#Chris Menzel

If we do this, that means that we only care about Beginof and Endof as they apply to activities. That is, axioms involving Beginof and Endof will all be the form:

(forall (?x ... ?z) (=> (Activity ?x)) (... (Beginof ?x) ... (... (Endof ?x) ...)))

or the like. That is, we will always qualify what we say about Beginof and Endof so that the only relevant cases involve their application

However, the question also came up of whether we want to require every activity to have a begin and end point. From a purely theoretical standpoint, I'm inclined to say "no". For it is easy to come up with a process specification that could only be realized by an infinite process -- just define a process with a loop that has an impossible exit condition. However, it is not clear to me that, in the contexts of PSL, it is necessary to allows the possibility of infinite processes. It seems plenty good enough just to say that specifications like the one above (loop with impossible exit condition) just have no possible realizations (cuz all processes are finite).

The advantage of doing so is simply that it enables us to continue to treat Beginof and Endof as functions rather than as relations (or, alternatively, it enables us to avoid introducing points at infinity, which, though elegant, do introduce some theoretical complications that seem to me better avoided). This is not a *huge* consideration, but treating them as functions does simplify matters.

So does anyone have any reasons why we should not simply define activities such that they have (finite) begin and end points? http://www.mel.nist.gov/psl/psl-secure/teams/semantics/hypermail/0355.html

#Austin Tate

I never understood the proposal for objects to time points directly associated. Thoiugh informally I can see that we might reffer to objects as having "lives" in this way. I can understand that the PROPERTIES (or state decriptions) of objects may alter in time, but that comes about due to their association with activities (create, modify, destroy, kill, paint, etc).

So I think I agree with Chris's proposed PSL solution while coming at it from a differnt perspective. Austin http://www.mel.nist.gov/psl/psl-secure/teams/semantics/hypermail/0356.html

#Chris Menzel

If I understand you, Austin, it seems that you are wanting to allow for infinite activities, but also suggesting that we assign such activities points at infinity as their begin and/or end points, where a point at infinity is a point that is infinitely distant from any other point. As a model, think of adding two extra elements to the set of integers (call them "omega" and "-omega"), one that is "greater than" all the finite integers (omega) and another that is "less than" all of the finite integers (-omega). So if an activity begins at a given timepoint (17, say) and has omega as an endpoint, then it has an infinite duration. Effectively, using a more standard model of time like the integers proper, this is the same as simply allowing activities that have no endpoints (or begin points) at all. So let me ask the question in a way that is independent of which model of time we use: do we want to require that all activities (i) have definite begin and endpoints, and (ii) are of finite duration?

I am inclined -- in the context of PIF and PSL, which are being designed to capture information about real world business, engineering, and manufacturing process -- to answer "yes", i.e., not to allow infinite activities.

> I never understood the proposal for objects to time points directly > associated. Thoiugh informally I can see that we might reffer to objects > as having "lives" in this way.

That was the idea, Austin; the original proposal formalized this informal way of thinking about objects. But it appears we are now in agreement it is best just to introduce create/destroy attributes and the like as needed, and not to apply Beginof and Endof to objects (or more accurately, not to care what those functions do when applied to objects). http://www.mel.nist.gov/psl/psl-secure/teams/semantics/hypermail/0357.html

#Chris Menzel

Pat wrote: > >[Chris wrote:] > >do we want to require that all activities (i) have definite > >begin and endpoints, and (ii) are of finite duration? > > > >I am inclined -- in the context of PIF and PSL, which are > >being designed to > >capture information about real world business, engineering, > >and manufacturingprocess -- to answer "yes", i.e., not to > >allow infinite activities. > > But why? This adds a constraint which seems unnecessary,...

But there is no real constraint there at all, Pat. (Though my putting the word "finite" in there threw an unnecessary wrench into the works; see below.) I am only suggesting that we require that all activities to have begin and end points, and that we not include points at infinity into the theory.

> ...and incidentally > raises the semantic spectre of making the language first-order > inexpressible.

Hmm, shades of those interlingua discussions involving you, me, Sowa, and Fritz Lehmann a few years ago. I said it then and I'll say it now: This is a specious worry. (I thought I'd convinced you at the time.) All first-order inexpressibility means is that you have introduced a concept into your system that you can't *precisely* pin down in a first-order language, in the sense that no number of axioms (even a nonrecursive set of them) will enable you to rule out unintended models. But look, we're already there: to do durations and calenders right we need an underlying theory of the integers with addition and multiplication. But, of course, the notion INTEGER, like FINITE (and indeed, ultimately for the same Godelian reasons), can't be pinned down in a first-order langauge. There will always be models of *any* set of axioms that are not isomorphic to the actual set of integers.

