Information Technology Reference
In-Depth Information
Denition 1.2.8.
Let
(
O; D
)
be a basic action scenario. 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 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
)
.
The second component of an interpretation may be depicted as a tree
(of innite depth) whose root node contains the initial state,
Res
([ ]), and
whose branches each characterize the supposed evolution of the world under
a particular sequence of actions; see Fig. 1.1.
Interpretations always tell us the exact result of performing any possible
action sequence. It is therefore straightforward to determine whether an ob-
servation is true with regard to a particular interpretation: First of all, it can
be true only if the state is dened which results from performing the sequence
of actions in question. If, moreover, the fluent formula in question is true in
that state, then the observation itself is true.
9
Denition 1.2.9.
Let
(
;Res
)
be an interpretation for a basic action sce-
nario
(
O; D
)
. An observation F
after [
a
1
;:::;a
n
]
(n
0
)is
true
in Res
i Res
([
a
1
;:::;a
n
])
is dened and F is true in Res
([
a
1
;:::;a
n
])
.
For the purpose of illustration, the reader may verify that the two obser-
vations (1.4) from above are true in the interpretation depicted in Fig. 1.1,
but not the observation
up
(
s
1
) after [
toggle
(
s
1
)
;
switch-one-up
], nor the
observation
:
light
after [
toggle
(
s
2
)
;
switch-one-up
] because the result
of the latter action sequence is undened.
Among all possible interpretations for the underlying action domain we
are especially interested in those which satisfy all observations of a specic
scenario. As indicated, these are called the models of the scenario. Models
help us dene what can be concluded from a scenario description, namely,
any observation which is true in
all
models of a domain.
Denition 1.2.10.
Let
(
O; D
)
be a basic action scenario. A
model
of this
scenario is an interpretation
(
;Res
)
such that each o 2O is true in Res.
An observation o is
entailed
, written Oj
=
D
o,i o is true in all models
of
(
O; D
)
.
This completes the introduction to our basic action theory. Now that we
have put together all necessary formal concepts, let us give a fully formalized
9
An observation
F
after
a
is considered false in an interpretation
Res
when-
ever
Res
(
a
) is undened (the alternative would be to consider it undened,
too), because we take the observation as implicitly asserting that the action
sequence
a
be executable.
