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(
s
)
^ Caused
(
f
1
;v
1
;s
)
^ ::: ^ Caused
(
f
n
;v
n
;s
)
Caused
(
f; v; s
) (2.48)
where
Caused
(
f; v; s
) should be read as \fluent
f
is caused to take on truth-
value
v
in situation
s
," and where
(
s
) describes properties of situation
s
.
Notice the distinction between a context, i.e.,
, and explicitly occurring
eects
f
1
;:::;f
n
. E.g., a formalization of the electric circuit involving two
switches and a light bulb of Fig. 2.3 would include the specication
Holds
(
up
(
s
2
)
;s
)
^ Caused
(
up
(
s
1
)
; True;s
)
Caused
(
light
; True;s
) (2.49)
along with the action denition
23
:Holds
(
up
(
s
1
)
;s
)
Caused
(
up
(
s
1
)
; True; Do
(
toggle
(
s
1
)
;s
))
(2.50)
The general axiom of persistence used in this context is
:9v: Caused
(
f; v;Do
(
a; s
))
(
Holds
(
f;Do
(
a; s
))
Holds
(
f; s
) )
(2.51)
This axiom is of course useless unless an instance
Caused
(
F; V; Do
(
A; S
))
is provably false whenever it does not follow from the domain-specic ax-
ioms both for direct eects of actions (like in (2.50)) and for causal de-
pendencies (like in (2.49)). This is formally achieved by minimizing the
set of true instances of
Caused
via so-called circumscription [77]. For
instance, suppose a situation
S
0
be specied by
:Holds
(
up
(
s
1
)
;S
0
)
^
Holds
(
up
(
s
2
)
;S
0
)
^:Holds
(
light
;S
0
) , and let
S
1
=
Do
(
toggle
(
s
1
)
;S
0
).
Then the axioms (2.49) and (2.50) entail
Caused
(
up
(
s
1
)
; True;S
1
) , hence
Caused
(
light
; True;S
1
). They do not entail
9v: Caused
(
up
(
s
2
)
;v;S
1
). Mini-
mizing
Caused
therefore allows to conclude
:9v: Caused
(
(
s
2
)
;v;S
1
) and,
up
hence,
Holds
(
up
(
s
2
)
;S
1
) according to axiom (2.51).
An alternative causality-based method to address the Ramication Prob-
lem, which also includes the distinction between context and triggering eect,
is [47].
The causality-oriented approaches cited so far all intrinsically follow the
policy of minimizing change. This amounts to rejecting any potential suc-
cessor state whose distance to the original state is strictly greater than the
distance of another proper successor state. With our electric circuit involving
the light detecting device (Example 2.6.3) we have seen that the applica-
bility of this paradigm is limited. Causal relationships and their successive
application to preliminary successor states constitute the rst approach that
does not follow the principle of minimizing change [113];
24
the notion of
steady state constraints has been introduced in [116]. In [93], an extension
to \Features-and-Fluents" was developed as an alternative to the aforemen-
tioned [92] for categorization-based approaches. The newly proposed action
23
For the sake of simplicity we neglect the concept of action preconditions here.
24
First published as [109]. The approach [71], which also supports non-minimal
successor states, emerged at the same time but proved erroneous in producing
lots of `magic' indirect eects.