The point carries over directly to activities, of course, since all the stuff about the infinite, points at infinity, etc will all be expressed, if at all, in terms of the underlying theory of the integers. In particular, when I proposed that activities all have finite durations, I was making a proposal about what we wanted our *intended* models to look like. Thus, I was suggesting that it be a theorem of PIF/PSL that, relative to some clock ?c, for every activity ?a, there be some non-negative integer ?n such that the duration from (Beginof ?a) to (Endof ?a) is ?n beats of ?c. (That's all I meant when I said that activities should be of finite duration -- I wasn't proposing that we go second-order.) Now, of course, we are intending to quantify over integers here, so there will surely be nonstandard models of PSL such that, e.g., for some activity ?a, (Beginof ?a) is a standard integer (i.e., timepoint) but (Endof ?a) is a nonstandard integer -- i.e., a timepoint that is ?n beats from (Beginof ?a), where ?n is some nonstandard "positive integer".

But so what? We know what the integers look like, and we just want to make sure that we've got axioms that are good enough to enable us to prove basic properties about *them* (and, consequently, durations and the like). The fact that those axioms are incomplete just doesn't really have any serious bearing on that. PIF/PSL is to be, among other things, a medium for exchanging process information. Hence, we need to be sure of at least two things: when humans read a PSL/PIF process specification, the basic terminology be sufficiently axiomatized that they will be able to understand what is meant (e.g., by "integer", "duration, etc), and the axioms we will use will do that. We don't have to rule out all intended models to get people to understand the intended meaning of a process specification. Second, if there is any automated reasoning involved, we have to make sure the reasoner has enough information to deduce interesting, useful truths. Incompleteness doesn't impact that, as far as I can see.

> (People think that 'finite' is a simple concept, but > actually the only safe way to define it mathematically is as 'not > infinite', where 'infinite' is defined as being isomorphic to a proper > subset;...

/* begin set theory geek discussion */

Actually, that's the definition of a *Dedekind-infinite* set. (I take it by "isomorphic" here you mean "equinumerous".) For truly interesting reasons, this is not generally used as the primary definition of "infinite" in set theory. The primary definition is a bit less direct and elegant than that of Dedekind-infinite. First, by the axiom of infinity we know that there is a set that contains the empty set 0 and, furthermore, contains x U {x} whenever it contains x. (Hence, it contains 0, {0}, {0,{0}}, {0,{0},{0,{0}}}, etc.) By power set and various other axioms there is a smallest such set (which turns out to be the set of von Neumann ordinals, the standard representation of the natural numbers in set theory). Call that set N. An infinite set is now defined to be a set that has a subset equinumerous to N.

Now we have the following Interesting Fact: To prove that every Dedekind-infinite set is infinite you need to assume the axiom of choice. (And given choice, the proof that infinite = Dedekind-infinite is very cool.)

The reason the primary definition is the *primary* definition, then, is because you generally want to define a notion with as little underlying baggage as possible, and the axiom of choice, in they eyes at least some (philosophically oriented) mathematicians is pretty serious baggage. Also, though, intuitively, there is *no doubt* that the standard definition gives you a dead-to-rights sense of infinite: if you are at least as big as the set of natural numbers, well then by god yer infinite. The idea of being as big as one of your own proper subsets, by contrast, makes the mind boggle *just a bit*. And, the fact that you need choice to prove this notion of infinity equivalent to the primary definition does show that something else or something more is going on there...

/* end set theory geek discussion */

> Why not be as catholic as we can manage?

It's not clear to me that for its intended users the notion of an infinite activity is useful. I am most willing to be set straight on this point.

> I can't see any *harm* > in allowing infinite activities, ...

That is also not clear to me. Might not some users might find the idea difficult or mysterious? And it *does* induce complications. First, you have to introduce special constants into your language for +inf and -inf, which you then have to axiomatize, e.g., you have to say that all other timepoints are before +inf, etc. Furthermore, do not these points *force* us to say that time is infinite in both directions? I though we wanted to remain agnostic on that point. This is certainly not in the catholic spirit you appeal to above. And one you are stuck with infinite time, you have to add awkward qualifications like, for every timepoint ?t there is another before, *unless* of course ?t is -inf, etc etc. This all I take to count as harm in allowing infinite activities.

