Issue 98033 | Editor: Erik Sandewall | 1.4.1998 |
Today |
The discussion about the ontology of time started with a question to Jixin Ma. Today Jixin comes back into the discussion, together with an additional contribution by Pat Hayes.
Debates |
What follows is our response to the arguments about the ontology of time from Pat Hayes, Ray Reiter, and John McCarthy.
Response to John
The example of car accelerating demonstrates the need of time points for time ontology.
A similar example is throwing a ball up into the air. The motion of the ball can be modelled by a quantity space of three elements: going-up, stationary, and going-down. Intuitively, there are intervals for going up and going down. However, there is no interval, no matter how small, over which the ball is neither going up nor going down. The property of being stationary is naturally associated with a point, rather than any interval (including Allen and Hayes' moment), a "landmark" point which separates two other intervals.
Response to Ray
During all the years that the debate has raged about time points vs
intervals, we devotees of the sitcalc have never seen it as an
issue.
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Response to Pat
...For example, the 1990 AIJ critique of Allen's account by Galton
(wrongly) assumes that Allen's intervals are sets of points on the
real line.
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there does seem to be a simple, basic, account which can be
extended in various ways to produce all the other alternatives, and
this core theory is the one I was referring to.
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Anyway, yes. There does seem to be such a simple, basic core theory. For general treatments, in Ma and Knight's CJ 94 paper,[j-cj-37-114] a time theory is proposed (as an extention to Allen and Hayes' interval-based one) which takes both intervals and points as primitive on the same footing - neither intervals have to be constructed out of points, nor points have to be created as the places where intervals meet each other, or as some limiting construction of intervals. The temporal order is simply characterised in terms of a single relation "Meets" between intervals/points. 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.
(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
The way I prefer, myself, is to say that propositions hold only
during intervals, so that it is simply ill-formed to assert a
proposition of a single point; but to allow the possibility of
pointlike intervals, of the form ((pointlike i) iff ((begin i) = (end i))) |
Yes, it's true. And, it seems that, all these can be reached
equivalently by simply taking pointlike interval
one can, for example, say something like
((illuminated i) or (dark i))) implies (not (pointlike i))so that the light is neither on nor off at the switching point. In this theory, every proposition has a 'reference interval' during which it is true, and a proposition might not be true of subintervals of its reference interval. (Though some propositions might be. This kind of distinction has often been made in the linguistic literature. Note however that this intuition is basically incompatible with the idea that an interval is identical to the set of the points it contains.) |
This can be distinguished by applying
Also, it seems that, in Pat's formulation, for expressing that
interval
We certainly need something corresponding to 'points', I agree. I
meant only that the formal theory can be crafted in the way Ive
outlined above, or alternatively by identifying the pointlike
intervals with their endpoints, and allowing a proposition to hold
at a single point. This is in many ways more intuitively
transparent but it is formally a bit more awkward, since pointlike
isnt definable any more, and one has to put in special axioms
forbidding points to meet each other. The 'reference interval' of a
proposition could now be a single point in the theory. This is
essentially the theory that Allen and I described in our 1985
paper [c-ijcai-85-528], though it takes a little work to see it.
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On the other hand, allowing a proposition to holds at a single point doesn't necessarily make pointlike un-definable. It depends on if one would impose some extra constraints, such as
((illuminated i) or (dark i))) implies (not (pointlike i))as introduced by Pat for the light switching example, which actually leads to the assertion that the light is neither on nor off AT the switching point.
Actually, in the later version of Allen and Hayes's theory that appears in 1989 [j-ci-5-225], an awkward axiom is proposed to forbid moments to meet each other. It is interesting to note that, although moments are quite like points (moments are non-decomposable), they still have positive duration (they are not pointlike). Moments are included in Allen and Hayes' time ontology, while points are not. One of the reasons that such an axiom is awkward is that it doesn't catch the intuition in common-sense usage of time. In fact, in many applications, one would like to take some quantity as the basic unit of time. E.g., we may take a second as the basic unit. In other words, seconds are treated as moments - they cannot be decomposed into smaller units. However, for a given second, we may still want to express the next one, that is, a second can meet another second, although they are both non-decomposable.
