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Ontologies for time
In ENRAC 11.7 (98054), Erik wrote:
| | I had actually asked for something more concrete: a scenario that can't
be expressed, or a scenario query that requires inordinately long
completion time, for example. Is it possible to be more specific about
in what sense it "will be more difficult" to represent time points that
stand between intervals?
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As I put in my former arguments, for many practical cases (e.g.,
in cognitive robotics perspectives), the Dividing Instant Problem
may be just simply dealt with by means of taking point-based
intervals as semi-open. However, from the perspectives of philosophy,
or commonsense reasoning, this approach seems arbitrary in deciding
which end the intervals should include/exclude. The choice has to be
artificial and hence unjustified. For example, there is nothing to
justify the reason why the light MUST BE off (or on) at the switching
point, although, practically, one may just artificially take a choice
which will make the robot work well in most cases (But who can
prove, even practically, there is no problem for ALL cases?).
As for the sense of "it will be difficult to represent time points in
an appropriate structure so that they can stand between intervals
conveniently", what I mean here is, if all time intervals are taken
as semi-open and therefore they can sit next to one another, it would
br difficult to allow time point to stand between intervals, since it
is not appropriate to represent time points with the semi-open
structure.
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However, it seems that by some careful and proper treatments, we may
also reach the same results by defining the concepts of intervals
based on points. The key point here is, in addition to the concept of
lower and upper bounds for point-based intervals, the concept of left
type and right type for intervals needs to be addressed as well. What
follows is the skeleton of the structure:
P is a partially-ordered set of points;
Type is a two-member set {open, closed} ;
An interval i is defined as a quaternion seq(p1, p2, l, r) such that:
p1 < p2
l and r belong to Type
if p1 = p2 then l = r = closed
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Yes, but how does this differ from the standard notions of open and
closed intervals? It would appear that
(a,b,open,open) = (a,b)
(a,b,open,close) = (a,b]
and so on. Are there some models where your fourtuple intervals can not
be reduced to standard definitions, such as
(a,b) = {x | a < x < b}
and so on? Do those "nonstandard" models have some interesting properties?
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First of all, in the quaternion form (p1, p2, l, r), the left-type,
l, and right-type, r, are allowed to be unspecified.
Therefore, without knowing what is the exactly value of r' and l'',
we can still have Meets((p1',p2',l',r'), (p1'',p2'',l'',r'')) if we
just have r' =/ l'' . This allows one to express the case that the
light changes from state "Off" to state "On" (but without modelling
the state of the light at the switching point), that is the case 3 in
the classfication given in my former contributions to the ETAI.
However, if we insist on that the light is "On" at the switching
point p2, then we have r' = open and l'' = closed . This is the case
1. Similarly, we can express case 2, that is the light is "Off" at
the switching point.
Secondly, for general treatments, intervals don't necessarily have
to be sets of points, though they can be specified as such a model,
e.g., as Erik suggested, seq(a, b) = {xmidaltxltb} .
In fact, we can have
various other models. For instance, we may specify that all the
bounding points of time intervals are integers, while the internal
points are all the reals bounded by the bounding points. In fact, we
can have models which can be discrete, dense, or neither discrete nor
dense. Also, we may have models which can be linear (one or both
directions), parallel, circular, bounded (one or both sides). And so
on.
In addition, both the point-based time theory using the quaternion
structure, and the time theory that takes both intervals and points
as primitive allow the expression of the case (case 4?) where a
left-open interval Meets a point which, in turn, Meets a right-open
interval (This open/closed nature would be expressed by the Meets
relation when both intervals and points are taken as primitive).
What follows is an example of using the time model in modelling some
critical problems:
Gelfond et al noted some of the limitations of the Situation
calculus, and specially proposed an approach to represent actions
with delayed effects. They use an action called Wait to deal with
the delay between action A and its effect. That is:
Result(S0, A+Wait) , where the duration of Wait equals the time delay.
However, if we consider some critical cases, such as that one that
throwing a ball into the air: While the ball is going up (say for
just 8 seconds), the velocity is not zero (and again, not zero when
the ball is going down). Only at the apex (the stationary point)
where the ball is neither going up nor going down, the velocity
becomes zero. Now, how to express the (delayed) effects of the action
of throwing the ball? Specially, using the approach of Gelfond et al,
one may use Result(S0, Throw+Wait) to represent the situation 8
seconds after throwing the ball (here, we assume the duration of Wait
is 8 seconds). However, in this result situation, is the velocity
zero or not? The answer is not unique.
In fact, there are two
situations, one is the situation where the ball is at the stationary
point, another is the situation immediately after the stationary
point. Both of them satisfy that the Wait action lasts for 8 seconds.
Gelfond's approach seems unable to distinquish these two delayed
effects. I have put this question to Pat in my former discussions
with him, and he suspected (and I have the same feeling) that the
real problem with the idea of Gelfond et al, like many applications of
sitcalc, is that it doesn't give one a good way to distinguish
'instantaneous' situations from 'stable' ones (where eg the block is
on the table).
However, if we associate actions with "typelised" point-based
intervals, we may sucessfully express them. For the former, the
associated interval's right type is "open", i.e., excluding the
stationary point; while for the latter, the associated interval's
right type has to be "closed", i.e., including the stationary point.
For both cases, the left type may be unknown.
Jixin
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