ab
)
2 F
and
:Initially
(
:f
ab
)
2 F
Induced orderings will be used below to verify the constituent properties for
potential extensions being prioritized extensions. It is easy to verify that the
standard extensions of the default theory (
W
(
O;D
)
;D
D
) are always potential
extensions.
Lemma 3.6.1.
Let
(
O;D
)
=(
W
(
O;D
)
;D
D
;<
D
)
be the axiomatization of
some qualication scenario, then each (standard) extension of
(
W
(
O;D
)
;D
D
)
is a potential extension of
(
O;D
)
.
Proof.
Let
E
be an extension of (
D
D
;W
(
O;D
)
), and let
1.
Γ
0
=
W
(
O;D
)
;
2.
Γ
1
=
Th
(
Γ
0
)
[f!
:
:
!
!
2 D
D
; :! 62 Eg
; and
3.
Γ
2
=
Th
(
Γ
1
).
Since all possibly applicable defaults in
D
D
have been applied to com-
pute
Γ
1
and since
E
is extension, we know that
Γ
2
=
E
. By construc-
tion,
Γ
2
, hence
E
, is subset of some potential extension. To see, then,
why it equals a potential extension, observe rst that
E
=
Th
(
E
) and
W
(
O;D
)
2 E
. It remains to verify that for every
f
ab
2F
ab
,
E
includes
either
Initially
(
:f
ab
)or
Initially
(
:f
ab
). Let
f
ab
2F
ab
be an abnormal-
ity fluent. From
:
Initially
(
:f
ab
)
Initially
(
:f
ab
)
2 D
D
and the construction of
Γ
1
,we
know that either
Initially
(
:f
ab
)
2 Γ
1
, hence
Initially
(
:f
ab
)
2 E
, or else
:Initially
(
:f
ab
)
2 E
.
Qed.
The notion of correspondence between interpretations for qualication
scenarios and prioritized extensions generalizes to potential extension in
the obvious way|a potential extension
E
=
Th
(
F
) and an interpretation
(
;Res
) correspond i the condition (3.14) holds for all
f
ab
2F
ab
. No-
tice that each interpretation has a unique corresponding potential extension,
whereas there may be multiple interpretations corresponding to a single po-
tential extension. For the latter does not necessarily x all states resulting
from the performance of action sequences. Notice further that whenever
E
is
consistent then there exists a corresponding interpretation which is a model
of the qualication scenario at hand. This is granted by Corollary 3.6.1, for
if
E
is consistent it admits a (classical) model
|which then corresponds
to some model of the scenario.