True, and indeed the Allen relations only have their usual
transitivity properties when applied to intervals which are
nonpointlike and forward-oriented. Of course both these are
properties expressible in the theory, so that the Allen
transitivity relationships can be stated there, suitably qualified.
(When the alternative extension axioms are added, the
qualifications become tautologous.)
...BTW, the claim that "meets" and "before" being exclusive is "intuitive" depends on how one's intuition is formed. Part of what I learned by having to construct alternative formalisations is that intuition is very malleable. Having gotten used to pointlike intervals, I dont find this exclusivity condition at all intuitive.
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But these cases only make sense if one thinks of interval and
points in the usual mathematical way, which is exactly what Im
suggesting we don't need to do. We can get almost everything we
need just from the ordering structure: we don't need to get all
tied up in distinguishing cases which can only be formally stated
by using all the machinery of real analysis.
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Again I was careless in using the word "knowledge", sorry. I
should have said: in order to answer the question whether the light
is on or off, one has to specify the interval with respect to which
this question is posed. On this view, the truth or otherwise of a
proposition is only meaningful with respect to certain intervals. I
dont mean that the facts are determined by knowing more about the
details of the interval, but that the question is a different
question when asked about one interval than when asked about
another, and for some intervals in may be simply meaningless. Is
the light on or off at (exactly) 3.00 pm? The only way to answer
this is to find a suitable non-pointlike interval of light or
darkness completely surrounding 3.00 pm, because 'being on' is the
kind of proposition that requires a nonpointlike reference
interval.
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This has nothing to do with whether an interval is open or closed:
in fact, there is no such distinction in this theory. It only
arises in a much more complicated extension which includes set
theory and an extensionality axiom for intervals.
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interval I is left-open at point P iff
interval I is right-open at point P iff
interval I is left-closed at point P iff there is
an interval I' such that
interval I is right-closed at point P iff there is
an interval I' such that |
That's all, and it seems quite intuitive. For instance, with the
knowledge
It is important to note that the above definition about the open and
closed nature of intervals is given in terms of only the knowledge of
the
Jixin
References:
c-aaai-82-197 | Marc Vilain. A System for Reasoning about Time. Proc. AAAI National Conference on Artificial Intelligence, 1982, pp. 197-201. |
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-aij-42-159 | Anthony Galton. A critical examination of Allen's theory of action and time. Artificial Intelligence Journal, vol. 42 (1990), pp. 159-188. |
j-amai-14-251 | Javier Pinto and Ray Reiter. Reasoning about Time in the Situation Calculus. Annals of Mathematics and Artificial Intelligence, vol. 14 (1995), pp. 251-268. |
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. |
John McCarthy wrote:
McCarthy and Hayes 1969 paper [n-mi-4-463] used time as a fluent
on situations, i.e. time(s). ...
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My previous message gave a reason for including time points in a
theory of events and actions. The theory could be founded so as to
regard them as degenerate intervals, but I don't see any advantage in
that, although I suppose the idea stems from the fact that people and
robots can't measure time precisely.
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That idea - that intervals are approximations to points, and the length of an interval represents a degree of ignorance about the location of a point - gives a rather different ontology. In that case, for example, it doesnt really make sense to be able to refer to the precise endpoints or meeting-points of intervals (since if one can, then absolute precision about timepoints comes for free.) The Allen set of thirteen relations reduces to just six (before, overlap, inside, and inverses) since those that require endpoints to be exactly identified (meets, starts, ends, equal, endby, startby, meetby) are undefinable (except in an infinite limit.) This is the theory called 'approximate-point' in my time catalog. There arent any points in this theory, of course, though they could be defined if one added enough mathematical machinery to be able to talk about limits of infinite sequences.
Pat Hayes
References:
n-mi-4-463 | John McCarthy and Pat Hayes. Some Philosophical Problems from the Standpoint of Artificial Intelligence. [postscript] Machine Intelligence, vol. 4 (1969), pp. 463-502. |