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2.11 Frame Problem
The frame problem, first described by McCarthy and Hayes, arises when we
attempt to describe the effects of actions or events using logic. Using classical
logic, if we describe what changes when a particular kind of action is performed
or a particular kind of event occurs, we also have to describe what doea not
change. Otherwise, we cannot use the description to draw any useful conclusions.
When we use the apparatus of first-order logic to describe the effects of
actions, the description of what does not change is considerably large than the
description of what does change. However, when we describe the effects of
actions, we should be able to concentrate on what changes, and be able to take
what does not change for granted. Therefore, the frame problem is the problem of
constructing a formal framework that enables us to do just this (Shanahan 1997).
2.11.1
Frame Axiom
Firstly let's consider the Block World example and represent it with the situation
calculus. Let Σ be the conjunction of the following formulas of the initial
situation S
0
.
Holds(On(C,Table), S
0
)
Holds(On(B,C), S
0
)
Holds(On(A,B), S
0
)
Holds(On(D,Table), S
0
)
Holds(Clear(A), S
0
)
Holds(Clear(D), S
0
)
Holds(Clear(Table), S
0
)
Suppose there is a single action Move and this action is described by the
following effect axioms:
Holds(On(
x
,
y
), Result(Move(
x, y
),
s
))
â
Holds(Clear(
x
),
s
) ∧
Holds(Clear(
y
),
s
) ∧
x
≠
y
∧
x
≠Table
Holds(Clear(
z
), Result(Move(
x, y
),
s
))
â
Holds(Clear(
x
),
s
) ∧
Holds(Clear(
y
),
s
) ∧ Holds(On(
x,z
),
s
) ∧
x
≠
y
∧
y
≠
z
