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Ontologies for time
Answer to Jixin's contribution to this discussion on 1.4:
There is really little point in arguing about which theories are more
'intuitive' unless one is more precise about what one's intuitions are.
There are two fundamental problems with arguments like this. First,
intuitions are malleable, and one can get used to various ways of thinking
about time (and no doubt many other topics) so that they seem 'intuitive'.
Second, our untutored intuition seems to be quite able to work with
different pictures of time which are in fact incompatible with one another.
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.)
Jixin wrote:
| | 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.
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Yes, I agree. However, notice that there is a coherent frame of mind which
would deny this. According to this intuition, which is similar to Newton's
old idea of the infinitesimal, one would say that there are no points, but
some intervals are so small that they can be treated like points at a
sufficiently larger scale. In this perspective, it would be false to claim
that there was no interval at which the velocity is zero; rather, one
would say that the interval was infinitesimal. (If you want to deny the
reasonableness of this perspective, first reflect on the fact that it is
nearer to physical reality than any model based on the real line.)
I agree with Jixin here.
| | ...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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After re-reading Galton's paper [j-aij-42-159],
as we understand, Galton's
arguments are in general based on the assumption that Allen's
intervals are primitive, rather than sets of points on the real line.
In fact, the main revision Galton proposes to Allen's theory is a
diversification of the temporal ontology to include both intervals and
points. That is, in Galton's revised theory, intervals are still
taken as primitive.
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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.
| | Having pointed out this, however, as shown in Ma,
Knight and Petrides' 1994 paper [j-cj-37-114], Galton's
determination to define
points in terms of the "meeting places" of intervals does not, as he
claims, axiomatise points on the same footing as intervals, and hence
that some problems still remain in these revisions.
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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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Does this core theory refer that one in which "intervals are uniquely
defined by their endpoints (which are also the points they fit
between) and two intervals meet just when the endpoint of the first is
the startpoint of the second"? Or Allen's one? - It seems the former
one.
Anyway, yes. There does seem to be such a simple, basic core theory.
For general treatments, in Ma and Knight's CJ94 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.
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This theory seems to be similar to that outlined in my 1990 paper
with Allen, [j-ci-5-225] (and given at greater length in a U of
Rochester tech report of the same date.)
But there is little point in bickering about who said what
first, as almost all this discussion (including for example Allens 13
relations) can be found in publications written in the last century, if one
looks hard enough. All AI work in this area (including my own) is like
children playing in a sandbox. The theories and idea themselves are a much
more interesting topic.
One technical point, about 'primitive'. One of the things I realised when
working with James on this stuff was that if ones axioms about points were
minimally adequate it was trivial to define interval in terms of points;
and one can also define points in terms of intervals, although that
construction is less obvious. (I was immensely pleased with it until being
told that it was well-known in algebra, and first described by A. N.
Whitehead around 1910.) Moreover, these definitions are mutually
transparent, in the sense that if one starts with points, defines
intervals, then redefines points, one gets an isomorphic model; and vice
versa. So to argue about which of points or intervals are 'primitive' seems
rather pointless. We need them both in our ontology. If one likes
conceptual sparseness, one can make either one rest on the other as a
foundation; or one can declare that they are both 'primitive'. It makes no
real difference to anything.
| | 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.
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See above. But in any case this doesnt 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 <T-left, T-right>
where T-left < T-right , to get
that one Pat prefers.
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Not quite right. In my simple theory, T-left isnt
before T-right , it
equals it.
....
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Yes, it's true. And, it seems that, all these can be reached
equivalently by simply taking pointlike interval <t,t> as identical
with point t in the case where both intervals and points are included
in the time ontology.
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Yes, that is another alternative way to formalise things.
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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.)
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| | This can be distinguished by applying Holds_In
and Holds_On (that is
Allen's Holds, see Galton's 1990 paper [j-aij-42-159]).
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Yes, exactly, although there is no need to use this formal strategy, as I
explain in the time catalog section 1. Briefly, HoldsIn P i is true just
when i is a subinterval of a reference interval j
where HoldsOn 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'.)
| | ....Actually, as shown in Ma and
Knight's 1996 paper [j-cj-37-114], to characterise the intuitive relationship
between Holds-On and Holds-In , in the case where intervals are
allowed, some extra axiom is needed.
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I will check this paper to see what you mean in detail, thanks.
| | Also, it seems that, in Pat's formulation, for expressing that
interval <a,b> is a subinterval of interval <c,d> , one would have
c < a < b < d . In this case, we get that pointlike intervals <a,a>
and <b,b> (or equivalently, points a and b ) fall in
interval <c,b> .
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Yes; but note that if the theory uses reference intervals, that fact that P
holds for an interval I doesnt imply that it holds for every point (still
less every interval) in I . So this is quite consistent:
- Ball is rising for interval <a,b>
- Ball is falling for <b,c>
- Ball is stationary for <b,b>
You can consistently add that rising and falling are true for all
nonpointlike subintervals and every properly contained subinterval of the
reference interval.
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On the one hand, many cases suggest the need of allowing a proposition
to holds at a single point. For instance, see the example of throwing
a ball up into the air described earlier in the response to John
MaCarthy.
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.
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In my theory it leads to the conclusion that <a,a> does not exist (or, is
not a reference interval for 'illumination'), where a is the switching
point. All the distinctions between kinds of point - ones where something
is true and ones where something is switching - can be cast into a typology
of intervals. (This example illustrates why I like to distinguish between
the point a - which undoubtedly exists, is where the intervals meet, has a
clock time, etc. - and the interval <a,a> , which, if it existed, might be
an embarrassment.)
| | 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 reason that such an axiom is awkward is that it doesn't
catch the intuition in common-sense using of time.
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I agree. This was awkward in our old paper, and stemmed from our reluctance
to accept the idea of intervals which could meet themselves. I'm now
reconciled to that idea: in fact, it seems inevitable, much as the
existence of zero seems inevitable once one allows subtraction.
| | 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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The cases make sense not only if one thinks of intervals and points in
the usual mathematical way. In fact, all the three cases are
demonstrated under the assumption that both intervals and points are
treated as primitive, rather than in the usual mathematical way.
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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.
| | ... 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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But it seems that there are also some other kind of proposition to
which one cannot assign any nonpointlike reference interval. For
instance, in the throwing ball up into the air example, proposition
"the ball is stationary" can only be true at points, and for any point
we cannot find any non-pointlike interval (completely) surrounding it
over which the ball is stationary.
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Yes, exactly. Some properties can accept pointlike reference intervals,
some can't. Like Galton's distinction between 'at rest' and 'motionless'
(former requires nonpointlike, latter doesnt.) Thats the point. Notice the
distinctions are now all about intervals. They arent between different ways
of being true, but are bread-and-butter distinctions between intervals,
expressible within the theory. The machinery of truth wrt an interval is
the same in both cases (and in others, eg 'intermittently true' and other
exotic variations.)
| | 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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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 of 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:
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's all, and it seems quite intuitive......
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 Meets relation. However, if one would like to specify intervals as
point-based ones, such a definition will be in agreement with the
conventional definition about the open and closed intervals.
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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.
Pat Hayes
PS. Maybe the most useful thing would be to put all these axiomatic
theories into some common place with a common syntax - we could use
vanilla-KIF - so people can compare and contrast them. I dont have enough,
er, time to offer to do this right now, im afraid, but will cooperate with
anyone who will volunteer.
References:
| 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-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. |
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