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, Red), S
0
)
In the real world we know that the color of a block will not be affected by
moving it. However, based on the current representation of
Holds(Color(
x
Σ and
∆, the
following result can not be reached at all:
∆∧Σ |= Holds(Color(A, Red), Result(Move( A, D), S
0
))
In order to get this result, we need to add the following frame axioms into ∆:
Holds(Color(
))
â
Holds(Color(
x, c
), Result(Move(
y, z
),
s
x ,c
),
s
)
))
â
Holds(Color(
Holds(Color(
x, c
1
), Result(Paint(
y, c
2
),
s
x ,c
1
),
s
)
∧
These axioms state that the color of any block x will not be affected by moving it,
and the color of x will not be affected by the painting of any other block.
Similarly, with respect to the action “Paint”, we need to introduce the
following frame axioms, which state that values of the fluent “On” and the fluent
“Clear” will not be affected by the action “Paint”:
Holds(On(
x
≠
y
x, y
), Result(Paint(
z, c
),
s
))
â
Holds(On(
x ,y
),
s
)
)
In general, because most fluents are unaffected by most actions, every time
we add a new fluent we are going to have to add roughly as many new frame
axioms as there are actions in the domain, and every time we add a new action
we are going to have to add roughly as many frame axioms as there are fluents in
the domain. Therefore, if ther are n fluents and m actions for a domain, then the
total number of frame axioms will be of the order of n×m (Shanahan, 1997).
In the following we consider a more succinct approach of expressing the
information in the frame axioms. This approach is remaining in the realm of
classical first-order logic and do not need to modify the situation calculus
significantly. Firstly, note that all frame axioms have a similar form of the
following:
Holds(Clear(
x
), Result(Paint(
y, c
),
s
))
â
Holds(Clear(
x
),
s
) ∧ Π
Where f is a fluent, a is an action, and Π is a conjunction. So, we can use the
following frame axiom:
Holds(
Holds(
f
, Result(
a, s
))
â
Holds(
f, s
f
, Result(
a, s
))
â
Holds(
f, s
) ∧ ¬Affects(
a, f, s
)
