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2
T
rue
(
;t
)
^ 2
T
rue
(
:
;t
)
^ 38x:
T
rue
(
:
(
x
)
;t
)
ignite
runs
in
2
T
rue
(
;t
+ 1)
(3.16)
runs
where
2
T
rue
(
`; t
) should be read as \at time
t
fluent literal
`
provably
holds" and
3
T
rue
(
`; t
) as \at time
t
fluent literal
`
may or may not hold."
Thus the rst of the two implications states that if it is known that some
action
(
x
) holds, and that possibly
:
heavy
(
x
) holds at that time, then
in
(
x
) holds at time
t
+ 1. Likewise, if
it is known that the action
ignite
occurs at time
t
, that
:
runs
holds, and
that possibly
8x: :
(
x
) occurs at time
t
, that
:
insert
in
in
(
x
) holds at that time, then
runs
holds at time
t
+1.
Observe how regular action preconditions, like
:
, must
provably hold whereas abnormal disqualifying conditions, like
heavy
(
x
) and
9x:
in
(
x
), are assumed away whenever the opposite does not provably hold.
Now, suppose given
2
T
rue
(
:
(
x
) and
:
in
runs
(
)
;
1)
^2
T
rue
(
:
;
1) in conjunction
in
pt
runs
with the action occurrences
2
T
rue
(
;
2).
Then chronological ignorance tells us that
3
T
rue
(
:
heavy
(
pt
)
;
1) holds
since nothing is known about
(
)
;
1)
^ 2
T
rue
(
insert
pt
ignite
T
rue
(
:
heavy
(
pt
)
;
1) itself. Hence, (3.15)
(
x
)
;
2).
12
Thus
the antecedent of the implication (3.16) is false (for
t
= 2) and, consequently,
the second action,
ignite
, is correctly concluded unqualied. Notice that
this being the unique suggested course of events relies on the chronologi-
cal order in which minimization is performed. Otherwise, it could equally
well be concluded that
38x:
T
rue
(
:
implies
2
T
rue
(
(
)
;
2), which gives us
:38x:
T
rue
(
:
in
pt
in
(
x
)
;
2) holds, for, in the rst place,
nothing is known about
8x:
T
rue
(
:
in
(
x
)
;
2) itself. This in turn entails
:3
T
rue
(
:
heavy
(
pt
)
;
1), i.e.,
2
T
rue
(
heavy
(
pt
)
;
1), since (3.15) is logi-
cally equivalent to
in
2
T
rue
(
insert
(
x
)
;t
)
^ 2
T
rue
(
:
in
(
x
)
;t
)
^ 3
T
rue
(
:
in
(
x
)
;t
+1)
2
T
rue
(
heavy
(
x
)
;t
)
This alternative conclusion corresponds to the counter-intuitive model ob-
tained by global minimization of abnormalities (c.f. Section 3.2) but, as in-
dicated, it is not supported by chronological ignorance.
The interesting, albeit informal, reason for chronological ignorance com-
ing to the desired conclusion in this and similar cases is a certain respect of
causality hidden in this method. By minimizing chronologically, one tends to
minimize causes rather than eects, which is the right thing to do, simply
because causes generally precede their eects. On the other hand, the appli-
cability of chronological minimization is known to be intrinsically restricted
to domains and scenarios which do not involve indeterminate information.
This has been shown for a variety of aspects of non-determinism; see, e.g.,
[60, 88, 104]. Informally speaking, the problem reveals whenever indetermi-
nate information provides sucient evidence for an abnormality without, by
virtue of not being deterministic, necessitating it. Putting o abnormalities
for as long as possible then ignores uncertain evidence and, in so doing,
12
As usual in modal logic,
:2
T
rue
(
`; t
) is equivalent to
3
T
rue
(
:`; t
).
