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Re: [ontology-summit] Reasoners and the life cycle

To: ontology-summit@xxxxxxxxxxxxxxxx
From: John F Sowa <sowa@xxxxxxxxxxx>
Date: Thu, 27 Dec 2012 23:15:56 -0500
Message-id: <50DD1CFC.4020905@xxxxxxxxxxx>
Michael and Fabian,    (01)

Many terms have a lot of overlap that can create confusions and
misunderstandings:  underspecified, general, vague, accurate,
approximate, detailed, simplified, and abstracted.    (02)

For some of these terms, it's possible to give formal definitions:    (03)

  - A theory X is more general (or less specialized) than a theory Y
    iff every model of Y is also a model of X.    (04)

  - A theory X is underspecified relative to a theory Y iff every axiom
    of X is also true of Y, but Y also has axioms (or implications)
    that are not implied by the axioms of X.    (05)

If the theories are expressed in first-order logic (or any subset
of FOL), the terms 'general' and 'underspecified' are synonyms:
X is more underspecified than Y iff X is more general than Y.    (06)

The other terms in the above list are harder to define formally.
They also depend on the level of *granularity*, which is also
related to the context, the point of view, the purpose of an
application, and the measuring instruments involved.    (07)

I recommend an article on granularity by Jerry Hobbs:    (08)

    http://www.isi.edu/~hobbs/granularity-web.pdf    (09)

Hobbs gave a good example of a road at various levels of granularity
for different purposes or points of view.  From page 1 of the article:    (010)

> We look at the world under various grain sizes and abstract from it
> only those things that serve our present interests. Thus, when we
> are planning a trip, it is sufficient to think of a road as a one-
> dimensional curve. When we are crossing a road, we must think of
> it as a surface, and when we are digging up the pavement, it becomes
> a volume for us...
> Our ability to conceptualize the world at different granularities
> and to switch among these granularities is fundamental to our
> intelligence and flexibility...    (011)

The axioms required for the different views at different levels
of granularity are *mutually inconsistent*.  But it's possible to
have a very general or underspecified theory of a road that is
consistent with all the special cases:  1-D, 2-D, or 3-D.    (012)

> Something can be accurate enough for one purpose, but woefully
> inadequate for others.  Ideally both representations can seamlessly
> live together zooming in and out to the appropriate level of detail...    (013)

The approach that I recommend is to use the general or underspecified
theory for interoperability among a wide range of applications.    (014)

For the road example, you can talk about a road from Boston to New York
without worrying about the granularity.  But when you're driving along
the road or fixing a pothole, you need to choose one or another of the
inconsistent specializations.    (015)

> "ontology x is accurate" iff all axioms in x are true.
> "ontology x is complete (given a set of requirements)" iff
>   (a) x contains all  classes, relations that are needed to represent
>       a piece of reality sufficiently to meet the requirements, and
>   (b) x contains sufficient axioms that are needed to answer queries
>       that are relevant for the requirements (given an appropriate theorem 
>prover)    (016)

The underspecified theory is the only one that can be considered true
for all conditions and points of view.  It can be useful for some
simple applications, such as finding a road between two cities.  But
it is inadequate for the details of driving or repairing potholes.    (017)

There are also issues about accuracy that are more subtle than just
the number of decimal places in the numbers.  For example, the distance
along a curve in a road is shorter if you're driving in the inner lane
or the outer lane.  If you switch lanes to pass another car, you may
increase the distance you travel -- unless you are lucky enough to
switch from the outer lane of a curve to the inner lane.    (018)

For such reasons, it's meaningless to specify the distance along
a highway with more than 3 or 4 decimal digits.    (019)

John    (020)

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