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Ontologies for time
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.
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in answer to my remarks:
| | 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.
...
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and
| | A more mundane example is given by temporal databases, which usually assume
in their basic ontology that time is discrete:...
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If I understand what John is saying, then I completely agree with him. (If
he intended to disagree with me, then maybe I dont understand his point.)
However, note that both Jixin and I are trying to give a theory which is as
elaboration-tolerant as possible, without being completely vacuous.
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 ...
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Well, you can; but to do so is at best unhelpful, and at worst arrogant.
If someone fails to understand you and asks for clarification, to simply
repeat yourself is obviously unlikely to give them the clarification they
need.
| | 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.
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May I ask in return if "everyone" here is meant to refer to everyone in
Pisa, or to a broader community? If the former, my advice is to travel
more; if the latter, then you are simply wrong.
| | The word "continuity", even at the ontological level, can not be read
as "continuous with some exception".
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There is an entire mathematical theory of punctuated continua, ie spaces
which are continuous everywhere except for a non-dense set of points. Such
structures even arise naturally from purely continuous phenomana in, for
example, catastrophe theory.
The formal trick, you see, is to alter the axiom so that instead of reading
'for all points...' it reads 'there exists a set S such that for all points
not in S ...'. The result is also an axiom, believe it or not.
| | 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".
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The axiom of completeness states:
Let be A and B non empty subsets of S such that a < b
for all a in A and b in B . Then exists xi in S such that
a < xi < b for all a in A and b in B .
Now, the set S , that is your domain, may consists as well either of
time-points or time-intervals; S holds real numbers on the former
case, intervals from the real line on the latter case.
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You havent said what < means for intervals. If it means that the
endpoint of a is point- < the first point of b , then this axiom seems
false; for consider a point p and the set A = { <p1, p> } for any
p1 < p and B = { <p, p2> } for any p2 > p . This satisfies your
premise, but there is no
interval between any of the intervals in A and any interval in B (unless
you allow intervals consisting of a single point.) But in any case, you are
here assuming that the real line is your intended model. But this axiom
doesnt characterize the real line. Its true on the rationals, for one
thing, but thats not all. For example, here is a nonstandard model of your
axiom: interpret points as pairs <n,q> where n is an integer and q is a
rational number or the symbol " i ", and say that <a,b> < <c,d> just when
a < c v (a = c ^ d = "i") v (a = c ^ b < d) .
This amounts to N copies of
Q laid end-to-end with points at infinity placed between them. It satisfies
your axiom. Im sure that anyone with a little imagination can easily cook
up lots more such nonstandard worlds.
| | Premise: It is evident that if you assume the axiom of completeness,
the domain S can just be continuous, while if you do not assume the
axiom of completeness then S is necessarily discrete.
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If you do not assume the axiom, then S may be discrete, continuous or any
mixture. Did you mean to say, if you assume the negation of the axiom?
But the negation of your axiom simply says that some point of discontinuity
exists; it does not impose a discrete structure on the whole of S . It is
much more difficult to axiomatise a discrete structure than a dense one; in
fact, it cannot be done in first-order logic.
| | Question: Suppose that you define your neither continuous nor discrete
Temporal Structure. What is your domain S ? What is your
replacement for the axiom of completeness?
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See above, but modify the domain to exclude the "i" symbols. This structure
( N copies of Q ) is dense almost everywhere, but your axiom fails to hold
when the sets A and B are infinite in a particular way. There are 2|Q
subsets in the power set of this domain, and only N|2 of them fail your
axiom, so by almost any standard it is true 'almost' everywhere. (Another
interesting example is got by reversing N and Q , so that one has Q copies
of N laid end-to-end. This fails your axiom 'locally', ie when the subests
are only finitely separated, but satisfies it for sufficiently separated
sets. It is like a discrete space which changes to a dense one when the
scale is reduced sufficiently. See J.F.A.K.Van Benthem, The Logic of
Time, [mb-Benthem-83] for a lovely discussion of such examples.
| | Why cannot time be continuous in some places but discontinuous at others?
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Places? If we shall understand time like (physicists) understands the
space, ...
