Issue 98065 | Editor: Erik Sandewall | 25.8.1998 |
Today |
Today, Murray Shanahan's answer to the questions by Paolo Eduardo Santos about his submitted article, and Jixin Ma's answer to Erik Sandewall's latest message about the ontologies of time.
ETAI Publications |
Additional debate contributions have been received for the following article(s). Please click the title of the article to link to the interaction page, containing both new and old contributions to the discussion.
Murray Shanahan
A Logical Account of the Common Sense Informatic Situation for a Mobile Robot
Debates |
In ENRAC 20.8 (98064), Erik wrote:
One of the cases you mention is where two intervals Meet in direct
succession; one is where the first interval Meets a point which in
turn Meets a second interval; in the two remaining cases one or the
other interval includes that point. - I am afraid there's a
misunderstanding here, since I was referring to the domain used in
each of the interpretations. For each particular interpretation, it
must certainly be determined whether or not there is a point between
the two intervals. Therefore, different scenarios will sometimes
differ with respect to their domains for the type of "point" (and
maybe also for the type "interval"?) if one insists on dealing with
dividing instant situations by using domains where for certain
clocktimes there is no corresponding (time)point. Sometimes,
different models for the same scenario will also differ in that
respect.
Now to the examples. I will take for granted that we talk about
timepoints and intervals that are related along the lines of Pat's
core theory, only with the adjustment that intervals are not
entirely determined by their endpoints: there can be up to four
intervals for each pair of endpoints, because you allow these
intervals to be either open or closed at each end. (The interval
will then be defined as closed if there exists a point beginning
resp. ending it, otherwise it's open).
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As I argued in my former message, all the above cases can be accommodated by the single time theory (model) -- all these scenarios may (but not necessarily) appear somewhere over the time lines (even if the time itself is further characterised as linear) without the need of any futher specifications. They are not conflict with each other.
You refer to an example by Galton where a Green light and a Red
light both switch On at the same time. This is somewhat
counterintuitive - I would have thought that one goes Off when the
other one goes On - but that doesn't matter.
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Holds(GreenOn, I) | ||
Holds(GreenOn, P) | ||
Holds(GreenOff, J) | ||
Holds(RedOff, I) | ||
Holds(RedOn, P) | ||
Holds(RedOn, J) | ||
Meets(I, P) | ||
Meets(P, J) |
As for the case where no information about the Switching point is given at all, i.e., none of the two lights is asserted as On or Off at the switching point P, the axioms wil be as simple as:
Holds(GreenOn, I) | ||
Holds(GreenOff, J) | ||
Holds(RedOff, I) | ||
Holds(RedOn, J) | ||
Meets(I, J) |
You propose the following scenario description for the case where we
have decided to consider the Green light to be On at the dividing
instant, and we have decided to keep that open for the Red light:
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Duration(I1) = Duration(I2) |
Duration(P+J2) = Duration(J2) |
Holds(GreenOff, I2) | ||
Holds(GreenOn, P) | ||
Holds(GreenOn, J2) | ||
Holds(RedOff, I1) | ||
Holds(RedOn, J1) | ||
Meets(I2, P) | ||
Meets(P, J2) | ||
Meets(I1, J1) | ||
I1+J1 = I2+P+J2 | ||
Duration(I1) = Duration(I2) |
(a) "GreenOff" Meets "GreenOn", i.e.,
(b) Since, together with the rest axioms, axiom
(c) The switching point P satisfies the "GreenOn" property which is
specified as P Starts the GreenOn interval (i.e.,
My two examples come out in similar ways. For example A, you write: |
Yeah, for the modelling of the throwing of a ball, it requires that there exists a point referring to the apex. However, the fact that Jim turned the switch does not necessarily imply that there must not be any such point, especially if one insists that "at a moment (point?) when it (the ball) reaches the top of its trajectory, he (Jim) turns the switch". |
But if (in a particular model) such a point exists for the clocktime where Jim turned the switch, then it must be determined (in that same model) whether the switch is on or off at that point, and you have your Dividing Instant Problem back again. For example B, you write: |
I don't agree with the claim that "a point both exists and does not exist at the clocktime whent he winner finishes his last cone and the bell rings". Again, I think this claim was reached by means of confusing two cases, that is, the case that an interval "Meets" a point, and the case that an interval was "Finished-by" a point. |
Not really. If you wish to avoid a dividing instant situation by
using a punctuated time domain (for each of the models, so that
there is no dividing instant problem in any of the models), then you
must exclude models where that timepoint is present. It can't be
present explicitly, and it can't be present implicitly by being the
ending or beginning of an interval, because in all of those cases
you end up assigning the truthvalue that you considered arbitrary.
The only way of complying is to have two successive open intervals
without any point between them. (That is, an interval not ending in
a point, and a subsequent Meeting interval not beginning in a
point). However, this in turn contradicts the assumption that the
Bell rings, since it was assumed the Bell rings at (time)points.
Therefore, the only possible models are those where the Bell rings
without the cones having been finished, and you obtain the
conclusion I indicated.
The bottom line is, therefore, that it is futile to try to impose
noncommitment for dividing instants on the level of the models and
by using nonstandard time domains such as "punctuated time". In
those cases where we wish to express that we don't know or don't
care whether a certain proposition is true or false at a point of
change, it's sufficient to use the multiple models approach while
admitting "standard" time (integers or reals, by preference). Then
we don't need any theory of time at all besides high-school or (at
most) college math.
All of this presumes of course standard two-valued logic, where
models can only assign the truth-value true or false. You may obtain
another perspective by going to e.g. three-valued logic, where
everything can be undetermined besides true or false. But, as H.C.
Andersen once said, that is another story.
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Jixin