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3.6.2 Introducing Nonmonotonicity
The entailment relation of an action theory being a nonmonotonic one has
been shown a fundamental characteristics of the Qualication Problem. Any
axiomatization suitable for qualication domains and scenarios must there-
fore be based on some nonmonotonic extension to classical logic. Speaking less
abstractly, this going beyond classical logic is necessary in order that abnor-
malities can be minimized, i.e., assumed away by default. We will meet this
requirement by embedding our current Fluent Calculus-based axiomatization
into a so-called default theory. Hence the general nonmonotonic framework
to be employed is
Default Logic
, or rather, to be more precise, a conceptual
extension of the original approach called
Prioritized Default Logic
. The latter
is vital for reflecting possible degrees of abnormality when minimizing. For
a formal introduction to both Default Logic and its prioritized variant see
Annotation 3.3.
Given a qualication scenario (
O; D
), on the basis of the Fluent Calculus-
axiomatization
W
(
O;D
)
we construct a prioritized default theory as follows.
The classical logic formulas
W
(
O;D
)
constitute the background knowledge
(which, by the way, resolves the mystery of why this denotation has been
chosen). The various assumptions of `normality' are formalized as default
rules. For each abnormality fluent
f
ab
2F
ab
, let
f
ab
denote the default rule
:
8s
[
Result
([ ]
;s
)
:Holds
(
f
ab
;s
)]
8s
[
Result
([ ]
;s
)
:Holds
(
f
ab
;s
)]
(3.12)
which is used to express the default assumptions that
f
ab
be false initially.
That is to say, as long as it is consistent to assume that
f
ab
does not hold in
the initial state, we do make this assumption. Let
D
D
denote the set of all de-
fault rules thus obtained from domain
D
. Furthermore, let a partial ordering
<
D
be dened on
D
D
so that
f
ab
<
D
f
ab
0
whenever
f
ab
<f
ab
0
according
to the partial ordering dened on the set
F
ab
of abnormality fluents.
This completes the axiomatization of qualication scenarios (
O; D
)as
prioritized default theories
(
O;D
)
=(
W
(
O;D
)
;D
D
;<
D
). Before we enter the
proof of correctness, let us check on it by an example.
Example 3.6.2.
Let
D
be the qualication domain of Example 3.2.1. Then
D
D
consists of the default rules
in(pot)
,
mysterious(ignite)
, and
disq
(ignite)
:
:
8s
[
Result
([ ]
;s
)
:Holds
(
in
(
pt
)
;s
)]
8s
[
Result
([ ]
;s
)
:Holds
(
(
)
;s
)]
in
pt
:
8s
[
Result
([ ]
;s
)
:Holds
(
)
;s
)]
8s
[
Result
([ ]
;s
)
:Holds
(
mysterious
(
ignite
)
;s
)]
:
8s
[
Result
([ ]
;s
)
:Holds
(
disq
(
(
mysterious
ignite
(3.13)
)
;s
)]
8s
[
Result
([ ]
;s
)
:Holds
(
disq
(
ignite
)
;s
)]
ignite
where
in(pot)
<
D
mysterious(ignite)
due to
(
)
<
(
).
in
pt
mysterious
ignite
