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More precisely, given a default theory <∆, Σ>, the corresponding extensions
can be constructed accordin to the following algorithm:
S' := { }
S := Σ ∪ logical consequences of Σ
While S ≠ S'
S' := S
Choose any φ
1
, φ
2
, φ
3
and τ
1
, …, τ
n
such that
φ
(
x
φ
)
:
φ
(
x
)
1
2
in ∆ and φ
1
(τ
1
…τ
n
)∈S and ¬φ
2
(τ
1
…τ
n
)∉S
(
x
)
3
S := S ∪ φ
3
(τ
1
…τ
n
)
S := S ∪ logical consequences of S
End While
This algorithm will not terminate in many cases, since the default rules will
be applicable aninfinite number of times. However, in such cases each
intermediate S is a subset of some extension, so we can still use the algorithm to
obtain useful information by executing the loop a finite number of times.
Now, we can apply this algorithm to the default theory representing the Yale
shooting problem. We start with the set S = Σ ∪ the logical consequences of Σ.
Then we might choose to add:
¬Ab(Wait, Loaded, Result(Load, S
0
))
The default rule permits this since it is consistent with S. Next we add in the
logical consequences of the new addition: the following is added according to
(F
2
):
Holds(Loaded, Result(Wait, Result(Load, S
0
)))
and correspondingly the following is added according to (Y
2
):
¬Holds(Alive, Result(Shoot, Result(Wait, Result(Load, S0)))).
We would need to iterate the algorithm forever to obtain an extension, but we
have already shown that extensions exist that include the above intended
consequence.
Now for anomalous extensions, which we get simply by making a different
choice when we apply the default rule. Again we start with the set S = Σ ∪ the
