From: Jerry Hobbs (hobbs@ai.sri.com)
Date: 05/01/02
Below are some responses to George Ferguson's comments on the time ontology. I should have a rewrite out tomorrow. If anyone on this mailing list doesn't want to be on it, please just say so and I'll remove you. I promise not to sell your email address to a Viagra marketeer in retaliation. (Now I've just lost everyone who filters out messages mentioning Viagra.) -- Jerry > -------------------------------------------------------------------------- > Overall, we like this initial formulation. It manages to capture > generalities without excluding different people's theories. We tested > the common ontology by seeing whether we could extend it with > additional axioms to produce the Interval Temporal Logic as described > in (Allen, 1984) and (allen & Hayes, 1989). This seems like a good > exercise not only because it is near and dear to our hearts, but also > because it's a widely-used and extensivley studied representation of > time. > > Our comments in this note are divided into three sections: the first > addresses issues in the development of the temporal logic through > Section 2, the second notes a few fairly minor points that we came > across in later sections of the proposal, and, finally, we describe > our mapping or extension of the proposed logic into the Allen ITL. > > -------------------------------------------------------------------------- > Comments on Temporal Logic: > > 1. INTERVAL-BETWEEN ought more properly to be called > TEMPORAL-ENTITY-BETWEEN, given the definition near the end of 2.1. Sounds reasonable. How about TIME-BETWEEN? > 2. Near the end of 2.1, you comment that "the ontology is silent about > whether intervals are uniquely determined by their starts and > ends." That is, INTERVAL-EQUALS is not (necessarily) true equality. > While this is certainly mathematically possible, it is different > from the standard use of equality in the interval algebra (which > uses only true equality). Also, in later examples you describe > intervals as, for example "[10:00, 11:00]". We suspect that most > people would expect that, for example, "Raining([10:00, 11:00])" > and "~Raining([10:00, 11:00])" was logically inconsistent, although > this would not necessarily be the case if intervals are not > uniquely determined by their endpoints. So we are in the camp that > would like to strengthen this. I have rewritten the treatment to make explicit this property and where it is required. I also compare it with Total Ordering and Convexity. On David Israel's suggestion I called it Extensional Collapse, but if you have a better name for it I would be happy to use it. > 3. In 2.3 on the interval relations, it would appear that several of > the definitions are counter-intuitive unless restricted to proper > intervals (those not of the form "[t,t]"). For example, a > degenerate interval [t,t] INT-EQUALS itself, INT-MEETS itself, and > is INT-MET-BY itself! Thus "Meets(X,Y) <-> ~MetBy(X,Y)" is not a > theorem of these axioms. In fact, unless restricted to proper > intervals, none of the standard antonyms of the interval algebra > are theorems. And of particular interest in planning (cf. Allen & > Koomen 1987), the degenerate interval is INT-DISJOINT from itself. It sounds like there are enough problems trying to keep the interval calculus open to length-0 intervals that one should just stipulate that the intervals have to be proper. Since what I was trying to do in Section 2.3 was link the standard interval calculus to the logic of instants, I'm happy to modify the treatment of intervals the way you suggest. > 4. Regarding the axioms for DURING, AT-TIME, and HOLDS: Up to this > point, if we wanted to have a theory based solely on intervals, we > could simply ignore the point-oriented aspects of the proposed > ontology. However, with the proposed definitions for HOLDS, we are > forced to accept that eventualities occur/hold at points if they > hold over intervals. > > Your axioms are (t is a point, T is an interval): > > holds(e, T) <=> during(e, T) > holds(e, t) <=> at-time(e,t) > during(e, T) & inside(t,T) => at-time(e, t) > > These imply: > > holds(e,T) & inside(t, T) => holds(e, t) > > Now you could say that we can just ignore this HOLDS predicate, and > define our own, and we won't get into trouble. This may be the > case, but we think this would be confusing. We think it would be a > better strategy to keep the DURING and AT-TIME definitions, and let > individual researchers define their own HOLDS and OCCURS predicates > however they wish. An example of this is given below, where we > remove the first two axioms above, and actually strengthen the > third. I like your fix described below. > -------------------------------------------------------------------------- > Comments on Later Sections (dates, etc.): > > These comments based on a quick overview of the later parts of the > document. I'm afraid we haven't had time to work through them in > detail yet (nor are we really experts on this apsect of temporal > representation anyway). I am redoing the second on "Hath" in terms of "concatenation" and a 3-argument "Hath", but I'll respond briefly to these comments. > 5. Section 3.2 on "Hath": saying that "x is composed of the disjoint > union of N intervals of type u" is speaking somewhat loosely, it > would seem. The meaning seems to be that x is "N unit intervals > with respect to the TemporalUnit u". That is, *Day* is not really a > "type". I'll clarify this. > Also here, the definition of Hath does not require that the > component intervals be contiguous. Perhaps this comes out in the > axioms and could be added to the english gloss along with the > previous change. I'm pretty sure this does come out in the axioms. It should pretty clearly in the new treatment. > 6. The axioms for Hath have a couple of typos (unless we're missing > something). The two that say that "every element of S has an > element that precedes and follows it" use "x" in their innermost > formula, when it seems "s" is intended: "there exists a y2 which is > a member of s (not x), for which int-meets(y1,y2)" (and similarly > for y1 in the second axiom. Thanks. Typos corrected. > 7. The comment following this states that if time is linearly ordered, > the E quantifier can be replaced by E!. Isn't this only true if > intervals are uniquely determined by their endpoints? Turns out it's true if Convexity holds. That's in the new treatment. > 8. The final axiom for Hath has "duration(y1,u)" as a conjunct, but > since DURATION is a function, it would seem that "= 1" is missing, > to make y1 a unit interval w.r.t. the TemporalUnit u (as in the > second axiom for Hath). Right. Thanks. Fixed. > And regarding this axiom, it isn't clear to us how it helps with > the election example used as motivation for granularity. I've given up trying to build in granularity here. > -------------------------------------------------------------------------- > Relationship between Proposed Ontology and Interval Temporal Logic: > > As described at the outset, we tried to figure out how to extend the > proposed common ontology into the Interval Temporal Logic of (Allen, > 1984) and (Allen & Hates, I hope this isn't a Freudian typo. Pat is really a very nice guy once you get to know him. > ... 1989). Our sense is that this serves both to > firm up our understanding of the proposed ontology, as well as being > potentially valuable in its own right as a mapping into a widely-known > formalism. We use the original proposal's conventions of implicit > quanitification and typing (T for intervals, t for points), and we > number our axioms "AF<n>". > > First, per comment #2 above, we make intervals uniquely defined by > their endpoints (ie., INT-EQUALS is true equality): > > AF1. int-equals(T, T') => T=T' > > Second, per comment #3 above, we allow only true intervals: > > AF2. interval(T) => proper-interval(T) > This should be unnecessary if I modify my treatment of the interval calculus to be restricted to proper intervals. However, as a specialization it should be consistent. > Per comment #4 above, we remove the common ontology axioms about > HOLDS. We can define our notion of HOLDS (and OCCURS) in our more > specialized ontology as implying DURING, with additional constraints > (HOLDS is homogenous, OCCURS is anti-homogeneous): > > AF3. interval(T) & holds(e, T) => during(e, T) > AF4. holds(e, T1) & int-contains(T2, T1) => holds(e, T2) > AF5. interval(T) & occurs(e, T) => during(e, T) > AF6. occurs(e, T1) & int-contains(T2, T1) => ~occurs(e, T2) This is good. This is a nice example of the sort of event ontology that the time ontology should support. > Finally, we are left with specifying the relationship between > intervals and points (in particular, the relationship between > eventualities holding/occuring over intervals and at points). The > original axiom is: > > during(e, T) & inside(t, T) => at-time(e,t) > > We would in fact strengthen this to the following: > > AF7. at-time(e,t) <=> Exists T' . inside(t,T') & during(e, T') > > In other words, at-time(e,t) means that e is "in progress" at t (as in > the progressive aspect statement "He was running at 3:15"). Note that > this axiom entails the original, but we probably wouldn't want it in > the common ontology as it would interfere with what some point-based > theories might want to say about the relationship between points and > intervals. We believe that once the axioms about HOLDS are removed > from the common ontology, this axiom is not needed either (although it > wouldn't hurt our formulation to leave it there, since we will just > strengthen it). I agree that Axiom AF7 is a good example of a strengthening that a specialize theory can add. I like this fix. > > -------------------------------------------------------------------------- > References: > > Allen, J.F. (1984). Towards a general theory of action and time. > Artificial Intelligence 23, pp. 123-154. > > Allen, J.F. and P.J. Hayes (1989). Moments and points in an > interval-based temporal logic. Computational Intelligence 5, pp. > 225-238. > > Allen, J.F. and G. Ferguson (1997). Actions and events in interval > temporal logic. In Oliveiro Stock (ed.), Spatial and Temporal > Reasoning, Kluwer Academic Publishers, pp. 205-245. > > --------------------------------------------------------------------------
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