> ...and the 'endpoint at infinity' way of > putting it has the practical advantage that one can then make end-point-of > and begin-point-of into functions, which fit more nicely into the OO way of > talking.

Yes, the desire to let Beginof and Endof be functions is part of the motivation for stipulating that activities be finite. And if we don't *need* infinite activities we get the benefits of the OO way of talking w/o the complications.

> Also, theres no need to talk about the *distance* from other > points: all that matters is that its *later* than all the others; its a > strictly ordinal concept.

You lost me here. > > Suppose someone wants to just ignore the endpoint of an activity, or to > distinguish between activities that are clearly terminable from those that > might not be. It seems natural to be able to say that some just dont have > endpoints, or that their endpoints are later than all other endpoints;

Well *that* certainly doesn't seem to require points at infinity. In these cases you just have finitely many timepoints and the fellows in question end at the last one.

> ie, as my son once said to me, they might as well be infinite.

Clever boy. He'll appreciate the finite model above. http://www.mel.nist.gov/psl/psl-secure/teams/semantics/hypermail/0358.html

#Austin Tate

At 15:10 -0500 28/8/97, Chris Menzel wrote: >If I understand you, Austin, it seems that you are wanting to allow for >infinite activities, but also suggesting that we assign such activities >points at infinity as their begin and/or end points, where a point at >infinity is a point that is infinitely distant from any other point.

Thats right Chris, if I have understood you correctly. To elaborate...

Time points are just abstract entities. Each activity has such an abstract entity associted with as its "begin-time-point" and its "end-time-point". There can be relationships directly between time points of various kinds - a simple one of which is temporally before. For all activities their begin point is before their end point. There can also be relationships between time points and one or more calendars. Our definition of calendar allows for a start time for the calendar and a temporal increment.

What I was saying was that if we had a celendar representing the start time as something at or before the start of the times of interest to us (zero or minus infinity on the "tick" scale for that calender), then an activity that lasts forever, and always existed would have its begin point associated with the start point on such a calender and the end point associated with the plus infinity tick point on the calender.

>As a >model, think of adding two extra elements to the set of integers (call them >"omega" and "-omega"), one that is "greater than" all the finite integers >(omega) and another that is "less than" all of the finite integers (-omega). >So if an activity begins at a given timepoint (17, say) and has omega as an >endpoint, then it has an infinite duration.

>Effectively, using a more >standard model of time like the integers proper, this is the same as simply >allowing activities that have no endpoints (or begin points) at all.

Not how I see begin and end. Begin and end are the times at which activities begin and end., ALL activities have temporal scoipe in other words. Just because we CALL these points begin and end doe snot mean the activities do begin and end. Something started once in our model may be assumed to run forever for example. So on some appropriate calender its temporal scope would be say 8.40am on 29th August 1997 and its end time would be the end of all time (plus infinity on the calender chosen).

>So let >me ask the question in a way that is independent of which model of time we >use: do we want to require that all activities (i) have definite begin and >endpoints, and (ii) are of finite duration?

answers, yes to first question, no to second (though in any model we might approximate infinity with a VERY VERY large integer of course, and use time point tick/calender reasoning so as to stay within the interval and not use that chosen figure. Thats the usual trick in an implementation.

>I am inclined -- in the context of PIF and PSL, which are being designed to >capture information about real world business, engineering, and manufacturing >process -- to answer "yes", i.e., not to allow infinite activities.

I am not, and it would cause us many problems with thjings we already model. When does the traffic light sequemces that I model end for example. I would have to invent some end time in the model - building up a future millenium problem. Bad news. The model can and should allow the initinty points I have suggested (you called them omega points) so as to capture what peolle want to say.

I think we are confusing implementation or technicalities of formalising this with the semantics we are trying to capture - which ared very straightforward. http://www.mel.nist.gov/psl/psl-secure/teams/semantics/hypermail/0359.html

#Austin Tate

As I said though those are implementation issues. Anyone being given our process models would just see real calender point and the token infinity and be able to model this in their own system correctly. That is what PSL should support. http://www.mel.nist.gov/psl/psl-secure/teams/semantics/hypermail/0360.html

#Chris Menzel

Yes, I do understand all that, Austin. I have no problem with the *coherence* of the idea of timepoints at infinity. My original question was whether we needed infinite activities. I believe you've given a convincing example of the need for them with traffic light sequences. But that still leaves open the question of whether we should model these with timepoints at infinity, or simply do away with the constraint that every activity have begin and end timepoints. That issue is the focus of this note.

