|Issue 98035||Editor: Erik Sandewall||15.4.1998|
The discussion between Pat Hayes and Jixin Ma on the ontology of time continues:
|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
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.
|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.
| Some advantages of this time theory are:
(1) It retains Allen's appealing characteristics of treating
intervals as primitive which overcomes the Dividing Instant
| 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.
|(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 |
| Not quite right. In my simple theory, |
|Yes, exactly, although there is no need to use this formal strategy,
as I explain in the time catalog section 1. Briefly,
Again, it is largely an aesthetic judgement, but I find Galton's
| 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.
|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:
| 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
On the other hand, if the theory allows distinct points to
Also, in our theory,
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)?
|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.