Discussion on Ontologies for Time
| Erik Sandewall | Introduction |
The present ENRAC discussion about ontology of time started with a remark by Pat Hayes on 13.3 in the discussion about the paper by Grünwald at the Commonsense workshop, [c-fcs-98-42]. It had been preceded by a brief discussion on a similar topic in the discussion about the paper by Knight, Peng and Ma at the same workshop, [c-fcs-98-183]. Furthermore, Pat Hayes had touched on the same topic in a contribution to the panel debate on theory evaluation in October, 1997 (see ENRAC issue for October 1997, page 83). Beginning in March of 1998 we had an animated discussion on this topic. The debate contributions follow.
References:
| c-fcs-98-183 | Brian Knight, Taoxin Peng, and Jixin Ma. Reasoning about Change over Time: Actions, Events, and their Effects. [abstract] [postscript] [discussion] Proc. Formalization of Commonsense Reasoning, 1998, pp. 183-197. |
| c-fcs-98-42 | Peter Grünwald. Ramifications and sufficient causes. [abstract] [postscript] [discussion] Proc. Formalization of Commonsense Reasoning, 1998, pp. 42-55. |
| 13.3 | Jixin Ma |
Dear Erik,
Since, as you mentioned in the Newsletter ENRAC 12.3 (98026), Pat Hayes' opinion about instantaneous changes has a close relation to our previous work (In fact, Pat did raise the similar question at the Commonsense workshop), I would like to make the following claims/arguments:
(1) For general treatment, both intervals and points are needed.
(2) To overcome the so-called Dividing Instant Problem, that is the problem in specifying whether intervals is "open" or "closed" at their ending-points, both intervals and points should be treated as primitive on the same footing. Neither intervals are constructed out of points, nor points are defined as the "meeting place" of intervals. Points have zero duration and are non-decomposable, while interval have positive duration and are either decomposable or non-decomposable (moments). Intervals Meets/Met-by points or other intervals. Therefore, although conceptually there is no definition of the ending-points for intervals, one may still say if an interval is open or closed at a point when the corresponding knowledge is available. E.g., if we know interval I Meets point P1, we may say I is (right) open at P1; if we know that interval I1 Meets Interval I2 and point P2 Meets I2, then we may say I1 is (right) closed at P2. In fact, this interpretation is consistent with the conventional definition about the closed and open nature of intervals that are constructed out of points such as reals or rationals. (For full details of the axiomatization of such a time structure based on both intervals and points as primitive, see [j-cj-37-114]).
(3) Now consider the classical example of switching on a light. The
arguments really depend on what knowledge is given/available for such
a case. First of all, one can image there is an interval I immediately
before the switching point P, and another interval J which is
immediately after P. That is:
By Commonsense, at any time, the light is either on or off, and cannot
be both on and off. In other words, one should be able to express the
example in terms of two adjacent intervals, I1 and I2, where over I1
the light is off and over I2 the light is on, that is
The question now is that, by taking both intervals and points as primitive temporal objects, on the one hand, we can talk about time points such as the switching point P. However, on the other hand, can we still successfully express the Commonsense knowledge for the above example, without bearing the DIP? The answer is YES, since it really depends on what knowledge is given/available. In fact, there are three possible cases:
Case a) We have no knowledge about the state of the Light at the switching point P, though we may insist that there is a switching point, but we don't know if the LightOff interval I1, or the LightOn interval I2 is open or closed at the switching point P. What we know is just that the light changes from state "Off" to state "On". Hence, such a case can be simply expressed as (1):
| Meets(I1, I2) ^ HOLDS(¬ LightOn, I1) ^ HOLDS(LightOn, I2) |
Case b) We do know, or we impose (by some reason for the specified
application) that the Light is on at the switching point P, that is,
Case c) As an alternative to b), we may know, or we may impose that
the Light is still off at the switching point P, that is,
Jixin
References:
| j-cj-37-114 | Jixin Ma and Brian Knight. A General Temporal Theory. Computer Journal, vol. 37 (1994), pp. 114-123. |
| 14.3 | Pat Hayes |
Im largely in agreement with Jixin about points and intervals, although I dont think its got anything to do with knowledge, and I think there's a simpler way to say it all.
First, just forget about whether intervals are open or closed.
This issue arises only if we insist (as the standard mathematical
account of the continuum does) that an interval is a set of points.
But if we take points and intervals as basic, there is no need to do
this. Points, as
Allen suggested long ago, can be thought of as places where intervals
meet each other, not as the substance out of which intervals are
constructed. It took me a long time to see how powerful this idea is.
The question of which interval 'contains' the meeting point is
meaningless. This gives a very simple, elegant formulation in which
points are totally ordered, 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. That's all the structure one needs. Truths hold during
intervals. One can allow instantaneous intervals, whose endpoints are
identical and which have no duration. One can, if one wishes, identify
the interval
(Instantaneous intervals have the odd property of meeting themselves, by the way; in fact this is a way to characterise them without mentioning points explicitly. It is also perfectly consistent to have 'backward' intervals whose end is earlier than their beginning, and which have negative durations. Axiomatic details can be found in a rather long document available as two postscript files
http://www.coginst.uwf.edu/~phayes/TimeCatalog1.ps
http://www.coginst.uwf.edu/~phayes/TimeCatalog2.ps
One can cast the whole theory in terms of a single three-place relation
MEETS-AT between two intervals and a point, much as Allen's original
theory can be cast in terms of MEETS.)
In this theory, to talk of the set of points 'in' an interval requires
one to specify what it means for a point to be 'in' an interval. If a
point is later than the beginning and earlier than the end, its clearly
in the interval, but we have some freedom with the endpoints. One could
insist that interval endpoints are 'in' the interval. But this is now
OK, since truths hold not at points but during intervals, so the
apparent contradiction of the light being both on and off at the
splitting point simply doesnt arise. The light isn't either on or off
at a single point: if you want to know whether the light was on or not,
you have to say which interval you are talking about. P may be true
during
Jixin says that "one cannot talk about anything about the switching
point P, which is intuitively there anyway." Well, the point is
certainly there, and we can talk about it (for example, its relation to
other points and intervals) but the question is whether it makes sense
to say that something is true at it. Some truths may be instantaneous,
ie true only at points; others make sense only when asserted to hold
during noninstantaneous intervals. Lights being on or off, for example,
might be enduring, while changes in illumination, or isolated flashes,
can be instantaneous. So for example suppose it is dark during
interval
Pat Hayes
| 17.3 | Jixin Ma |
After reading Pat's answers to our claims/arguments about the ontology for time, we would like to raise the following questions/arguments:
1. First of all, it is not clear what's the exact role that time points play in Pat's formulation, although, according to Erik's understanding, Pat Hayes "argues in favour of an ontology for time where intervals are the only elementary concept and timepoints play a secondary role". As Pat points out in his answers (in agreement with our opinion as stated in our claims), "if we take points and intervals as basic, there is no need to do this", i.e., deal with the question of whether intervals are open or closed. However, it is not clear what's the exact meaning of "taking points and intervals as basic". Are they both taken as primitive temporal objects, or, as Allen suggests, points are thought as places where intervals meet each other?
2. Pat argues that "the question is whether it makes sense to say that something is true at points". However, his argument is quite confused: in the first place, he claims "truths hold not at points but during intervals" (as for the case when one insists that interval endpoints are "in" the interval). Later, he states "Some truths may be instantaneous, ie true only at points; others make sense only when asserted to hold during noninstantaneous intervals". So, what's the answer to the question "whether it makes sense to say that something is true at points"?
3. Pat's claims that one may identify interval
I. As noted by Pat himself, "an instantaneous interval meets itself", though the "basic" points are totally ordered. How to characterise the relation between them? Pat's gives a suggestion: to characterise them without mentioning points explicitly. Then, what's the relationship between points and intervals?