My argument against timepoints at infinity is simple: to sanction them would introduce unnecessary complexities into the PIF/PSL core. I mentioned a few of these complexities in my reply to Pat. It doesn't seem to me to be at all trivial to modify Pat's axioms (in his Beckmann report) for durations, clocks, calendars, etc to accomodate timepoints at infinity. (Pat, can you confirm, deny, elaborate, pontificate?)

> >model, think of adding two extra elements to the set of integers (call > >them > >"omega" and "-omega"), one that is "greater than" all the finite integers > >(omega) and another that is "less than" all of the finite integers > >(-omega). > >So if an activity begins at a given timepoint (17, say) and has omega as > >an endpoint, then it has an infinite duration. > > >Effectively, using a more > >standard model of time like the integers proper, this is the same as > >simply allowing activities that have no endpoints (or begin points) > >at all. > > Not how I see begin and end. Begin and end are the times at which > activities begin and end.

We are in full agreement on that one, Austin!

> ALL activities have temporal scoipe in other > words.

Activities that have no end point, say, have temporal scope as well, in any reasonable sense of temporal scope I can think of (unless of course you *define* temporal scope in a way that requires begin and end points, which seems to beg the question); you can say precisely which points they occupy. There just doesn't happen to be a point at which such activities end.

> Just because we CALL these points begin and end ...

Which points?

> ...does not mean the activities do begin and end.

But you just said that begin and end points "are the times at which activities begin and end." I don't see how to reconcile that with this last comment.

> Something started once in our model may be > assumed to run forever for example.

Then why not just say that it has no end point?

> So on some appropriate calender its > temporal scope would be say 8.40am on 29th August 1997 and its end time > would be the end of all time (plus infinity on the calender chosen).

Or we just say there is no end timepoint at all. Now, as far as our *specification language* goes, it might be quite desirable to enable users to specify that an activity has no end point with some such construct as (= (Endof ACT) +inf) in which "Endof" is treated as a function symbol, and which could be thought of as a *macro* for

(not (exists ?t (Endof ACT ?t)),

i.e., for the assertion that ACT has no end point. My point is that there is no need to introduce an actual point at infinity "beyond" all ordinary timepoints to correspond to the term "+inf".

> >So let > >me ask the question in a way that is independent of which model of > >time we > >use: do we want to require that all activities (i) have definite begin > > and endpoints, and (ii) are of finite duration? > > answers, yes to first question, no to second (though in any model we might > approximate infinity with a VERY VERY large integer of course, and use time > point tick/calender reasoning so as to stay within the interval and not use > that chosen figure. Thats the usual trick in an implementation.

Of course we need to leave implementation out of it; that is another matter entirely stemming from the inherently finite nature of our machines. As noted, your traffic light example completely convinces me that you are right about (ii): we should not stipulate that activities are finite, as that would force people to model certain activities unnaturally. But I still see no reason to introduce the complexities of timepoints at infinity into our theoretical apparatus (except as a purely linguistic construct, as above).

> I think we are confusing implementation or technicalities of formalising > this with the semantics we are trying to capture ...

I believe both of us are quite clear that we are not talking about implementation here, Austin. As to the idea of confusing the technicalities of formalization with capturing semantics, I'm not sure what you have in mind; for what else is it to capturing semantic intuitions than to formalize them? Is that not the whole premise behind the formalization of the PIF/PSL core? Now, agreed, there might be more than one way of formalizing a semantic intuition. Indeed, that is exactly the issue here. I'm just trying pick the easiest approach, since it has direct bearing on the shape (and complexity) of the PIF/PSL core.

> ...which are very straightforward.

I agree the intuitions are straightforward, but your (and Pat's) way of capturing those intuitions is not.

To sum up, here again is my argument for the approach I've been defending.

1. The PIF/PSL core must allow for the possibility of infinite activities.

2. There are two options that are equally effective theoretically. We can either

a) Require all activities to have begin and end points, and introduce timepoints at infinity to represent the begin and end points of infinite activities; or

b) Do not require all activities to have begin and end points. Infinite activities are represented simply by the fact that they lack a begin or end point (or both).

3. The first option requires the introduction of two new theoretical objects -- -inf and +inf -- and axioms for those objects, as well as the introduction of some significant complications to the core theory of timepoints, durations, clocks, etc.

