At 1:10 PM 0500 1/23/08, Patrick Cassidy wrote:
>PatH,
> Thanks for the answer.
> The implementation of Time that I would like to use is one that I
>couldn't find in the Time Catalog  though I may not have interpreted all of
>the axioms correctly. What I would like is a representation of Time as
>isomorphic to the real line (01)
You can't get isomorphism in a FO axiomatization.
Why do you want continuity? Seems like density is
enough for interval/point reasoning, and for
calculus you need more than continuity anyway. (02)
>with:
> (1) open, closed and semiopen intervals, all distinguishable and specified
>by their endpoints (whether or not the endpoint is included in the interval) (03)
Right. This would be the way to do it. But do you
really want all these distinctions? It means that
any attribution of truth to an interval is forced
to make decisions about truth at the endpoints,
and often  Id say almost always  there is no
way to decide such questions, no actual fact of
the matter. Certainly there is no perceptual or
engineering/measurement way to distinguish an
open from a closed interval. (04)
> (2) a time point as a subtype of a closed time interval, and identical to a
>closed interval of zero length. (05)
Is there any utility in saying a point is a kind
of interval? And what do you say about open
intervals of zero length? (06)
> (3) two time intervals closed at the adjoining ends both overlap and meet
>(I think this is not permitted in Allen's algebra (07)
Right, it is not. This would wreck all the
algebraic timeinterval reasoners. Why do you
want to do this? (08)
>);
> Intervals that both overlap and meet must both be closed at the meeting
>ends and have a single time point as the overlap interval. (09)
Makes sense. But do you want to allow any other
things to be both points and intervals? (010)
> (4) dividing a time interval at a point gives rise (depending on the
>dividing operation) to either (a) two time intervals, each having a closed
>end with the point of division (and conversely, joining two time intervals
>that meet at closed ends merges the common point, which is a single ordinary
>point); or (b) two intervals open at the point of division; or (c) one
>interval (the earlier one) open and one (the later one) closed at the point
>of division. Operations a and b can divide an interval into two equal
>parts, if the starting interval ends are both closed or both open.
>Operation c can equally divide a closedopen interval. (011)
This seems baroquely complicated (how many
dividing operations must one keep track of??) and
also arbitrary: why does the dividing point get
absorbed by the later rather than the earlier
interval? (012)
When you say equal parts: does the closed
interval [a b] have the same duration as the open
interval ( a b) ? Yes, surely. So why aren't
these intervals 'equal' in the required sense (ie
when judging duration.) (013)
> (5) catenating two intervals both open at the joining end creates the point
>of juncture. (014)
? creates? What does that mean? (015)
You need to be careful. Consider the tossed ball.
There are two open meeting intervals where its
vertical velocity is nonzero. But at the point
where they meet it is zero. You don't want the
truth of the proposition (vertical velocity =/=
0) to 'leak' into that point when they are
concatenated. (016)
BTW, this is a real issue. Points like this are
central to qualitative physics reasoners and
planners, which have many real applications in
industry now. (017)
> (6) Two intervals open at the adjacent ends also meet if those ends are
>defined by a common point (018)
Now you have the problem that a pair of points
does not uniquely determine the interval between
them, which runs counter to all intuition and
will destroy many calendar and intervalreasoning
systems. (019)
> (7) time points are both 'part of' and 'contained in' time intervals. (020)
OK (021)
> (8) the default time interval type (if defaults are needed (022)
I have no idea what it would mean, so lets ignore this. (023)
>) is one closed
>at the lower end and open at the upper end 
> (This type is used to specify AM or PM in a day).
> (9) Events that occur at a single time point imply a preceding interval
>open at that point, in which the event state differs from the state at (and
>for some interval after) that point. Though unrealistic for physical
>events (024)
I think its realistic for them too. A light comes
on: even if the process of coming on takes some
small interval of time, that interval must have a
first point. (025)
>, this can be used to specify socially defined events, such as someone
>becoming president at a particular time point.
>
> But I didn't see (and may have missed) a single theory that has all of
>those requirements in the time catalog. Is such a theory actually in the
>time catalog  or elsewhere? (026)
Probably not in the catalog in that exact form,
no. I only considered actual axiomatic theories.
Its impossible to specify the real line in FOL,
for example: you need the notion of continuity,
which is inherently secondorder. But this might
be one possible model of some of the theories, I
havn't checked. The vector continuum treats <b b>
as being both a point and an interval, but it
modifies the notion of 'meet' to preserve an
Allenstyle algebra: you have to say that these
pointintervals meet themselves. In that theory,
<a b> does not overlap <b,b>: it meets it. (027)
> If not, is it somehow internally logically
>inconsistent? (028)
Im sure its not inconsistent, but I am quite
unconvinced that it would be particularly useful
(and certainly not universally acceptable.) (029)
PaT (030)
>
> PatC
>
>Patrick Cassidy
>MICRA, Inc.