II. How to define other relationships between intervals like those
introduced by Allen? For instance, it is intuitive to say that
III. By saying
4. Pat argues that "I'm largely in agreement with Jixin about points and intervals, although I dont think its got anything to do with knowledge". But it does. In fact, as pointed out by Pat himself, "if you want to know whether the light was on or not, you have to say which interval you are talking about". In other words, if the (additional) knowledge of "which interval you are talking about" is given (e.g., in terms of which interval is open/closed at the switching point, or in terms of the corresponding meets relations - "knowledge"?), we can say whether the light was on or not.
5. Pat also argues that his formulation is simpler (and elegant). In what aspects, compared with which formulation? It seems that it still needs a lot of axioms to characterise the formal structure, especially when issues such as density, linearity, boundness, etc, are to be addressed.
Jixin & Brian
| 27.3 | Pat Hayes |
Sorry, I wasnt sufficiently clear, and my carelessness in using intuitive phrasing led to misunderstanding.
First, in my view there is no single answer to many of the issues that Jixin raises. One can make various choices, each internally consistent but not consistent with the others. (That is why I called the cited paper a 'catalog' of time theories, rather than a single theory of time.) This freedom means that one must be clear which alternative one is using, as confusion follows when one tries to put together bits and pieces of incompatible views. (For example, the critique of Allen's account by Galton in [j-aij-42-159] in 1990 (wrongly) assumes that Allen's intervals are sets of points on the real line.) Having said this, however, 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.
Second, I dont agree with Erik's introduction of my note (14.3) as putting intervals before points. As Allen and I showed some time ago, the choice is arbitrary, since points can be transparently defined in an interval theory and vice versa, so the choice of either one as somehow more basic is, er, pointless; and one gets a more useful account simply by allowing them both as primitive. (Actually, if anything, the simple theory I outlined seems more to rely on points as basic, since an interval there is completely defined by its two endpoints and has no other structure, and all the temporal relations between intervals can be inferred from the total ordering of points.)
Jixin asks:
| However, it is not clear what's the exact meaning of "taking points and intervals as basic". Are they both taken as primitive temporal objects, or, as Allen suggests, points are thought as places where intervals meet each other? |
Both. These arent incompatible alternatives. The basic idea in the 'simple' theory is essentially Allen's, that points are meeting-places. Still, there's no harm in being able to mention these meeting-places as real objects, and doing so makes it easier to say quite a lot of things, such as 'when' some change happens. Clock times seem to be associated more naturally with points than intervals, for example.
| 2. Pat argues that "the question is whether it makes sense to say that something is true at points". However, his argument is quite confused: in the first place, he claims "truths hold not at points but during intervals" (as for the case when one insists that interval endpoints are "in" the interval). Later, he states "Some truths may be instantaneous, ie true only at points; others make sense only when asserted to hold during noninstantaneous intervals". |
(In the above I was careless at the place marked by boldface, sorry. I should have said 'pointlike interval'. It gets hard to speak about this stuff clearly in English, since I need to distinguish our intuitive notion of 'point' from the way that a particular theory encodes this intuition, and different theories do it differently. I will use scare-quotes to refer to the intuitive concept.)
| So, what's the answer to the question "whether it makes sense to say that something is true at points"? |
There is no (single) answer: one can craft the theory to suit various
different intuitions on matters like these. 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
This doesnt require us to say that every point fills an interval, notice:
since 'interval' is a basic predicate, it is perfectly consistent to say
| pointlike(i) <-> begin(i) = end(i) |
one can, for example, say something like
| illuminated(i) v dark(i) ·-> ¬ 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.)
|
3. Pat's claims that one may identify 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 IJCAI paper [c-ijcai-85-528], though it takes a little work to see it.
|
Let's consider the case that we do
(in fact, for general treatments, we do need them). For this case,
Pat's states that
if |
Yes, exactly. Interval relations are comletely determined by endpoint orderings,and Allen's huge transitivity table can be painstakingly derived from the assumption of total ordering. That's all it amounts to, in fact.
|
Below
are some problems with this formulation:
I. As noted by Pat himself, "an instantaneous interval meets itself", though the "basic" points are totally ordered. How to characterise the relation between them? Pat's gives a suggestion: to characterise them without mentioning points explicitly. Then, what's the relationship between points and intervals? |
The relations are quite simple and transparent: intervals lie between endpoints, and points have intervals extending between them. Self-meeting is the interval-interval relation corresponding to equality in the point ordering. Again, if one has an intuitive objection to self-meeting intervals, then one can take the second alternative mentioned earlier. (All these alternatives are got by extending the basic theory.)
|
II. How to define other relationships between intervals like those
introduced by Allen? For instance, it is intuitive to say
that |
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.
|
III. By saying |
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.
| 4. Pat argues that "I'm largely in agreement with Jixin about points and intervals, although I dont think its got anything to do with knowledge". But it does. In fact, as pointed out by Pat himself, "if you want to know whether the light was on or not, you have to say which interval you are talking about". In other words, if the (additional) knowledge of "which interval you are talking about" is given (e.g., in terms of which interval is open/closed at the switching point, or in terms of the corresponding meets relations - "knowledge"?), we can say whether the light was on or not. |
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.
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.
| 5. Pat also argues that his formulation is simpler (and elegant). In
what aspects, compared with which formulation?
|
| It seems that it still
needs a lot of axioms to characterise the formal structure, especially
when issues such as density, linearity, boundness, etc, are to be
addressed.
|
The theory is basically linear in its nature, since it assumes timepoints are totally ordered. One can easily weaken it to allow partial orders, but then the extensions involving density, etc.,, get rather tricker. I think the universe is deterministic in any case, so linearity doesnt bother me :-)
Pat Hayes
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-aij-42-159 | Galton. A critical examination of Allen's theory of action and time. Artificial Intelligence, vol. 42 (1990), pp. 159-188. |
| 29.3 | Ray Reiter |
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. Here's why I think this is so.
In the sitcalc, a fluent (
| LightOn(do(switchOn, do(switchOff, S0))) |
| ¬ LightOn(do(switchOff, do(switchOn, S0))) |
In the sitcalc with explicit time, the first might become
| LightOn(do(switchOn(3.14), do(switchOff(1.41), S0))) |
meaning that as a result of the action history consisting of first
switching off the light at time 1.41, then switching on the light at time 3.14,
the light will be on. Notice that there is no way of expressing the claim that
the light is, or is not on at time 3.14 (or 3.5), independently of the
situation leading up to this time. On
the other hand, time based formalisms do allow one to write
Now, one could rightly object to the above account because it provides only for fluent truth values at discrete time points, namely at the action occurrence times. So we are tempted to understand
| LightOn(do(switchOn(3.14), do(switchOff(1.41), S0))) |
to mean that the light is on at time 3.14, but it tells us nothing about time 3.5 say. This is particularly bad for (functional) fluents that vary continuously with time, for example, the location of a falling object. To handle this, introduce a time argument for fluents, in addition to their situation argument. For the light, one can write:
| LightOnT(t, s) <-> LightOn(s) ^ t > start(s). |
Here,
An instance of this would be
| LightOnT(t, do(switchOn(3.14), do(switchOff(1.41), S0))) <-> t > 3.14 |
Here we have committed to the light being on at exactly the time of the switchOn action, and forever thereafter, relative to the history
| do(switchOn(3.14), do(switchOff(1.41), S0)) |
| LightOnT(3.14, do(switchOn(3.14), do(switchOff(1.41), S0))) |
| ¬ LightOnT(3.14, do(switchOff(1.41), S0)) |
without contradiction. This seems to be precisely the point at which purely time-based formalisms run into difficulties, and the sitcalc version of this problem illustrates the role that explicit situation arguments play in resolving these difficulties.
