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Its rst component violates
C
s
, which is why rst an acceptable state with
this regard is to be found by means of
R
s
. This can only be achieved by
applying, in either order, the two instances of
:
down
(
x
) causes
up
(
x
)if
>
.
The resulting state-eect pair is
(
f
(
)
; :
g ;
f:
down
(
lhs
)
; :
down
(
rhs
)
;
up
(
lhs
)
;
up
(
rhs
)
g
)
(
)
;
(
)
; :
(
)
; :
up
lhs
down
lhs
up
rhs
down
rhs
stain
No further causal relationship, steady or non-steady, is applicable. The rst
component satises the entire set of state constraints
C
, hence constitutes
the unique successor state.
Since now we nally arrived at a satisfactory solution to the Ramication
Problem, let us extend the formal concepts of action domains, scenarios, and
entailment so as to be able to specify and reason about domains involving
indirect eects of actions.
Denition 2.7.2.
A
ramication domain
D is a 6-tuple
(
E; F; A; L; C; R
)
where
(
E; F; A; L
)
constitutes a basic action domain, C is a set of state
constraints with a designated subset C
s
of steady constraints, and R is a set
of causal relationships with a designated subset R
s
of steady relationships.
The
transition model
of D is a mapping from pairs of an acceptable state
and an action into (possibly empty) sets of states such that S
0
2
(
S; a
)
i
S
0
is successor of S and a.
As solutions to the Ramication Problem are only concerned with the so-
phistication of transition models, all concepts beyond can be adopted without
modication. For the sake of completeness, let us reiterate them.
Denition 2.7.3.
A
ramication scenario
is a pair
(
O; D
)
where D is a
ramication domain and O is a set of observations. An
interpretation
for
(
O; D
)
is a pair
(
;Res
)
where is the transition model of D and Res is
a partial function which maps nite (possibly empty) action sequences to
acceptable states and which satises the following:
1. Res
([ ])
is dened.
2. For any sequence a
=[
a
1
;:::;a
k−
1
;a
k
]
of actions (k>
0
),
a) Res
(
a
)
is dened if and only if Res
([
a
1
;:::;a
k−
1
])
is dened and
(
Res
([
a
1
;:::;a
k−
1
])
;a
k
)
is not empty, and
b) Res
(
a
)
2
(
Res
([
a
1
;:::;a
k−
1
])
;a
k
)
.
A
model
of a ramication scenario
(
O; D
)
is an interpretation
(
;Res
)
such that each o 2O is true in Res, and an observation o is
entailed
,
written Oj
=
D
o, i it is true in all models.
As a small exercise, the reader may consider the ramication domain mod-
eling the electric circuit with the relay of Fig. 2.4, along with the scenario
given by these two observations:
