Information Technology Reference
In-Depth Information
Here
(
s
1
) is the eect whose appearance triggers indirect eect
light
in the context dened by the atomic fluent formula
up
(
s
2
). Of course, the
analogue is true as well in this particular domain, viz.
up
up
(
s
2
) causes
if
up
(
s
1
)
(2.3)
light
Yet this symmetry is not always true, as we will see later. In any case, the
two causal relationships express dierent things: The former applies whenever
up
(
s
1
) became true with
(
s
2
) already being true, as opposed to the latter,
up
which applies whenever
s
1
) holds.
6
Causal relationships shall be employed to generate additional, indirect ef-
fects of actions after having obtained the direct eects through the application
of an action law. Formally, causal relationships operate on state-eect pairs
(
S; E
) where
S
is some current `intermediate' state, a preliminary successor,
for instance, and
E
contains all direct and indirect eects that have been
generated so far. As an example, recall that
S
0
=
f
up
(
s
1
)
;
up
(
s
2
)
; :
light
g
is the preliminary successor of toggling the rst switch in the state depicted
in Fig. 2.3. State
S
0
(
s
2
) became true while
(
up
up
s
1
)
g
. The
causal relationship (2.2) above is applicable to (
S
0
;E
) on account of both
up
(
s
1
) being part of eect
E
and
up
(
s
2
), the context, being true in
S
0
.
This relationship implies that
S
0
is obtained through direct eect
E
=
f
(
up
should be modied so as to account for the
indirect eect
g
.
In addition, we augment
E
by our new eect,
light
. Altogether the re-
sult of applying the causal relationship to (
S
0
; f
up
(
s
1
)
g
) is the state-eect
pair (
S
00
; f
up
(
s
1
)
;
light
g
). Notice that our second causal relationship, (2.3),
should not be applicable to (
S
0
;E
) despite
, which yields the state
S
00
=
f
(
s
1
)
;
(
s
2
)
;
light
up
up
light
s
1
) being true in
S
0
, because
up
(
eect
up
(
s
2
) is not contained in
E
.
The reason for maintaining the second component,
E
, is that identical
intermediate states (such as
S
0
) can often be reached by dierent eects, each
of which may require diverse, sometimes opposite treatment, as the following
example illustrates.
Example 2.4.1.
Suppose two switches
s
1
;
s
2
be tightly coupled by a spring
so that they are always in the same position; see Fig. 2.5. This is reflected by
the state constraint
s
2
). As a consequence, closing or opening
either switch has the indirect eect that the other switch closes or opens,
respectively, as well to release the tension of the spring. This is expressed by
these four causal relationships:
up
(
s
1
)
up
(
up
(
s
1
)
causes
up
(
s
2
) f
>
(
s
2
)
causes
(
s
1
) f
>
up
up
(2.4)
:
up
(
s
1
)
causes
:
up
(
s
2
) f
>
:
up
(
s
2
)
causes
:
up
(
s
1
) f
>
6
If both switches get closed as direct or indirect eects, then of course either
relationship is applicabl and the result is the same, namely, light.