Now, we can axiomatize falling objects:
| positionT(t, s) = position(s)+ | ||
| velocity(s)*(t-start(s))+0.5*g*(t-start(s))2 |
| 29.3 | John McCarthy |
When my car accelerates, there is a time point at which it passes 65 miles per hour. It is awkward to describe this point in a language not providing for time points.
| 31.3 | Graham White |
John McCarthy wrote:
| When my car accelerates, there is a time point at which it passes 65
miles per hour. It is awkward to describe this point in a language
not providing for time points.
|
Graham White
| 31.3 | John McCarthy |
McCarthy and Hayes (1969) used time as a fluent on situations, i.e. time(s). One motivation was that people, and perhaps future robots, often do not know the time with sufficient resolution to compare two situations, e.g. Ray Reiter's recent message with times 1.41 and 3.14. A second motivation for making situations primary was to make it correspond to human common sense. Many people who can reason about the consequences of actions in situations perfectly well do not know about real numbers, and some don't know about numbers at all. The falling body example was also in that paper with time as a fluent. Galileo did know about real numbers.
It's not clear that either of these considerations is of basic importance for AI.
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.
| 31.3 | Pat Hayes |
Responses to Ray Reiter and John McCarthy. Ray wrote:
|
In the sitcalc, a fluent (
In the sitcalc with explicit time, the first might become
meaning that as a result of the action history consisting of first switching off the light at time 1.41, then switching on the light at time 3.14, the light will be on. Notice that there is no way of expressing the claim that the light is, or is not on at time 3.14 (or 3.5), independently of the situation leading up to this time. |
I think of Ray's 'sequences of actions' as alternative ways the temporal universe might be, ie possible timelines (or histories, as Ray sometimes calls them.) The point/interval controversy is about reasoning within, or with respect to, one of these possible timelines; sitcalc gets this muddled up with reasoning about alternative futures for the partial timeline up to the present. (Think of the tree of accessible situations in a state's future: the distinction is between reasoning about a single branch, and comparing two different branches.)
| On the other hand, time based formalisms do allow one to write
|
|
Now, one could rightly object to the above account because it provides only for
fluent truth values at discrete time points, namely at the action occurrence
times. So we are tempted to understand
to mean that the light is on at time 3.14, but it tells us nothing about time 3.5 say. This is particularly bad for (functional) fluents that vary continuously with time, for example, the location of a falling object. To handle this, introduce a time argument for fluents, in addition to their situation argument. |
Hold on! What kinds of things are these 'times' supposed to be? They seem to be something like clock-times, ie temporal coordinates (maybe understood with respect to a global clock of some kind.) OK, but notice that this isn't what I mean by a 'timepoint'. There are at least six distinct notions of 'time' (physical dimension, time-plenum, time-interval, time-point, time-coordinate and duration.) I think the nearest thing in Reiter's ontology to what I call a time-point is something like the pairing of a clock-time with a situation ('3.14 in situation s').
|
For the light, one can write:
Here, An instance of this would be
Here we have committed to the light being on at exactly the time
of the
without contradiction... |
This seems to be the half-open-interval solution, where intervals contain their endpoints but not their starting points. This makes sense for the sitcalc, which focusses on the results of actions, but seems ad-hoc and unintuitive in a broader context. (Also, BTW, the idea that one can ever say that some finite list of actions is all the actions that have occurred seems quite unrealistic. After all, people's fingers probably pushed the switch and something somewhere was generating electricity. Surely one should be able to actually infer this from a reasonably accurate common-sense description of light-switching.)
| ...This seems to be precisely the point at which purely
time-based formalisms run into difficulties, and the sitcalc version
of this problem illustrates the role that explicit situation arguments
play in resolving these difficulties.
|
The problem is that even if we stick to talking about a single timeline (eg the unique past, or one alternative future) there still seems to be an intuitive difficulty about timepoints like the time when a light came on. The solution I suggested - that is, truth at a point has to be defined relative to a reference interval containing the point (which is not my idea, let me add) - is similar in many ways to Reiter's , except it applies not just across timelines but also within a single one.
John wrote:
| When my car accelerates, there is a time point at which it passes 65
miles per hour. It is awkward to describe this point in a language
not providing for time points.
|
The 65mph example is logically similar to the point at the top of a
trajectory when the vertical velocity is zero. Examples like this appeal to
a basic intuition about continuous change, that it has no 'jumps', so if it
is
(continuous X i) =df
(forall (y)(implies (between (X (begin i)) y (X (end y)))
(exists t) (and (in t i) (= y (X t))))
(strictlycontinuous X i) =df
(forall (j) (implies (subint j i)(continuous X j)))
where subint is the Allen union {begins, inside, ends}. (This assumes that the timeline itself is dense; if not, strictlycontinuous is trivially true everywhere.) Other conditions like monotonicity and so forth also transcribe directly from their usual mathematical formulations.
Pat Hayes
| 1.4 | Jixin Ma |
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.
|
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.
|
| 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.
|
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.
|
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.
|
| 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.
|
| 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.
|
| 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.
|
|
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. |
| 1.4 | Pat Hayes |
John McCarthy wrote:
| McCarthy and Hayes 1969 paper [n-mi-4-463] used time as a fluent
on situations, i.e. time(s). ...
|
| 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.
|
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. |
| 13.4 | Pat Hayes |
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.
|
| Response To Ray: (---)
|
| 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. |
|
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.
|
| 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. |
| 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. |
|
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.
|
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.
|
| (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
|
....
|
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
|
| ....Actually, as shown in Ma and
Knight's 1996 paper [j-cj-37-114], to characterise the intuitive relationship
between
|
| Also, it seems that, in Pat's formulation, for expressing that
interval
|
- 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.
|
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.
|
In my theory it leads to the conclusion that
| 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.
|
| 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. |
|
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.
|
| ... 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. |
|
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.
|
| 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.
|
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:
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. |
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
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. |
| 15.4 | Jixin Ma |
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.)
|
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.
|
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.
|
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
|
| Not quite right. In my simple theory,
|
Pat wrote
| 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
|
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.
|
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:
|
|
| 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)?
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. |
| 18.4 | Sergio Brandano |
Salve.
The following are some fragments from the current discussion:
From Pat Hayes - ENRAC 14.3.1998
| instantaneous intervals completely. It is also quite consistent to have
arbitrary amounts of density, discreteness, etc.; for example, one can
say that time is continuous except in a certain class of 'momentary'
intervals whose ends are distinct but have no interior points.
|
| 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.
|
Here in Pisa, we write ``continuity'' and we read ``axiom of completeness'', which is what everyone commonly means when speaking about (the founding notion of) continuity. I really find it difficult to believe that you like to make an exception in this sense, also because the hat here is ``formal (temporal) reasoning''. It also seems to me that any temporal structure must necessarily fail to be persuasive if on one hand it includes the notion of continuity and on the other it refuses it; how can time be continuous ... with some exception? Either it is continuous, or it is not! That is, either the Basic Time Structure assumes the axiom of completeness, or it does not!!
In fact, in this discussion I have not yet seen any explanation why an alternative notion of continuous structure is needed at all? I am not asking you to argue about your own notion, I just ask you to give a convincing argument on the need of a notion which is an alternative to the classical one, such as: ``the problem P of temporal reasoning about actions and change can not be solved adopting the axiom of completeness'', or ``the axiom of completeness is too strong an assumption for our purposes; axiom A is better suited, because...''.
Sergio
| 21.4 | Jixin Ma |
To Pat
As an addition to my response (ENRAC 15.4 (98035)) to Pat's suggestion of "simply map Allen and Hayes' moments to Ma and Knight's points":
The constraint that "moments cannot meet each" will lead to the conclusion that we can have neither a completely discrete nor a completely dense system which contains both moments and decomposable intervals. However, if we revise Allen and Hayes' system to include both points and intervals (including moments), and impose the "not-meet-each-other" constraint on points only, rather than on moments, this objection does not apply.
To Sergio
| Here in Pisa, we write ``continuity'' and we read ``axiom of
completeness'', which is what everyone commonly means when speaking
about (the founding notion of) continuity. I really find it difficult to
believe that you like to make an exception in this sense, also because the
hat here is ``formal (temporal) reasoning''. It also seems to me that any
temporal structure must necessarily fail to be persuasive if on one hand
it includes the notion of continuity and on the other it refuses it;
how can time be continuous ... with some exception? Either it is
continuous, or it is not! That is, either the Basic Time Structure
assumes the axiom of completeness, or it does not!!
