Ontologies for time
Pat wrote:
| Jixin's own intuition, for example, seems to agree with McCarthy's
that time is continuous, and yet also finds the idea of contiguous
atomic 'moments' (intervals with no interior points) quite
acceptable. But you can't have it both ways: if moments can meet
each other, then there might not be a point where the speed is
exactly 60mph, or the ball is exactly at the top of the trajectory.
If time itself is discrete, then the idea of continuous change is
meaningless. Appealing to a kind of raw intuition to decide what
axioms 'feel' right lands one in contradictions. (That was one
motivation for the axiom in Allen's and my theory, which Jixin found
"awkward", that moments could not meet. The other was wanting to be
able to treat moments as being pointlike. That was a mistake, I'll
happily concede.)
| First of all, as Pat noted (see below), our theory is similar (in
fact, it is specially stated to be an extension/revision) to that of
Allen and Hayes ([c-ijcai-85-528], [j-ci-5-225]).
As an extension, it allows not only
intervals and moments, but also points (however, it dose not
necessarily have to include points or moments). As a revision, Allen
and Hayes' constraint that moments could not meet each other is
replaced by "a point cannot meet another point", that is there must
be an interval (or a moment) standing between any two points (if
these two point are explicitly expressed). However, basically, the
theory doesn't commit the time structure as (left-, right-,
left-and-right) linear, (left-, right-, left-and-right) branching,
continuous or discrete, etc., although all these can be characterised
by means of some corresponding extra axioms.
Yes, in this time theory, atomic moments (AND points) are acceptable.
However, they are just ACCEPTABLE, but not necessarily to be
everywhere over the time. The theory only claims that a time element
is either an interval (or specially, a moment) or a point. If one
insists on using moments/points somewhere over the time, they can be
explicitly expressed there. For somewhere else over the time, it may
be the case that each time element is a decomposable interval. That
is, there may be no moments/points at all. It is also consistent to
have a time structure where each time element is either an
decomposable interval or a point, or even a time structure where each
time element is a decomposable interval. In fact, generally speaking,
the basic time structure may be neither dense nor discrete anywhere,
or may be continuous over some parts and discrete over other parts.
This depends on what you want to express and what extra axiom you
would impose.
Pat wrote
| Galton's intuitions are clearly based on thinking of intervals as
sets of points. He takes it as simply obvious, for example, that
there is a distinction between open and closed intervals.
| I still cannot find where Galton claimed this. Actually, before I
answered Pat's last contribution (Newsletter ENRAC 27.3 (98030), I had
re-read Galton's paper and phoned Galton to confirm about this. Yes, a
distinction between open and closed intervals did appear in Galton's
paper, but I found it was only used to demonstrate the corresponding
problem in thinking of intervals as sets of points, and hence as the
reason for him (and for James and Pat?) to take intervals as
primitive. Anyway, this issue doesn't really affect this discussion.
Pat wrote
| Some advantages of this time theory are:
(1) It retains Allen's appealing characteristics of treating
intervals as primitive which overcomes the Dividing Instant
Problem.
|
| See above. But in any case this doesn't overcome the problem.
Allen's treatment allows lights to just come on, but it doesnt
provide anywhere for the ball to be motionless.
| So, that is why we need to extend Allen and Hayes' treatment by
allowing time points as primitive as well. On the one hand,
treating time intervals as primitive avoids the question of whether
intervals are open or closed; on the other hand, allowing time points
(as primitive) provides means for expressing instantaneous phenomena
such as "the ball is motionless at a point".
Pat wrote
| (2) It includes time points into the temporal ontology and therefore
makes it possible to express some instantaneous phenomenon, and
adequate and convenient for reasoning correctly about continuous
change. (3) It is so basic that it can be specified in various ways to
subsume others. For instance, one may simply take the set of points
as empty to get Allen's interval time theory, or specify each
interval, say T, as <T-left, T-right>
where T-left < T-right ,
to get that one Pat prefers.
|
| Not quite right. In my simple theory, T-left isnt before
T-right , it equals it.
| But, the relation T-left < T-right does already
include T-left = T-right , that is T-left equals T-right . In
the case T-left < T-right ,
T is an interval; in the case T-left = T-right ,
T is a pointlike
interval.
Pat wrote
| Yes, exactly, although there is no need to use this formal strategy,
as I explain in the time catalog section 1. Briefly,
Holds-%In P i
is true just when i is a subinterval of a reference interval j
where Holds-%On P j .
Again, it is largely an aesthetic judgement, but I find Galton's
hold-on vs. hold-in distinction awkward and unintuitive. (It
suggests that there are two different 'ways to be true'.)
| Yes. And in Ma et al's paper [j-cj-37-114],
some examples are given to show the
problem with Galton's distinction between hold-on
and hold-in . Also,
it is claimed that the fundamental reason for the problem is that
Galton wanted to characterise the fact that a proposition holds for an
interval in terms of that the proposition holds for every point within
the interval.
