**From:** George Ferguson (*ferguson@cs.rochester.edu*)

**Date:** 04/26/02

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I've attached our comments on the proposed abstract ontology of time. I apologize for taking so long to get this out. George -- Dr. George Ferguson Research Scientist Dept. of Computer Science EMAIL: ferguson@cs.rochester.edu University of Rochester WWW: http://www.cs.rochester.edu/u/ferguson/ Rochester, NY USA 14627-0226 TEL: (585) 275-5766 FAX: (585) 273-4556 Overall, we like this intial 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. 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. 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. 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. -------------------------------------------------------------------------- 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). 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". 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. 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. 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? 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). And regarding this axiom, it isn't clear to us how it helps with the election example used as motivation for granularity. -------------------------------------------------------------------------- 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, 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) 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) 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). --------------------------------------------------------------------------

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