In fact, in this discussion I have not yet seen any explanation why
an alternative notion of continuous structure is needed at all? I am not
asking you to argue about your own notion, I just ask you to give a
convincing argument on the need of a notion which is an alternative to
the classical one, such as: ``the problem P of temporal reasoning about
actions and change can not be solved adopting the axiom of completeness'',
or ``the axiom of completeness is too strong an assumption for our purposes;
axiom A is better suited, because...''.
|
| in Pisa, we write ``continuity'' and we read ``axiom of completeness'',
which is what everyone commonly means when speaking'
about (the founding notion of) continuity.
|
As for general treatments, the Basic Time Structure does not have to impose the axiom of density or discreteness (Similar arguments apply to issues such as linear/non-linear, bounded/un-bounded). Therefore, the time structure as a whole may be continuous or discrete, or neither continuous nor discrete.
Now, "why an alternative notion of continuous structure is needed at all"? It has been noted that, temporal knowledge in the domain of artifical intelligence, including "temporal reasoning about actions and change", is usually imcomplete, and using time intervals in many cases is more convenient and more in-keeping with common sense of temporal concepts than to use the classical abstraction of points. In fact, the notion of time intervals (or periods) has been introduced for a long time in the literature. In addition, in order to overcome/bypass the annoying question of whether various intervals are open or closed, various approached have been proposed. An example is Allen's interval-based time theory. As for these time theories, the old (classical?) notion of continuity no longer simply applies.
Jixin
| 21.4 | Pat Hayes |
Sergio Brandano wrote:
| The following are some fragments from the current discussion: From Pat Hayes - ENRAC 14.3.1998 |
| instantaneous intervals completely. It is also quite consistent to have arbitrary amounts of density, discreteness, etc.; for example, one can say that time is continuous except in a certain class of 'momentary' intervals whose ends are distinct but have no interior points. |
| From Jixin Ma - ENRAC 15.4.1998 |
| 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. |
|
Pat and Jixin, what do you mean when you write ``continuous''?
Here in Pisa, we write ``continuity'' and we read ``axiom of
completeness'', which is what everyone commonly means when speaking
about (the founding notion of) continuity. I really find it difficult to
believe that you like to make an exception in this sense, also because the
hat here is ``formal (temporal) reasoning''. It also seems to me that any
temporal structure must necessarily fail to be persuasive if on one hand
it includes the notion of continuity and on the other it refuses it;
how can time be continuous ... with some exception? Either it is
continuous, or it is not! That is, either the Basic Time Structure
assumes the axiom of completeness, or it does not!!
|
You talk about a 'founding notion' of continuity as being that captured by the axiom of completeness. Here, in my view, you commit a philosophical error (especially in Pisa!) There are intuitions about continuity which one can try to capture in various formal ways, but there is no 'founding notion' of continuity other than those intuitions. In the late 19th century, famous mathematicians objected strongly to the view of the continuum as consisting of a set of points, for example. This modern perspective, now taught in high schools, is a modern invention, not a 'founding' notion. It is more recent than the gasoline engine, yet people have had intuitions about smoothness, instantaneity and continuity for eons. (Whether or not one agrees with me on this admittedly controversial point, it seems unwise to identify a mathematical property such as continuity with any kind of axiom until one has verified that no other axiom will do as well; and as I am sure Sergio knows, there are many alternative ways to axiomatize continuity.)
In my view, axioms are tools which we can manipulate at will; they are not set in stone or somehow inevitable. Different formal accounts of time might be appropriate for different purposes or to capture different intuitions. (I agree with Jixin that it is useful to seek a common 'core' theory which can be extended in various ways to describe various possible more complex temporal structures; and that this theory will have to be rather weak.)
| In fact, in this discussion I have not yet seen any explanation why
an alternative notion of continuous structure is needed at all? I am not
asking you to argue about your own notion, I just ask you to give a
convincing argument on the need of a notion which is an alternative to
the classical one, such as: ``the problem P of temporal reasoning about
actions and change can not be solved adopting the axiom of completeness'',
or ``the axiom of completeness is too strong an assumption for our purposes;
axiom A is better suited, because...''.
|
A more mundane example is given by temporal databases, which usually assume in their basic ontology that time is discrete: for example, they routinely describe times as integers representing the number of milliseconds since the birth of Christ. (Of course, one can always insist that these are to be understood as being embedded in a continuum, but then what use is an axiom whose sole purpose is to insist that times exist which have no name and about which nothing can be asserted, other than that they exist?)
Pat Hayes
| 22.4 | John McCarthy |
Pat Hayes wrote
| In my view, axioms are tools which we can manipulate at will; they are not
set in stone or somehow inevitable. Different formal accounts of time might
be appropriate for different purposes or to capture different intuitions.
(I agree with Jixin that it is useful to seek a common 'core' theory which
can be extended in various ways to describe various possible more complex
temporal structures; and that this theory will have to be rather weak.)
|
| A more mundane example is given by temporal databases, which usually assume
in their basic ontology that time is discrete: for example, they routinely
describe times as integers representing the number of milliseconds since
the birth of Christ. (Of course, one can always insist that these are to be
understood as being embedded in a continuum, but then what use is an axiom
whose sole purpose is to insist that times exist which have no name and
about which nothing can be asserted, other than that they exist?)
|
As to the rhetorical "what use", suppose the theory is to tolerate the elaboration that two successive events, shooting Pat and his falling to the ground, occurred between successive ticks of the clock. If you guarantee that no such elaborations will be required or that you are willing to do major surgery on your theory should elaboration be required, then you are ok with a weak theory even if it is unextendable.
| 22.4 | Sergio Brandano |
In reply to Pat and Jixin.
I apologize for the length of this message, although it mainly consists of quoted text. As ``skin perception'', it seems to me my critics hits the target. The arguments of reply I received, in fact, are not as convincing as they were supposed to be. The details follow.
To Jixin
| First of all, what do you mean "the classical one"? (the classical
continuous time structure)? Does it refer to the classical physical
model of time, where the structure is a set of points which is
isomorphic to the real line?
|
| Here in Pisa, we write ``continuity'' and we read ``axiom of
completeness'', which is what everyone commonly means when speaking
about (the founding notion of) continuity.
At the ontological level, the notion of continuous time vi discrete time is closely related to questions "Is the set of time elements dense or not?", and " Are there really time atoms?".
|
| For a point-based model, the continuity is usually characterized as
"Between any two points, there is a third"; while for an
interval-based model (like that of Allen), it is characterized as
"Every interval can be decomposed into two adjacent sub-intervals".
|
Let be
Now, the set
| As for general treatments, the Basic Time Structure does not
have to impose the axiom of density or discreteness (Similar
arquements apply to issues such as linear/non-linear,
bounded/un-bounded). Therefore, the time structure as a whole may be
continuous or discrete, or neither continuous nor discrete.
|
| Now, "why an alternative notion of continuous structure is needed at
all"? It has been noted that, temporal knowledge in the domain of
aritifical intelligence, including "temporal reasoning about actions
and change", is usually imcomplete, and using time intervals in
many cases is more convenient and more in-keeping with common
sense of temporal concepts than to use the classical abstraction of
points. In fact, the notion of time intervals (or periods) has been
introduced for a long time in the literature. In addition, in order
to overcome/bypass the annoying question of if intervals are open or
closed, various approached have been proposed. An example is Allen's
interval-based time theory. As for these time theories, the old
(classical?) notion of continuity no longer simply applies.
|
Let me ask you a more stringent question.
