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a) Res ( a ) is dened if and only if Res ([ a 1 ;:::;a k− 1 ]) is dened;
( Res ([ a 1 ;:::;a k− 1 ]) ;a k ) is not empty; and no F disq ( a k ) 2Q
exists such that F is true in Res ([ a 1 ;:::;a k− 1 ]) .
b) Res ( a ) 2 ( Res ([ a 1 ;:::;a k− 1 ]) ;a k ) .
c) a 2 Ab if and only if Res ([ a 1 ;:::;a k− 1 ]) is dened and there is
some F disq ( a k ) such that F is true in Res ([ a 1 ;:::;a k− 1 ]) .
Put in words, whenever Res ([ a 1 ;:::;a k− 1 ]) entails the antecedent of a dis-
qualifying condition for action a k , then Res ([ a 1 ;:::;a k ]) is not dened and
Ab includes the sequence [ a 1 ;:::;a k ].
The appropriate preference criterion and entailment relation can then be
dened straightforwardly on the basis of comparing the additional compo-
nents, Ab . Informally speaking, the less action sequences are declared dis-
qualied by a model the less abnormal the latter. Entailment is then decided
on the basis of least abnormal models, which thus are the preferred ones.
Denition 3.2.4. Let ( O; D ) be a ramication scenario with transition
model and Q a set of disqualifying conditions. If I =( ;Res ; Ab ) and
I 0 =( ;Res 0 ; Ab 0 ) are interpretations with abnormalities for ( O; D ) , then I
is less abnormal than I 0 , written I I 0 ,i Ab Ab 0 .A model with abnor-
malities of ( O; D ) is an interpretation ( ;Res ; Ab ) such that each o 2O
is true in Res. A model is preferred i there is no other model which is less
abnormal. An observation o is entailed i it is true in all preferred models
with abnormalities of ( O; D ) .
Let us see how this account of abnormal action disqualications solves
our initial example.
Example 3.2.1. Let D be the ramication domain consisting of entity pt ,
fluent names
runs 0
in 1 , and action name
and
accompanied by
ignite
the action law
g . Furthermore, let
in ( x ) disq ( ignite ) be a disqualifying condition. Suppose be the tran-
sition model of D , and let O 1 consist of the single observation
ignite
transforms f:
runs
g into f
runs
: runs
after []
Two models M 1 =( ;Res 1 ; Ab 1 ) and M 2 =( ;Res 2 ; Ab 2 ) exist for
the scenario ( O 1 ; D ), namely, where Res 1 ([ ]) = f:
) g and
Res 2 ([ ]) = f: runs ; in ( pt ) g . Since the antecedent of the instance fx 7! pt g
of the underlying disqualifying condition,
; :
(
runs
in
pt
), is true in
Res 2 ([ ]), we have Ab 2 = f [ ignite ] g . In contrast, the rst model entails no
abnormal disqualication, that is, Ab 1 = fg . Hence, M 1 M 2 . Model M 1
being the unique preferred one, we see that our scenario entails the observa-
tion runs after [ ignite ].
Let, on the other hand, O 2 consist of the two observations
( x ) disq (
in
ignite
: runs
after
[]
in ( pt )
after
[]
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