4. The second option requires the introduction of no new objects, no changes to the existing core theory of timepoints, durations, clocks, etc. (we can use Pat's work directly), and only a couple trivial changes to the existing theory of activities.

5. It is unreasonable to introduce unnecessary theoretical complications.

6. The first option introduces far more complications into the PIF/PSL core than the second option.

7. Therefore, we should represent infinite activities by the second option.

If this argument is unsound, I am, as always, happy to be corrected. http://www.mel.nist.gov/psl/psl-secure/teams/semantics/hypermail/0361.html

#Chris Menzel

A good practice. The question is whether we want to insist in our core theory that that token correspond to an actual point at infinity, or whether (as I'v argued) there is a less complicated way to capture the semantics in question.

> and a specific integer associated with a calender point > where we want to be precise.

Right.

> Although the IMPLEMENTATION uses a very large > integer for infinity, our algorithms all know the value and reason > appropriately. It is NOT possible to state a finite time point on the > calkender which is the same as or larger than the initnity point, and all > the temporal map computations do not allow for overflow. > > As I said though those are implementation issues.

Granted, *those* are implementation issues; and entirely irrelevant to point in question (as you know).

> Anyone being given our > process models would just see real calender point and the token infinity > and be able to model this in their own system correctly. That is what PSL > should support.

No doubt about it.

Unless I am grossly mistaken, Austin (and it is prudent never to rule this possibility out!), the approach I'm arguing for gives you *everything* you want in regard to the specification of infinite processes, along with a substantial theoretical gain. http://www.mel.nist.gov/psl/psl-secure/teams/semantics/hypermail/0362.html

#Austin Tate

My argument is that ALL activities take place in time, and the begin and end points just are associated with thhe ends of that time interval. These can be initinity points ON A CALENDAR.

>Activities that have no end point, say, have temporal scope as well, in any >reasonable sense of temporal scope I can think of (unless of course you >*define* temporal scope in a way that requires begin and end points, which >seems to beg the question); you can say precisely which points they occupy. >There just doesn't happen to be a point at which such activities end.

Maybe it is a matter of DEFINITION Chris as you say.

>Then why not just say that it has no end point?

I would say it has a temporal scope whose end is infinity. Its just a definition I suppose.

>Now, agreed, there might be more than one way of formalizing a >semantic intuition. Indeed, that is exactly the issue here. I'm just >trying pick the easiest approach, since it has direct bearing on the shape >(and complexity) of the PIF/PSL core.

poinmt taken,. and I have been using one set of DEFINITIONS which I thought had simple appeal. The activities come with the idea of having them all having a begin and end point and the relationship of begin before end for all activities. It remains to be seen if this is so simple, or whether defining them atemporally (i.e. they MAY not have begin and end points and thus may not have the defined relationship of begin before end esxcept where both a begin and end are actually specified). Note that such a spec of the begin and end time poinmts is unnecessary in my suggestion. If asspec is not given, the outer envelope of all possible begin and end times is possible.

>I agree the intuitions are straightforward, but your (and Pat's) way of >capturing those intuitions is not.

well I am nbot sure aboit that. In fact what we are doing is pretty much the modelling approach I have seen used in a number of other frameworks including some SADT style techniques that I would have though related to IDEF methods for example. You may know more about that than I do?

>To sum up, here again is my argument for the approach I've been defending. > >1. The PIF/PSL core must allow for the possibility of infinite >activities.

I agree.

>2. There are two options that are equally effective theoretically. We >can either > > a) Require all activities to have begin and end points, and > introduce timepoints at infinity to represent the begin and > end points of infinite activities; or > > b) Do not require all activities to have begin and end points. > Infinite activities are represented simply by the fact that they > lack a begin or end point (or both).

I agree these seem to be the options.

>3. The first option requires the introduction of two new theoretical >objects -- -inf and +inf -- and axioms for those objects, as well as the >introduction of some significant complications to the core theory of >timepoints, durations, clocks, etc. > >4. The second option requires the introduction of no new objects, >no changes to the existing core theory of timepoints, durations, clocks, >etc. (we can use Pat's work directly), and only a couple trivial changes >to the existing theory of activities.

But lacks semantic reasoning power about all activities. Including the more usual case where the activities are not infinite. It also is a less natural modelling method in my mind. Others will have to comment to break the impasse here.

>5. It is unreasonable to introduce unnecessary theoretical complications.

well maybe:-) unless we want user communicable results. But I agree we could do that in suitable extensions. Hence I will not concede this one Chris:-)

>6. The first option introduces far more complications into the >PIF/PSL core than the second option.