Premise: It is evident that if you assume the axiom of completeness,
the domain
Question: Suppose that you define your neither continuous nor discrete
Temporal Structure. What is your domain
To Pat
| Why cannot time be continuous in some places but discontinuous at others?
|
| There is no mathematical objection to such a structure, and it has been
|
| You talk about a 'founding notion' of continuity as being that captured by
the axiom of completeness. Here, in my view, you commit a philosophical
error (especially in Pisa!) There are intuitions about continuity which
one can try to capture in various formal ways, but there is no 'founding
notion' of continuity other than those intuitions. ...
|
Concerning the intuition, let me remind that the student who discovered the square root of 2 was killed (down the cliff), and no one was allowed to speak about ... ``the fault of the god'' for long time. Humans' common sense, to me, is something we shall not call too much.
| century, famous mathematicians objected strongly to the view of the
continuum as consisting of a set of points, for example. This modern
perspective, now taught in high schools, is a modern invention, not a
'founding' notion. It is more recent than the gasoline engine, yet people
have had intuitions about smoothness, instantaneity and continuity for
eons. (Whether or not one agrees with me on this admittedly controversial
point, it seems unwise to identify a mathematical property such as
continuity with any kind of axiom until one has verified that no other
axiom will do as well; and as I am sure Sergio knows, there are many
alternative ways to axiomatize continuity.)
|
If another axiom exists, which does as well, then it is surely equivalent to the axiom of completeness, just because it does as well. Alternative notions are clearly equivalent, until we speak about continuous domains. The point here, instead, was whether one can have a continuous domain with exceptions, that is the claim I originally criticized.
| In my view, axioms are tools which we can manipulate at will; they are not
set in stone or somehow inevitable. Different formal accounts of time might
|
| be appropriate for different purposes or to capture different intuitions.
(I agree with Jixin that it is useful to seek a common 'core' theory which
can be extended in various ways to describe various possible more complex
temporal structures; and that this theory will have to be rather weak.)
|
Finally, concerning your examples:
| The 'dividing point' problem which gave rise to this discussion would do.
According to the modern account of the continuum, this point must exist,
and since all intervals consist of points, the light is therefore either on
or off at it. But it seems more natural, as well as formally simpler, to
just say that the question is meaningless; perhaps (though this is no
longer my own preference) because that point doesn't exist.
|
| A more mundane example is given by temporal databases, which usually assume
in their basic ontology that time is discrete: for example, they routinely
describe times as integers representing the number of milliseconds since
the birth of Christ. (Of course, one can always insist that these are to be
understood as being embedded in a continuum, but then what use is an axiom
whose sole purpose is to insist that times exist which have no name and
about which nothing can be asserted, other than that they exist?)
|
Best Regards Sergio
| 22.4 | Erik Sandewall |
Pat,
In answer to Sergio, you wrote
| Why cannot time be continuous in some places but discontinuous in
others?
|
The first problem is with respect to motivation. For what reasons would Time suddenly skip over potential timepoints? If the reason is, as you wrote, that
| The 'dividing point' problem which gave rise to this discussion would do.
According to the modern account of the continuum, this point must exist,
and since all intervals consist of points, the light is therefore either on
or off at it. But it seems more natural, as well as formally simpler, to
just say that the question is meaningless; perhaps (though this is no
longer my own preference) because that point doesn't exist.
|
The other problem is with respect to the axiomatizations. Since your
article "A catalog of temporal theories" characterizes the various
theories through axiomatizations, I thought I'd go back to that article
and check how you had done this formally. However I was not able to
find it; the closest I got was the denseness axiom on page 15. If the
intuitive notion is that time itself is continuous in some places but
not in others, wouldn't it be natural to start with an axiomatization
of continuous time (such as the real numbers) and then to proceed from
there? For example, a domain of piecewise continuous time could be
represented as a twotuple
Maybe I'm missing something - are constructs of this kind subsumed by the axioms in your report, or can they be inferred as theorems? Or why is this not the natural way of doing things?
Erik
| 23.4 | Jixin Ma |
To Sergio,
| First of all, what do you mean "the classical one"? (the classical continuous time structure)? Does it refer to the classical physical model of time, where the structure is a set of points which is isomorphic to the real line? |
| I can just quote myself ... |
| Here in Pisa, we write ``continuity'' and we read ``axiom of completeness'', which is what everyone commonly means when speaking about (the founding notion of) continuity. |
|
Concerning the core theory that you and Jixin are willing to obtain,
I already developed a Basic Time Structure which may be of interest.
It is as simple as I managed to design it, without un-useful
complications. The structure works well in my case. you are welcome
to read and comment my contribution, which may be found in my ETAI's
reference.
|
| At the ontological level, the notion of continuous time vi discrete time is closely related to questions "Is the set of time elements dense or not?", and " Are there really time atoms?". |
|
The word "continuity", even at the ontological level, can not be read
as "continuous with some exception".
|
| The axiom of completeness states:
Let be
|
Secondly, you take time-points as real numbers, and intervals "from" the real line. Are your intervals sets of real numbers limited by their end-points (real numbers)? If no, what are they? If yes, have you considered the dividing instant problem? This problem would be more obvious with your time structure when you try to impose the axiom of completeness (see below).
Thirdly, if the domain S consists of time-intervals, you need to
re-define (or revise, or, at least, explain) the
relation
By the way, may I take this as one of the "un-useful" complications with your time structure?
| As for general treatments, the Basic Time Structure does not have to impose the axiom of density or discreteness (Similar arquements apply to issues such as linear/non-linear, bounded/un-bounded). Therefore, the time structure as a whole may be continuous or discrete, or neither continuous nor discrete. |
|
I agree with your premise: the Basic Time Structure does not have to
impose the choice, in fact it leaves you free in that sense. As soon as
you make the choice, then you obtain either a continuous structure or
a discrete structure, just depending on this choice. I do not agree,
instead, with your conclusion. If I leave you the freedom to choose,
it does not mean the Structure is neither continuous nor discrete; it
simply means you still have to make the choice.
|
| Now, "why an alternative notion of continuous structure is needed at all"? It has been noted that, temporal knowledge in the domain of artifical intelligence, including "temporal reasoning about actions and change", is usually imcomplete, and using time intervals in many cases is more convenient and more in-keeping with common sense of temporal concepts than to use the classical abstraction of points. In fact, the notion of time intervals (or periods) has been introduced for a long time in the literature. In addition, in order to overcome/bypass the annoying question of if intervals are open or closed, various approached have been proposed. An example is Allen's interval-based time theory. As for these time theories, the old (classical?) notion of continuity no longer simply applies. |
|
My question referred to what is needed rather than convenient.
I understand it may be convenient, in some cases, to use intervals, but
this is not pertinent with my criticism, which still holds.
|
| Let me ask you a more stringent question.
Premise: It is evident that if you assume the axiom of completeness,
the domain
|
| Question: Suppose that you define your neither continuous nor discrete
Temporal Structure. What is your domain
|
The basic core theory doesn't commit itself to whether the time stucture is continuous or discrete. So, if you would like one which is neither continuous nor discrete, you don't need the axiom of completeness. Why do I need a replacement for it, anyway, if it is not supposed?
Extra axioms regarding dense/discrete, linear/non-linear, bounded/non-bounded time structure, etc. can be given (e.g., see Ma and Knight's 1994 paper [j-cj-37-114]). Specially, the characterisation of continuity does not have to be in the form of axiom of completeness. In addition, as shown above, in the case where time intervals are addressed, it becomes very complicated (if not impossible) to simply apply such an axiom.
As for example you would like to see, the DIP is a typical one, as I have shown in the above.
