Issue 98040 | Editor: Erik Sandewall | 24.4.1998 |
Today |
Due to an editing mistake, we yesterday only sent out the first half of Jixin Ma's contribution in the debate about the ontology of time. Today's issue also contains additional contributions by Pat Hayes and Sergio Brandano.
Debates |
Sergio Brandano wrote futhermore:
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. |
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 if 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 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 CJ94 paper). 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 appy 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, Sergio wrote in his reply to Pat:
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 "whehter 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 assume 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 not 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
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
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 be
|
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