Re: First cut at time ontology

From: George Ferguson (
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.
    Dr. George Ferguson           Research Scientist
    Dept. of Computer Science     EMAIL:
    University of Rochester       WWW:
    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
    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
    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
       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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