Also, in your reply to Pat you wrote:
| What properly formalizes the notion of continuity is the axiom of
completeness. Alternative notions are equivalent, until we speak
about continuous domains. The point was whether one can have a
continuous domain (that is the If another axiom exists, which does as well, then it is surely equivalent to the axiom of completeness, just because it does as well. Alternative notions are clearly equivalent, until we speak about continuous domains. The point here, instead, was whether one can have a continuous domain with exceptions, that is the claim I originally criticized.
|
Second, the question of "whether one can have a continuous domain with exceptions" depends on how do you understand the real meaning. It is important to note that neither Pat nor myself claims that one can have such a structure as you understood and hence described by "a continuous domain with exceptions". Of course, if you have already assumed that the domain as a whole is continuous, then it must be continuous - no exception! This is just like if you impose that "The traffic light was green throughout last week", then, of course, it was green any time during last week, no exception. Similarly, if you impose that "The traffic light was red throughout last week", then it was red any time during last week. Again, no exception. However, if you don't have either of them, why can't one have the case that over the last week, the traffic light was sometimes red, and sometimes green, and even sometimes yellow?
As I said earlier, when Pat talked about "continuous with exceptions", he actually meant that "except at those time moments, the time is continuous", or more specially, "except for time moments, each time interval is decomposable". I don't think he would actually assume, in the first place, the continuity of the whole domain, then expect there are some exceptions. Do I understand your meaning rightly, Pat?
Jixin
References:
| j-cj-37-114 | Jixin Ma and Brian Knight. A General Temporal Theory. Computer Journal, vol. 37 (1994), pp. 114-123. |
| 23.4 | Sergio Brandano |
Pat Hayes wrote (ENRAC 21.4.1998):
| Why cannot time be continuous in some places but discontinuous at others?
There is no mathematical objection to such a structure, and it has been
argued that a continuum punctuated by a sparse collection of points of
discontinuity might be a plausible mathematical picture of time which seems
to 'flow smoothly' except when things happen suddenly. (Similar arguments
can be made for describing spatial boundaries, by the way; and elementary
physics makes similar assumptions, where velocity is supposed to change
smoothly except when 'impact' occurs.)
|
Sergio
| 24.4 | Pat Hayes |
John McCarthy wrote
| If axioms are guaranteed to be used only in a particular program or
set of programs, they need be no stronger than necessary.
As to the rhetorical "what use", suppose the theory is to tolerate the
elaboration that two successive events, shooting Pat and his falling
to the ground, occurred between successive ticks of the clock. If you
guarantee that no such elaborations will be required or that you are
willing to do major surgery on your theory should elaboration be
required, then you are ok with a weak theory even if it is
unextendable.
|
| In my view, axioms are tools which we can manipulate at will; they are not
set in stone or somehow inevitable. Different formal accounts of time might
be appropriate for different purposes or to capture different intuitions.
...
|
| A more mundane example is given by temporal databases, which usually assume
in their basic ontology that time is discrete:...
|
Answers to Sergio Brandano
Sergio seems to be on a different planet, as his responses to both Jixin and I seem to quite miss the point of our debate, and often to be completely free of content.
| I can just quote myself ...
|
| Here in Pisa, we write ``continuity'' and we read ``axiom of
completeness'', which is what everyone commonly means when speaking
about (the founding notion of) continuity.
|
| The word "continuity", even at the ontological level, can not be read
as "continuous with some exception".
|
| For a point-based model, the continuity is usually characterized as "Between any two points, there is a third"; while for an interval-based model (like that of Allen), it is characterized as "Every interval can be decomposed into two adjacent sub-intervals". |
|
The axiom of completeness states:
Let be
|
| Premise: It is evident that if you assume the axiom of completeness,
the domain
|
| Question: Suppose that you define your neither continuous nor discrete
Temporal Structure. What is your domain
|
| Why cannot time be continuous in some places but discontinuous at others? |
|
Places? If we shall understand time like (physicists) understands the
space, ...
|
| There is no mathematical objection to such a structure, and it has been |
|
If a Temporal Structure exists in this sense, may I have a look at
its domain (that is at the
|
But this is a trivial challenge. It can be done for any set
P . . . . . . . . S ...........|\|\|\|\|\|\.........|\......
looks like this when 'straightened out':
............. .. .. .. .. .. .. .. .. .............. .............
(BTW, another way to describe this is that each point in the
(Aside to Jixin: this is the intuition behind the idea of replacing moments by points. The endpoints of a moment can be thought of as the result of this construction on a smaller set of points, and the construction can be reversed by identifying the endpoints of the moment, ie treating the moment as being pointlike. The result is a timeline with some points identified as being 'interval-like', ie capable of having something true at them. If moments never meet, then all the axioms of the Allen-Hayes theory apply to the S-line iff they applied to the original. This is why your theory and ours are essentially the same. )
Erik Sandewall wrote
| ..... I have no problems accepting that
a function of time may be piecewise continuous, or that it may be
undefined for some points along the time axis. However, it seems to
me that there are several problems with saying that time itself
is piecewise continuous (btw - do you mean piecewise dense?).
|
| The first problem is with respect to motivation. For what reasons would Time suddenly skip over potential timepoints? If the reason is, as you wrote, that |
| The 'dividing point' problem which gave rise to this discussion would do. According to the modern account of the continuum, this point must exist, and since all intervals consist of points, the light is therefore either on or off at it. But it seems more natural, as well as formally simpler, to just say that the question is meaningless; perhaps (though this is no longer my own preference) because that point doesn't exist. |
|
then exactly what events in the world would be allowed to contribute to
the continuity faults? Does the next time I hit a key on my keyboard
qualify?
|
| And what about the midpoint halfway between two continuity faults,
is it also a continuity fault, recursively?
|
| The other problem is with respect to the axiomatizations. Since your
article "A catalog of temporal theories" characterizes the various
theories through axiomatizations, I thought I'd go back to that article
and check how you had done this formally. However I was not able to
find it; the closest I got was the denseness axiom on page 15.
|
As to whether time really is discrete or continuous, etc., the only people who can answer questions like that are physicists, not we who merely craft ontologies.
| If the
intuitive notion is that time itself is continuous in some places but
not in others, wouldn't it be natural to start with an axiomatization
of continuous time (such as the real numbers) and then to proceed from
there? For example, a domain of piecewise continuous time could be
represented as a twotuple
|
| Maybe I'm missing something - are constructs of this kind subsumed by
the axioms in your report, or can they be inferred as theorems?
|
Pat
References:
| mb-Benthem-83 | Johan van Benthem. The Logic of Time. , 1983. |
| 24.4 | Sergio Brandano |
In reply to Jixin Ma (ENRAC 23.4.1998)
| So, you didn't refer "the classical one" to "the Basic Time
Structure" you developed, did you? If No, why did you develop it?
What is your convincing argument(s) on the need of such a structure?
Is it also an alternative to the classical one? (Sorry, I am here
using the similar question raised by youself to ask you, though I
don't have to). If Yes, I shouldn't ask this question.
|
By "the basic time structure" I mean a basic (minimal) time structure.
By "the time structure X" I mean the temporal structure we like to deal with. It is obtained from the basic time structure via additional axioms.
You also invited me to be more explicit with respect to the following sentence.
| The axiom of completeness states:
Let
|
An interval from the real-line is an ordered set of real numbers limited by its end-points, which are not necessarily included in the set.
Suppose
Suppose now that
I could not penetrate the rest of your message.
Best Regards
Sergio
| 3.5 | Pat Hayes |
A comment on Sergio's reply to Jixin:
| An interval from the real-line is an ordered set of real numbers limited
by its end-points, which are not necessarily included in the set.
|
| Suppose
|
| Suppose now that
|
Pat Hayes
| 3.5 | Jixin Ma |
To Sergio, who wrote:
| The (temporal) domain
|
Anyway, you have claimed that "The (temporal) domain S may
consist either of time-points or (exclusive-or) of time-intervals",
and "an interval from the real-line is an ordered set of real numbers
limited by its end-points, which are not necessarily included in the
set."
In this case, can your intervals be "pointlike"? That is, for an interval
I have shown in my former message that if your
domain
In fact, you define the (partial) relation "
By the way, it seems that your description of the axiom of completeness is not a first-order one.
