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any other, concrete cause. Then the unknown cause is used only as a last
resort when it comes to explaining an abnormality.
On the formal side, degrees of abnormality are represented by a par-
tial ordering among the `abnormality' fluents. If, for instance, we spec-
ify that
<
low-battery
, then
shall always
tank-empty
low-battery
be minimized|initially|with higher priority than
. Accord-
ingly, the concept of unknown causes is adequately modeled by dening
f
ab
<
mysterious
(
a
) for all other `abnormality' fluents
f
ab
. Property
mysterious
(
a
) is thus assigned the highest degree of abnormality. Being
a partial ordering, the comparison relation
<
may be indierent regarding
some pairs of fluents, in which case no preference is made for either of them.
For a formal introduction to partial orderings and related concepts see An-
notation 3.2. In the following, we extend our solution to the Qualication
Problem to the eect that dierent degrees of abnormality are supported.
tank-empty
A
binary relation R
on some set
A
is a subset of the Cartesian product
A A
,
that is,
R
is a set of pairs (
a; b
) where
a; b 2 A
.If (
a; b
)
2 R
, then this is also
written
aRb
. A binary relation
R
may obey the following properties:
8a 2 A: : aRa
(irreflexive)
8a; b 2 A:
(
aRb ^ bRa a
=
b
)
(antisymmetric)
8a; b; c 2 A:
(
aRb ^ bRc aRc
)
(transitive)
If it does, then the relation
R
is a
partial ordering
on
A
. If in addition
R
satises
8a; b 2 A:
(
aRb _ bRa
), then
R
is
strict
. A strict ordering
R
0
is an
extension
of a partial ordering
R
i
R
0
R
, that is, whenever
aRb
then also
aR
0
b
. Let, for example,
A
be
f
tank-empty
;
low-battery
;
engine-problem
;
in
(
pt
)
;
mysterious
(
ignite
)
g
and suppose a partial ordering
<
be given by
tank-empty
<
low-battery
<
in
(
pt
)
<
mysterious
(
ignite
)
tank-empty
<
engine-problem
<
mysterious
(
ignite
)
Let
be
<
augmented by
low-battery
in
(
pt
)
engine-problem
, then
is one out of three possible strict orderings extending
<
.
Annotation 3.2.
Orderings.
Denition 3.5.1.
A
qualication domain
D is a plain qualication domain
augmented by a partial ordering < on the set of abnormality fluents. Accord-
ingly, a
qualication scenario (
O; D
)
consists of a set O of observations and
a qualication domain D.
Suppose M
=(
;Res
)
is a model of
(
O; D
)
. Then M is
preferred
i
there is a strict ordering which extends < and such that for all mod-