No its a trivial think, easily understood and very common in a range of systems. We would have to explain how to translate to such models anyway and would want to introduuce the equivelence in an extension starighaway if we excclude it form the core. So you don't win there I am afraid.

>7. Therefore, we should represent infinite activities by the second >option.

not QED due to points after 4 and 6.

We have thge options now though, very clearly puit Chris thanks. Someone else should give inpurts. Perhaps some of the suer requirements folks, or those familiar with other modelling frameworks. http://www.mel.nist.gov/psl/psl-secure/teams/semantics/hypermail/0363.html

#Chris Menzel

I certainly agree with that, Austin. Your emphasis on "ALL" makes me wonder if you think that somehow I deny it. To say a process never ends, and hence (as I would have it) has no end point is not at all to deny that it takes place in time, by my reckoning.

>and the begin and >end points just are associated with thhe ends of that time interval. These >can be initinity points ON A CALENDAR.

*Can* be, I agree, but also *needn't* be.

>>Activities that have no end point, say, have temporal scope as well, in any >>reasonable sense of temporal scope I can think of (unless of course you >>*define* temporal scope in a way that requires begin and end points, which >>seems to beg the question); you can say precisely which points they occupy. >>There just doesn't happen to be a point at which such activities end. > >Maybe it is a matter of DEFINITION Chris as you say. > >>Then why not just say that it has no end point? > >I would say it has a temporal scope whose end is infinity. Its just a >definition I suppose.

Not *quite* a matter of definition, Austin, since the points-at-infinity option forces us to introduce some extra ontology, plus make some revisions that complicate our existing theory. But I do agree with your implication here that there is no deep issue that divides us. I have agreed whole-heartedly with you about the need for infinite activities. The only thing we disagree on is exactly how to capture that in the core.

>>Now, agreed, there might be more than one way of formalizing a >>semantic intuition. Indeed, that is exactly the issue here. I'm just >>trying pick the easiest approach, since it has direct bearing on the shape >>(and complexity) of the PIF/PSL core. > >poinmt taken,. and I have been using one set of DEFINITIONS which I thought >had simple appeal.

I think it *does* have a simple appeal, Austin; I have never denied that. But it hides a fairly nasty theoretical bite when we try to capture it directly in the core theory.

>The activities come with the idea of having them all >having a begin and end point and the relationship of begin before end for >all activities. It remains to be seen if this is so simple, or whether >defining them atemporally (i.e. they MAY not have begin and end points and >thus may not have the defined relationship of begin before end esxcept >where both a begin and end are actually specified).

It is correct to say that on the view I'm suggesting you couldn't say that an infinite activity ends "at infinity" after all activities with "standard" end points. But do we not capture basically the same idea when we say that such an activity is still occurring after the end points of all the other activities?

And don't forget that, if we want, we can define our specification language in a way that allows people to define activities that have infinite begin or end timepoints; we would then just cash them out in the theory proper in terms of activities that have no begin or end timepoints.

I must note that I am not comfortable with your idea of defining "atemporal" activities, as that term suggests that such activities don't occur in time. They are *fully* temporal; they just never end (and/or, perhaps, never begin).

>>I agree the intuitions are straightforward, but your (and Pat's) way of >>capturing those intuitions is not. > >well I am nbot sure aboit that. In fact what we are doing is pretty much >the modelling approach I have seen used in a number of other frameworks >including some SADT style techniques that I would have though related to >IDEF methods for example. You may know more about that than I do?

In all seriousness, Austin, I would not presume to know more about modeling than you. Let's not lose sight of the context here: PSL. When I say the points-at-infinity approach is not as straightforward, I am speaking relative to PSL. We have a theory of activities that includes a theory of time. It is just a fact that the points-at-infinity approach will require the introduction of points at infinity, which will need to be axiomatized, and which will require nontrivial changes to our existing theories.

>>4. The second option requires the introduction of no new objects, >>no changes to the existing core theory of timepoints, durations, clocks, >>etc. (we can use Pat's work directly), and only a couple trivial changes >>to the existing theory of activities. > >But lacks semantic reasoning power about all activities...

Some sort of example would be helpful here. I don't see it (but would be happy to have you enlighten me, as you did vis-a-vis the need for infinite activities).

>...Including the >more usual case where the activities are not infinite.

?? The view I'm defending only has implications for infinite activities. How can it impact the "more usual case where the activities are not infinite"?

>It also is a less >natural modelling method in my mind.