Jixin
| 3.5 | Sergio Brandano |
In ENRAC 24.4.1998 I made a typing mistake. I wrote: "
In reply to Jixin Ma (ENRAC 23.4 and 24.4 1998) -- completion:
| So, you think intervals are not needed? Anyway, our arguments...
|
| ... about the convenience of using intervals are based on the belief
of the need of them.
|
| Premise: It is evident that if you assume the axiom of completeness,
the domain |
|
Wrong! Even if you do not asssume the axiom of completeness, it is
still not nessarily discrete.
|
Concerning the dividing instant problem, which seems to summarize what is left from your objections, please read below.
In reply to Pat Hayes (ENRAC 24.4.1998):
As posted in my original message, I have not yet seen any explanation why an alternative notion of continuous structure is needed at all?
Probably, in order to prevent any misunderstanding, I should have included an additional sentence like "... is needed at all, within the search of those non-monotonic logics which purpose is to formalize common sense reasoning when reasoning about actions and change", but I thought it was evident, as the title of this Newsletter reminds.
In particular, in the same message, I asked to give at least one convincing argument on the need of a notion which is an alternative to the classical one, along the lines: "the problem P of temporal reasoning about actions and change can not be solved adopting the axiom of completeness", or "the axiom of completeness is too strong an assumption for our purposes; axiom A is better suited, because..." ( >>>>star )
You and Jixin Ma proposed the "dividing instant problem", apropos of
the problem of switching on the light, and argued the axiom of
completeness inadequate for solving that problem. The formulation
I gave in ENRAC 24.4.1998, with today's minor adjustment, gives the
evidence on how the axiom of completeness is, instead, safe with
respect to the dividing instant problem. You and Jixin based your
argument on the fact that I do not allow the domain
You also gave other examples, but you did not explain how they
relate to the world of "Reasoning about Actions and Change". In
particular, and I somehow repeat myself, it is not evident that one
needs a temporal domain with non-homogeneous continuity (let me say it
is even less evident the need of the imaginary number
You also gave an informal argument on the plausibility of a temporal structure which formalizes the perceived smooth flux and perceived fast flux of time (ENRAC 21.4.1998). I refuted that plausibility with my contribution to ENRAC 23.4.1998.
(Is it really ``free of context'' to you ?)
Best Regards
Sergio
| 4.5 | Sergio Brandano |
In reply to Pat Hayes (ENRAC 3.5.1998)
| An interval from the real-line is an ordered set of real numbers limited
by its end-points, which are not necessarily included in the set.
|
| It seems from this that the set of intervals is supposed to include open,
half-open and closed intervals; is that right? (Or do you mean to say that
there may be some doubt about whether a particular interval does or does
not include its endpoints? If the latter, this is not the usual notion of
'interval' as used in real analysis, and you need to explain further.)
|
You posed a good question, which may call into the present debate the possible relations between epistemological and ontological assumptions, at least within the "Features and Fluents" framework.
If we assume the epistemological assumption
| Suppose |
|
|
You are right concerning the case whether
| Suppose now that
|
| Consider the closed intervals
|
Best regards
Sergio
| 7.5 | Jixin Ma |
In ENRAC 3.5 (980521), Sergio wrote:
| I am actually skeptic about the need of a temporal domain which
includes time-intervals. There are many convincing arguments that a
temporal domain consisting of time-points is good enough in many
different situations (Newtonian mechanics and Thermodynamics, for
instance, as well as Sandewall's underlying semantics
for
|
| ... about the convenience of using intervals are based on the
belief of the need of them.
|
| ... According to the standard scientific
methodology, in fact, we shall build on top of already existent
solutions, and be consistent. Just to make an example, suppose one
refuses a classical notion (continuity?), and encounters the
problems that this notion was used to solve (the dividing instant?);
it is surely not consistent to justify the need for a novel approach
via the claim that the problems he encountered can not be solved by
the notion he just refused.
|
Anyway, while I (and many others) have seen the convenience of using intervals, I can also see the need of them. In fact, there have been quite a lot of examples (many) in the literature that demonstrated the need of time-intervals (or time-periods). Haven't you ever encountered any one of them? Or you simply cannot see anyone of them is convincing?
All right, let's just have a look at the example of throwing a ball up into the air. As I showed in ENRAC 1.4 (98033) (one may disagree with this), the motion of the ball can be modelled by a quantity space of three elements: going-up, stationary, and going-down. Firstly, or at least, we can see here the convenience of using intervals. In fact, we can conveniently associate the property that "the ball changes its position" with some time-intervals. Secondly, let's see if we indeed need time-intervals. Without the notion of time-intervals (neither primitive nor derived from time-points), can you just associate such a property with time-points? Yes, we may associate it with a pair of points. However, this doesn't mean that the property holds at these points. What it really means is that the property holds for the time periods denoted by the pair of points. Aren't these time periods in fact time intervals?
It is important to note, up to now in the above, I just talked about the need of the notion of intervals. As for how to characterise intervals (e.g., are intervals taken as primitvie or derived structures from time-points?) is another important issue, and this issue, again, has been addressed in the literature for a long time.
The Point Is: while we were/are discussing/arguing about some broader issues on temporal ontology, you just jumped in and asked "why an alternative notion of continuous structure is needed at all?" First of all, the "continuity" (or more truly, density) is not the main issue we are talking about. The fundamental question is if we need to address and how to addess time intervals. Based on such a discussion, in the case that intervals are taken as temporal primitive, then, we are talking about how to characterise some corresponding issues including dense/discrete structures. But your questions and arguments/replies do not seem to follow this. As stated in the former replies from both Pat and myself, first of all, the dense structure does not have to be characterised in terms of the only form of the so-called "axiom of completeness". Also, in the case where time-intervals are involved (even they are still point-based, let alone in the case they are taken as primitive), such an axiom doesn't simply apply. In fact, I have shown this twice with different notations in this discussion. I will point out more problems in detail below in my response to your reply to Pat.
| Concerning the dividing instant problem, which seems to summarize
what is left from your objections, please read below.
|
| In reply to Pat Hayes (ENRAC 24.4.1998):
|
| As posted in my original message, I have not yet seen any
explanation why an alternative notion of continuous structure is
needed at all?
|
| You and Jixin Ma proposed the "dividing instant problem", apropos of
the problem of switching on the light, and argued the axiom of
completeness inadequate for solving that problem. The formulation I
gave in ENRAC 24.4.1998, with today's minor adjustment, gives the
evidence on how the axiom of completeness is, instead, safe with
respect to the dividing instant problem. You and Jixin based your
argument on the fact that I do not allow the domain
|
| The closed intervals
|
It follows that you do need alternation, doesn't it?
(Note that this is just for the case
when you construct intervals out of points. In the case where
intervals are taken as primitive, the need of such alternative is
indeed more conceptually necessary). However, your adjustment is not
enough, or you haven't reached the proper form for general treatments.
In fact, you need to address the issue regarding different cases. To see
this, you may just consider the difference between the case where at
least one of
Jixin
| 8.5 | Sergio Brandano |
In reply to Jixin Ma (ENRAC 7.5.1998)
| Pat's example becomes invalid only after you made the "minor
adjustment" that replaces the relation
|
The only one "minor adjustment" I made consists in the first four lines of my contribution to ENRAC 3.5.1998, where no inequality appears at all. Concerning the hypothesis, I remind you what I wrote in ENRAC 24.4.1998:
| Suppose now that
|
| So, you do need alternation, don't? (And this is just for the case ...
|
| ...when you construct intervals out of points. In the case where
intervals are taken as primitive, the need of such alternative is
indeed more conceptually necessary). However, your adjustment is not
enough, or you haven't reached the proper form for general treatments.
In fact, you need address the issue regarding different cases. To see
this, you may just consider the difference between the case where at
least one of
|
1.
2.
3.
where
Note I used
Pat's example:
- trivially fails when using
< , - trivially succeeds (
xi = [q, q] ) when using< .
| use
|
4.
where
So, the axiom of completeness has no problems with your examples.