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Yes, that is more or less what I have in mind. Do you propose to formalise
a theory of time which is incompatible with physics? (Why??)
| | There is no mathematical objection to such a structure, and it has been
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If a Temporal Structure exists in this sense, may I have a look at
its domain (that is at the S domain, as stated above)?
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See above examples and use your imagination.
But this is a trivial challenge. It can be done for any set S with a
(strict) ordering < . Select a subset P of S , and define a new order
relation « on S+P as follows: x « y iff
(x in S ^ y in P ^ x = y) xor x < y .
( + here is disjoint union, xor is exclusive-or). This inserts a
'twin' of each point in P just after it, with no points between them. If S
is dense/continuous/whatever, then this new structure is that too
everywhere except at points in P . If P is a dense subset of S , this
construction effectively makes two copies of the original set in that
region with a 'sawtooth' ordering that jumps back and forth between them,
inserting a discrete section into the originally dense ordering:
P . . . . . . . .
S ...........|\|\|\|\|\|\.........|\......
looks like this when 'straightened out':
............. .. .. .. .. .. .. .. .. .............. .............
(BTW, another way to describe this is that each point in the P -subset of S
is replaced by a 'two-sided' point.) If S is dense, then your axiom applies
everywhere except at points in P .
(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?).
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(Yes, most of this discussion is really about density rather than
continuity. Ive just let this ride for now.)
| | 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
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| | 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.
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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?
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It all depends on whether you find it useful to describe things that way.
The idea of intervals simply meeting seems to be a very useful way to
think about time, and it immediately gives rise to all these problems.
| | And what about the midpoint halfway between two continuity faults,
is it also a continuity fault, recursively?
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Not necessarily. (Im not sure quite what your point is here. Must there be
a waterfall exactly between two waterfalls?)
| | 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.
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Throughout the catalog, I give density and discreteness axioms. As I say in
the text, you can take your pick; or, if you like, you can say that time is
dense sometimes and discrete others, making obvious slight changes to the
axioms to make these assertions. The axioms in the theories of the catalog
are offered to you like pieces of an erector set. I make no committment to
their truth, only that they fit together properly.
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 <R,D> where R is the real numbers and
D is a "small" subset of it; the intention being that R-D is the
modified time domain in question. The notions of non-standard intervals
could then be constructed as the natural next step.
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Yes, that is a possible approach. However, (1) the real numbers are already
a very compicated domain to axiomatise, requiring such things as set theory
and notions of limits, etc..; I was looking for something much more
mundane; and (2) as Ive said repeatedly, the real line isnt a very good
model of our temporal intuitions, in my view, but comes along with a lot of
misleading assumptions which are not necessary for temporal reasoning.
| | Maybe I'm missing something - are constructs of this kind subsumed by
the axioms in your report, or can they be inferred as theorems?
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Neither. The time axioms are far too weak to be able to infer anything
about real analysis. However, it should be possible to construct models of
the time axioms using ordinary mathematical notions like the integers and
the reals, and indeed I try to do that for every theory in the catalog. At
the very least, this helps guide ones intuitions about just what it is ones
axioms really say, instead of what one hopes they ought to say.
Pat
References:
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.
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By "the classical one" I mean the classical notion of continuity.
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 A and B be non empty subsets of S such that a < b
for all a in A and b in B . Then there
exists xi in S such that
a < xi < b for all a in A and b in B .
Now, the set S , that is your domain, may consists as well either of
time-points or time-intervals; S holds real numbers on the former
case, intervals from the real line on the latter case.
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The (temporal) domain S , as I meant, may consist either of time-points
xor of time-intervals (exclusive "or").
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 S consists of intervals from the real line. Assume
<s1,t1> in A and <s2,t2> in B , intervals in S . We say that
<s1,t1> < <s2,t2> iff t1 < s2 . The strict order relation
< is an abbreviation for < ^ =/ .
Suppose now that <s1,t1> < <s2,t2> . The axiom of completeness states
the existence of xi in S
such that <s1,t1> < xi < <s2,t2> .
I reply in advance to your next question: "Why did you write < instead
of < ?". The reply is that < means "less or equal", that is
xi may
not be equal to t1 or s2 , but it can do so. Note that since
xi
belongs to S , then xi is an interval. This is also meant as a reply
to your question about the dividing instant problem.
I could not penetrate the rest of your message.
Best Regards
Sergio
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