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of the frame problem. Cirsumscription allows us to declare that the extensions of
certain predicates are to be minimized. For example, consider a formula set Γ
from which the formula P(
A
) can be deduced, but from which we cannot show
either P(
x
) or ¬P(
x
) for any
x
other than
A
. It follows from the circumscription of
Γ minimizing
P
, writted CIRC[Γ ; P
], that P(
x
) is false for any
x
unless Γ
demands that it is true. So, ¬P(
].
The way to apply circumscription to the frame problem is to minimize the
predicate Affects (McCarthy calls the predicate Ab rather than Affects). However,
minimizing Affects in a naïve way will yield counter-intuitive results.
According to the viewpoint of McCarthy, Hayes and Sandewall,
formalization of the common sense law of inertia is the pivot for a nonmonotonic
solution to the frame problem. Here the common sense law of inertia states that
inertia is normal and change is exceptional. One component of this common
sense law is the following default rule: normally, given any action (or event type)
and any fluent, the action does not affect the fluent.
Consider the following universal frame axiom:
B
) will follow from CIRC[Γ ; P
F 1 : [Holds(
f
, Result(
a, s
)) ↔ Holds(
f, s
)] â ¬ Affects(
a, f, s
)
(2.30)
It says that the fluents that hold after an action takes place are the same as those
that held beforehand, except for those the action affects. This leaves us the tricky
job of specifying exactly which fluents are not affected by exactly which actions,
which is the essence of the frame problem.
If we simply replace the Affects predicate by the predicate Ab proposed by
McCarthy, then we will get axioms of the following form:
F 2 : [Holds(
f
, Result(
a, s
)) ↔ Holds(
f, s
)] â ¬ Ab(
a, f, s
)
(2.31)
Let Σ be the conjunction of effect axioms, domain constraints and observation
sertences. Theobvious way to augment Σ with the common sense lae of inertia is
to conjoint it with (F 2 ) and then circumscribe it, minimizing Ab and allowing
Holds to vary. In other words, we consider CIRC[Σ∧(F 2 ); Ab ; Holds].
The approach that minimising Ab and allowing Holds to vary seems a sound
approach to solve the frame problem. However, as McDermott and Hanks
showed in 1968, this approach fails to generate the conclusions we require even
with extremely straightforward examples. They distilled the essence of the
difficulty into a simple example, the so-called Yale shooting problem.
In the Yale shooting problem, someone is killed by a gunshot. The
formalization is composed of:
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