Concerning the first part of your message, as you wrote in it, it was entirely based on the DIP problem and the above argument-examples.
| while I (and many others) have seen the convenience of using
intervals, I can also see the need of them. In fact, there have been
quite a lot of examples (MANY) in the literature that demonstrated the
need of time-intervals (or time-periods). Haven't you ever encountered
any one of them? Or you simply cannot see anyone of them is
convincing?
|
There exists at least one problem (within R.A.C) that needs to introduce intervals into the temporal domain?
The other problem is:
There exists at least one problem (within R.A.C.) that can not be solved with a continuous temporal domain, so that to justify a temporal domain with non-uniform continuity?
This debate aims at generality, surely does not aim at completeness of case examples. If many examples do exist, then this is the proper debate where at least the most representative of them should appear "naked" under the spotlight, for general benefit. On the other hand, I note that more than two weeks are now passed from my criticism, and no such representative example appeared.
Sergio
| 12.5 | Jixin Ma |
Reply to Sergio Brandano (ENRAC 8.5.1998)
| The only one "minor adjustment" I made consists in the first four
lines of my contribution to ENRAC 3.5.1998, where no inequality
appears at all. Concerning the hypothesis, I remind you what I wrote
in ENRAC 24.4.1998:
|
| The axiom of completeness imposes
|
| ...when you construct intervals out of points. In the case where
intervals are taken as primitive, the need of such alternative is
indeed more conceptually necessary). However, your adjustment is not
enough, or you haven't reached the proper form for general treatments.
In fact, you need address the issue regarding different cases. To see
this, you may just consider the difference between the case where at
least one of
|
|
... at least one is closed. So we have, since
Note I used Pat's example:
|
This is exactly what I wanted to show and have shown a few times now. That is, in the case there your domain S contains intervals, to fulfill the axiom of completeness, S has to contains singletons (single points) as well, not as you specially claimed that S contains points or (exclusive-or) intervals. My observation that Pat's example would be valid is under your assumption that the domain S refuses to take both intervals and singletons (points). I think when Pat gave the example, he also followed this assumption of yours. (Actually, Pat did specially claim that "unless you allow intervals consisting of a single point" when he gave the example in ENRAC 24.4.1998).
| use |
|
... the latter case. So we have, since
|
| The problem about intervals is whether one needs to introduce them
into the temporal domain, and the few argument-examples I encountered
are far from being convincing. Furthermore, in this debate, you and
Hayes proposed the DIP, and I refuted it.
|
Jixin
| 17.5 | Pat Hayes |
[S.B.]
| Suppose
|
| It follows then that for intervals,
|
|
|
Sergios point seems to be that one can describe the line in terms of
conventional open and closed intervals in such a way that no 'gaps' appear,
so that the 'interval' between the open intervals
In any case, many users of temporal ontologies do not wish to assume continuity or even density, for reasons of their own, and so a general-purpose temporal ontology should therefore not make such unnecessarily strong assumptions as Sergio's 'completeness' axiom. Temporal database technology usually assumes times are discrete, for example.
Pat Hayes
| 24.5 | Sergio Brandano |
In reply to Pat Hayes [ENRAC 17.5]
|
Sergios point seems to be that one can describe the line in terms of
conventional open and closed intervals in such a way that no 'gaps' appear,
so that the 'interval' between the open intervals
|
In my example, the temporal domain $S$ consists exclusively of
intervals from the real-line and no total order relation was imposed,
so that the interpretation for which one can describe the line with
that domain is surely not correct. If one restricts
| The question is whether this standard mathematical view of the line
is the most suitable for capturing linguistic intuition or for
action reasoning. For example, we want to be able to assert that a
light is off before time
|
\[ to\_switch(s,\tau,t) =
\cases{ {\rm off} & if $\tau = s$ \cr
{\rm on} \vee {\rm off} & if $\tau \in (s,t)$ \cr
{\rm on} & if $\tau = t$}
\]
where the interval
Now, assume the light is initially "off". Then it will remain "off"
until
| In any case, many users of temporal ontologies do not wish to assume
continuity or even density, for reasons of their own, and so a
general-purpose temporal ontology should therefore not make such
unnecessarily strong assumptions as Sergio's 'completeness'
axiom. Temporal database technology usually assumes times are
discrete, for example.
|
(b) The problem that comes when aiming at generality (as in this debate?), is one shall include all possible cases, instead of excluding those which are of no interest for someone. Personally, I like to consider problems within Newtonian physics as being an important part of Reasoning about Actions and Changes, where continuity plays an fundamental role when idealizing the physical world. Furthermore a continuous temporal-domain subsumes the case of integer time, as well as the case of rational time, so that it is general enough to embrace all possible cases.
Therefore, in my opinion, the axiom of
completeness is a good assumption, at least
within the classical point-based approach. The purpose of my example
with
Finally, I would like to resume Jixin's original statements in ENRAC 13.3:
| (1) For general treatment, both intervals and points are needed.
(2) To overcome the so-called Dividing Instant Problem, that is the
problem in specifying whether intervals is "open" or "closed" at their
ending-points, both intervals and points should be treated as
primitive on the same footing.
|
| Does there exist at least one problem (within R.A.C) that needs to
introduce intervals into the temporal domain?
|
Best Regards Sergio
| 24.5 | Erik Sandewall |
Pat,
You wrote (ENRAC 17.5]:
|
The question is whether this standard mathematical view of
the line is the most suitable for capturing linguistic intuition or for
action reasoning. For example, we want to be able to assert that a light is
off before time |
The most natural way of dealing with such a situation, it seems to me,
is to admit that one's axioms allow two kinds of models: those where
the light is on at time
The alternative that has occurred in the present discussion, and proposed
as a way of dealing with the "dividing instant problem"
is the use of a punctuated timeline where the domain
of timepoints is chosen e.g. as
Given that the standard view already exists, it seems worthwhile to understand what the concrete reasons would be for replacing it with the punctuated timeline ontology. Apart from the purely subjective reasons (that is, some people prefer to do it that way) I wonder what results have been obtained using punctuated or other nonstandard ontologies, and which are not trivially translatable back to the standard view. The following would seem to be interesting results for this purpose:
- Expressiveness in a concrete scenario. This ought to be a scenario that can't be properly expressed in the standard ontology, but which can be expressed in the nonstandard one.
- Algorithms or other implementation techniques. Is there some algorithm that works with a punctuated timeline and that outperforms those that don't?
- Validation and range of applicability results. These are results stating that or when a certain entailment method is correct, that is, it obtains exactly the intended models viz conclusions. Have some such results been obtained using nonstandard ontologies?
Please feel free to add more categories to the list. (In the methodology discussion at the NRAC workshop at IJCAI last year we got to more than ten such categories).
I realize of course the purely philosophical and/or logical interest in analyzing different possible concepts of time, but from an AI point of view there is no point in pursuing an approach if it doesn't deliver any results. I haven't been able to see any indications of such concrete results from the present discussion or from the articles that have been referenced in it. (That most work in temporal databases uses discrete time doesn't say anything about the choice between a standard or a punctuated real timeline, does it?)
Although I'd prefer to see algorithmic results or range of applicability results, a few words about expressiveness since after all that's also a relevant type of achievement. One nice scenario for hybrid change (that is, continuous and discrete) is the impact problem that Persson and Staflin used in their ECAI 1990 paper [c-ecai-90-497]. It goes like this: two solid spheres B and C are lying side by side on a horizontal surface, B to the left of C. A third sphere, A, comes rolling from the left and hits B. As we know from physics, the result of the impact is that A stops, B stays, and C starts moving towards the right. This particular exercise has been solved in 1990, but is there now some harder one that makes essential use of punctuated time ontology?
Erik
References:
| c-ecai-90-497 | Tommy Persson and Lennart Staflin. A Causation Theory for a Logic of Continuous Change. Proc. European Conference on Artificial Intelligence, 1990, pp. 497-502. Also available as Linköping technical report Nr. 90-18 [postscript]. |