Perhaps! But we are not defining a modeling method! We are trying to develop a framework capable of capturing any possible process information. Hence, we want a framework that is both theoretically powerful and as uncomplicated as possible relative to that task. I have been resisting your approach because it is demonstrably more complicated *relative to PSL* without buying us any gain in theoretical power.

If what you are worried about is *naturalness*, i.e., the accessibility of PIF/PSL for users, then as I pointed out in my previous note we can give people the *constructs* you want; we will just cash them out theoretically in the manner I suggest. Doesn't that satisfy the concerns of both of us?

>>5. It is unreasonable to introduce unnecessary theoretical complications. > >well maybe:-) unless we want user communicable results. But I agree we >could do that in suitable extensions. Hence I will not concede this one >Chris:-)

But Austin, if we can give a process specification in the friendlier fashion you suggest, either by suitable extensions or by constructs like the ones I've suggested, what reason is left for insisting on the more complicated underlying *theory*? Again, ordinary users don't even have to see that. This is the theory beneath the core we're talking about here, not the pretty face we put on it.

>>6. The first option introduces far more complications into the >>PIF/PSL core than the second option. > >No its a trivial think, easily understood and very common in a range of >systems.

The first option is *not* trivial RELATIVE TO PSL, for reasons I've given several times. (Sorry for shouting!) That is my concern here, not how common points at infinity are in other systems (and I would dispute how easily understood the notions are; in unformalized systems there are bound to be points of unclarity until the meaning of a point at infinity is fixed).

>We would have to explain how to translate to such models anyway >and would want to introduuce the equivelence in an extension starighaway if >we excclude it form the core. So you don't win there I am afraid.

But writing translators will not complicate the core in the least. Adding points at infinity to the core does.

>>7. Therefore, we should represent infinite activities by the second >>option. > >not QED due to points after 4 and 6.

Unless I've successfully fended off your counterpunches!

>We have thge options now though, very clearly puit Chris thanks. Someone >else should give inpurts. Perhaps some of the suer requirements folks, or >those familiar with other modelling frameworks.

I'm not sure I see the bearing of this, Austin, as it seems to me that you have already identified the one relevant requirement, viz., the need to represent infinite activities. So to me the only issue that remains is how best to represent them in our core theory. We may well well identify a further requirement to the effect that people need to be able to talk about points at infinity. But that requirment can be met irrespective of whether we actually put them in the core -- we simply give them the relevant constructs, and they will "compile" into the core just fine. To me, the *only* issue here is that, in the core theory (which users needen't see in detail), points at infinity are not needed, and only complicate matters. Again, this is NOT to say that we won't let users talk in terms of points at infinity! It is an issue for the underlying core theory only. Now, if that isn't enough, Austin, what more is needed? http://www.mel.nist.gov/psl/psl-secure/teams/semantics/hypermail/0364.html

#Pat Hayes

Austins has the merit of retaining a simple mapping between intervals and endpoints, in which every interval is uniquely defined by its unique endpoints; but by introducing the peculiar 'points', it may force many ordinary assertions about endpoints to be restricted explicitly to the finite ones. Chris' has the advantage that such restrictions are unnecessary, but it makes the point-interval mapping more complicated by forcing the 'infinite' cases to be described explicitly. I must confess that I find the difference to be largely aesthetic. Both have natural intuitions to justify them; they correspond to the projective line and the real line. Given this, why not allow BOTH? On a combined account, we have a set of p-timepoints (read projective timepoints) with a special subset called f-timepoints (finite-timepoints) and two other members called -w and w. Chris' intuitions are handled by quantifying over f-timepoints, and Austin's by quantifying over p-timepoints. Since w and -w arent f-timepoints, it follows that if an interval's endpoint is w then it doesnt have an endpoint in the set of f-timepoints, which is exactly what Chris' account requires; and vice versa (assuming endpoint to be a function). The only axiom we need, I think, is that forall p in f-timepoint, not( p<-w or w<p ). Together with the axiom of Chrisian infinity and the totality of the before-ordering, this ensures that forall p in f-timepoint, -w<p<w.

This has the merit, by the way, of later allowing other classes of timepoints, such as 'variable' ones, or uncertain ones, or whatever, while still retaining the integrity of the 'normal' f-timepoints.