Pat wrote
| Well, it depends on what axioms one assumes! Perhaps I have been
speaking too carelessly about the 'usual mathematical way'. Heres my
intuition: the standard account of the continuum seems forced to
resolve the dividing point problem by deciding which interval
contains the point, distinguishing open from closed intervals,
because it identifies an interval with a set of points. (So if
both intervals 'contain' the point, the intervals must intersect.)
One can take points as basic or intervals as basic or both
as primitive; that's irrelevant, but the crucial step is that
(set-of-points = interval) identification. Thats exactly what I want
to avoid. My point is only that if we abandon that idea (which is
only needed for the formal development of analysis within set
theory, a rather arcane matter for us), then there is a way to
formalise time (using both intervals and points as primitive, if you
like) which neatly avoids the problem.
| It is also exactly what the approach of treating both intervals and
points as primitive wants to avoid - it avoids to "identify" an
interval with a set of points. In addition, such an approach allows
expressions of all the three cases shown in my former discussion
(Newsletter ENRAC 13.3 (98027)), without thinking of intervals and
points in the usual mathematical way, or "identify an interval with a
set of points"
Pat wrote
| In the time theory where both intervals and points are taken as
primitive, we can (if we like) talk about the open and closed nature
of intervals with some knowledge being available. This kind
knowledge can be given in terms of the Meets relation, rather than
some "much more complicated extension which includes set theory and
an extensionality axiom for intervals". In fact, we can define that:
|
|
- interval I is left-open at point P iff
Meets(P, I)
interval I is right-open at point P iff Meets(I, P)
interval I is left-closed at point P iff there is an interval I'
such that Meets(I', I) ^ Meets(I', P)
interval I is right-closed at point P iff there is an interval I'
such that Meets(I, I') ^ Meets(P, I') ...
|
| That certainly seems to be an elegant device. (Though the
definitions have nothing to do with knowledge; all Jixin is saying
is that the definitions of open and closed can be given in terms of
MEETS . As Allen and I showed in our old paper, the entire theory
can be reduced to MEETS .) However, in order to be nontrivial, it
must be that points 'separate' meetings, ie if meets(I, P) and
meets(P, J) then ¬ meets(I, J) , right? For if not, all left-open
intervals are also left-closed, etc. This seems to make 'points'
similar to our old 'moments': in fact, if Jixin's theory predicts
meets(P, Q) ·-> P = Q for points P and Q , then I'll lay odds
it is isomorphic to our moments theory. One of the main observations
in our paper was that with the no-meets axiom, one can map moments
to points with no change to the theorems provable.
On the other hand, if the theory allows distinct points to MEET ,
I'd be interested to know how it is able to map smoothly to a
conventional account of the continuum, since that is provably
impossible. One-point closed intervals exist everywhere on the real
line, but no two of them are adjacent. Atomic adjacent times
(whatever we call them) are pretty much a definition of discrete
time models, and are incompatible with density, let alone
continuity.
| Yes. If Meets(I, P) ^ Meets(P, J) then ¬ Meets(I, J) . Actually, in
our theory, If Meets(I, P) ^ Meets(P, J) then Before(I, J) , where
Before(I, J) ·-> ¬ Meets(I, J) , since all the 13 relations are
exclusive to each other. Therefore, it is impossible for an interval
to be both left-open and left-closed.
Also, in our theory, Meets(P, Q) implies that at least one of P and Q
is an interval (or a moment) - they cannot be both points.
In addition, it is important to note that the constraint that a point
cannot meet another point makes it is possible to establish a
consistency checker for temporal database systems (see Knight and
Ma's 1992 paper [j-aicom-5-75]).
Yes. As claimed earlier in this dicussion and actually pointed out in
our published paper, our theory is in fact an extension to that of
Allen and Hayes. Our points are quite like Allen and Hayes' moments -
they cannot meet each other. However, on the one hand, points are
fundamentally different from moments - points have no duration while
moments do have, no matter how small they are. Therefore, it is more
convenient to use points than moments in modelling some instantaneous
phenomenon, especially in the case where duration reasoning is
involved. On the other hand, if moments are simply mapped
to points, how to express the real moments, i.e., non-decomposable
intervals with positives duration (like the "seconds" example given
in my last discussion)?
Jixin
References:
c-ijcai-85-528 | James Allen and Pat Hayes.
A Common-Sense Theory of Time.
Proc. International Joint Conference on Artificial Intelligence, 1985, pp. 528-531.
|
j-aicom-5-75 | Brian Knight and Jixin Ma.
A General Temporal Model Supporting Duration Reasoning.
AI Communications, vol. 5 (1992), pp. 75-84. |
j-ci-5-225 | James F. Allen and Patrick J. Hayes.
Moments and points in an interval-based temporal logic.
Computational Intelligence, vol. 5, pp. 225-238. |
j-cj-37-114 | Jixin Ma and Brian Knight.
A General Temporal Theory.
Computer Journal, vol. 37 (1994), pp. 114-123. |
|