>9085613416
>cell: 9085654053
>cassidy@xxxxxxxxx
>
>> Original Message
>> From: ontologforumbounces@xxxxxxxxxxxxxxxx [mailto:ontologforum
>> bounces@xxxxxxxxxxxxxxxx] On Behalf Of Pat Hayes
>> Sent: Wednesday, January 23, 2008 12:12 AM
>> To: [ontologforum]
>> Subject: Re: [ontologforum] Time representation
>>
>> At 11:31 PM 0500 1/22/08, Patrick Cassidy wrote:
>> >Just a question about one point (so to speak):
>> >
>> >>
>> >> Re PTim: I realize that calling an interval a point is
>> problematical.
>> >> But in anything that has to do with the physical world, there is no
>> >> way to specify a true point. Perhaps a better term would be "grain
>> >> in time", abbreviated "Grit".
>> >>
>> >
>> >Has anyone observed any problems of *logical consistency* in
>> classifying a
>> >point on a line as being identical to a closed interval of zero
>> length, with
>> >beginning and end points identical?
>>
>> I presume you mean, interval on the real line.
>> None: this is quite consistent. This model
>> (actually several versions of it) is discussed in
>> the 'time catalog'. But it does produce some
>> complications. For example, consider a ball
>> tossed into the air. At one point in its
> > trajectory, its vertical velocity is zero. If
>> this is a closed interval [b,b], then the
>> intervals of positive and negative vertical
>> velocity must be open (or semiopen) intervals
>> (a,b) and (b,c). Now, these intervals have the
>> same endpoint, so they apparently meet (in the
>> sense used by Allen's interval algebra). Yet
>> there is an interval *between* them, so they
>> apparently cannot meet. So this model seems to be
>> incompatible with the Allen algebra, which is a
>> major problem. The ontology called the 'vector
>> continuum' there takes this idea and develops it
>> axiomatically rather than presuming the real
>> line, and gives a temporal ontology which I think
>> is quite neat. But it has some apparent oddities,
>> eg it allows negative intervals (which turn out
>> to be quite useful.) And it isn't possible
>> (AFAIK) to interpret it in the real line.
>>
>> > Or are the 'point' and 'interval'
>> >classifications merely two different views of the same thing?
>>
>> Really, there is no simple answer to this
>> question. Try reading some of the options
>> described in the catalog. I think the various
>> intuitions are reasonably well explained there.
>>
>> PatH
>>
>> >
>> >PatC
>> >
>> >Patrick Cassidy
>> >MICRA, Inc.
>> >9085613416
>> >cell: 9085654053
>> >cassidy@xxxxxxxxx
>> >
>> >> Original Message
> > >> From: ontologforumbounces@xxxxxxxxxxxxxxxx [mailto:ontologforum
>> >> bounces@xxxxxxxxxxxxxxxx] On Behalf Of John F. Sowa
>> >> Sent: Monday, January 21, 2008 12:48 PM
>> >> To: [ontologforum]
>> >> Subject: Re: [ontologforum] Time representation
>> >>
>> >> Pat,
>> >>
>> >> The position I most strongly advocate is not a specific ontology,
>> >> but a framework of conventions for organizing a multiplicity
>> >> of special cases (not necessarily consistent with one another),
>> >> making the implicit relationships explicit, and providing tools
>> >> and guidelines for mixing and matching. The lattice of theories
>> >> is an example. Robert Kent's IFF is a much more ambitious example.
>> >>
>> >> I would recommend a fairly simple framework for starters, since
>> >> there's a danger of freezing halfbaked ideas before they're fully
>> >> baked. (RDF, for example, was hardly out of the oven before
>> >> Tim Bray tried, unsuccessfully, to pull it back in.)
>> >>
>> >> > Do you have any granularity axioms? That is one of the hardest
>> >> > ontological problems, in my experience.
>> >>
>> >> There are so many hard problems, it's hard to say which are harder.
>> >> But the idea of taking the least significant digit as the criterion
>> >> for implicit granularity is fairly common for experimental data
>> >> (unless some explicit margin of error is stated).
>> >>
>> >> Re PTim: I realize that calling an interval a point is
>> problematical.
>> >> But in anything that has to do with the physical world, there is no
>> >> way to specify a true point. Perhaps a better term would be "grain
>> >> in time", abbreviated "Grit".
>> >>
>> >> John
>> >>
>> >> PS re HTML email formats: Your note of 11:18 was in a readable
>> font
>> >> for Thunderbird, but your note of 11:37 appeared in a tiny, tiny
>> font.
>> >> I had to increase the font size by two steps to make it the same as
>> >> the previous note. But then the fonts for all other notes were too
>> >> big, and I had to decrease the default by two steps.
>> > >
>> >> At least each of your notes was entirely in one font size. I've
>> >> received some email in which each paragraph was in a progressively
>> >> smaller font. That's why I hate HTML email.
>> >>
>> >>
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>> >
>> >
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