Response to George Ferguson

From: Jerry Hobbs (
Date: 05/01/02

  • Next message: patrick hayes: "Re: Response to George Ferguson"
    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
    > 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
    > 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.
    > --------------------------------------------------------------------------

    This archive was generated by hypermail 2.1.4 : 05/01/02 EDT