Regarding Chris' question about all my time axioms, I havnt checked this in detail but I am pretty sure that it can all be handled by saying that w (and -w) is a moment, ie both a point and an interval. This allows it to meet itself, so that w is the point at which any interval of the form [p, w] meets w; and [w,w] is equal to w. This means that every interval still meets another interval and that the meeting-place is unique, which preserves the basic idea behind the 'vector continuum' axioms. What makes w special is that it has the unique property that it doesnt meet anything except itself, ie meets(w, x) implies w=x. Similarly meets(x, -w) implies x=w. The clock theory may need to be adjusted here and there, but I'll need to check them more carefully and get back.

Pat

PS. I bow to Chris on matters of infinity and Dedekind infinity. http://www.mel.nist.gov/psl/psl-secure/teams/semantics/hypermail/0365.html

#Michael Uschold

But lacks semantic reasoning power about all activities. Including the more usual case where the activities are not infinite. It also is a less natural modelling method in my mind. Others will have to comment to break the impasse here. http://www.mel.nist.gov/psl/psl-secure/teams/semantics/hypermail/0370.html

#Craig Schlenoff

Summary:

Much of the discussion focused on infinite activities. There seems to be general agreement that it is important to be able to represent an infinite activity (which I agree) but a little disagreement on how to do it (refer to Chris's and Pat's email for a summary of this discussion). May I suggest that for now, we take the seemingly simpler option (that of saying that an activity can have no end timepoint) and see how it goes. If problems arise, we can move on to the other option and including timepoints at infinity.

The also seemed to be general agreement that we should not have a beginof and endof timepoint associated with objects, although objects can have created and destroyed optional attributes. http://www.mel.nist.gov/psl/psl-secure/teams/semantics/hypermail/0373.html

#Pat Hayes

#Chris Menzel

As nice as this work is, Pat, I continue to suggest we retain our current theory and simply provide a PSL interface that permits users to talk in terms of points at infinity in practice, while translating that talk transparently into assertions about activities without begin or end points. http://www.mel.nist.gov/psl/psl-secure/teams/semantics/hypermail/0382.html

#Pat Hayes

>As nice as this work is, Pat, I continue to suggest we retain our current >theory and simply provide a PSL interface that permits users to talk in terms >of points at infinity in practice, while translating that talk transparently >into assertions about activities without begin or end points. >

Heres the trap you are heading towards. If one has intervals as a type, then being semi-infinite is a property of an interval. But PSL has disavowed the expressive advantages of intervals and decided to restrict itself to points as a way to describe temporal constraints. But some of your activities won't have points; so you have neatly excised from your vocabulary any way to naturally refer to any temporal properties of such endless intervals. This combination shoots you in the foot. Of course it will be possible to say what you want, but only by including extra clauses all over the place which handle the 'endless' cases in a special way and then say '..or else..' and then refer to timepoints; or by having 'endless' activities as a special category (don't forget the 'beginless' ones, and now you have to worry how to classify an 'endless' activity that suddenly stops for some reason.) If you follow through on your own ontological decisions and use points consistently, there would be no need for all this extra complication. Its very simple: every activity has a start timepoint and an end timepoint, the points are totally ordered, -w is before all finite times and w is after them. Thats about all you need to make the ordinary finite-time-point machinery extend naturally to the infinite cases. (Sometimes processes stop before one expects them to: that is, their actual stop time t is earlier than their predicted one, t'. If t' is w, this means that t must be a finite time, ie when the 'endless' activity broke down. All the reasoning goes through this transparently, as far as I can tell.)

But I won't belabor this matter any further. You'll find out eventually ;-)

Pat

PPS. Those time axioms are full of bugs, please dont attempt to use them as they stand. In fact, please erase them. http://www.mel.nist.gov/psl/psl-secure/teams/semantics/hypermail/0378.html

#Austin Tate

This is one of the concerns (specific exceptions needing to be dealt wqithin all reasoning systems that must use tye "endless" activities ratrher than what I saw as a simpler model of everything having ends - some of which are -omega and +omega.

>If you follow through on your own ontological >decisions and use points consistently, there would be no need for all this >extra complication. Its very simple: every activity has a start timepoint >and an end timepoint, the points are totally ordered, -w is before all >finite times and w is after them. Thats about all you need to make the >ordinary finite-time-point machinery extend naturally to the infinite >cases.

Sounds simple to me. But Pat PLEASE lets use the agreed begin/end pairing (start/finish is fine too but not the term we used in pSL),. but not mmix them up to have start/end - an old favourite moan of mine I am afraid. http://www.mel.nist.gov/psl/psl-secure/teams/semantics/hypermail/0387.html