Information Technology Reference
In-Depth Information
Prior to dening what conclusions an action scenario allows, we need to
clarify under which circumstances a particular observation can be said to be
true. This obviously depends on what state supposedly results from executing
the action sequence in question. If that state is xed, then deciding whether
the fluent formula itself holds is straightforward, following the standard in-
terpretation of the logical connectives.
Denition 1.2.7.
Let E and F be sets of entities and fluent names, re-
spectively, and let S be a state. The notion of a closed formula being
true
(resp.
false
)in S is inductively dened as follows:
1. > is true and ? is false in S;
2. a fluent literal ` is true in S i ` 2 S;
3. :F is true in S i F is false in S;
4. F ^ G is true in S i F and G are true in S;
5. F _ G is true in S i either F or G is true in S (or both);
6. F G is true in S i F is false in S or G is true in S (or both);
7. F G is true in S i F and G are true in S, or else F and G are
false in S;
8. 9x: F is true in S i there exists some e 2E such that F fx 7! eg is
true in S;
9. 8x: F is true in S i for each e 2E, F fx 7! eg is true in S.
Here, F fx 7! eg denotes the fluent formula resulting from replacing in F
all free occurrences of variable x by entity e.
As an example consider the formula
9x: :
up
(
x
)
:
light
, which is true
in the state
f:
up
(
s
1
)
;
up
(
s
2
)
; :
light
g
(since
:
light
is true) and also in
f
up
(
s
1
)
;
up
(
s
2
)
;
light
g
(since
9x: :
up
(
x
) is false), but the formula is false
in, e.g.,
f
g
.
As indicated, the observations that describe a scenario usually provide
only incomplete information as to the entire state of aairs. This is espe-
cially true if non-deterministic actions are considered because then complete
information means to know the actual result of any possible sequence of non-
deterministic actions. Thus the normal case is that there is more than just one
unique state of aairs that ts a scenario description. Following a standard
terminology in logic, we call any possible state of aairs an
interpretation
,
and if the latter matches a scenario description it is called a
model
thereof.
We have already seen examples involving reasoning about hypothetical
developments of the world. An interpretation therefore must not just tell
us exactly what happens during the execution of one particular sequence of
actions. Rather it needs to provide this information as to any possible course
of events. Of course we assume the world always evolves according to the
underlying action laws. That is to say, whenever some state
S
results from
performing some action sequence, and some further action
a
is executed,
then the result should be a successor of
S
and
a
.
up
(
s
1
)
; :
up
(
s
2
)
;
light
