Ferenc, Ali, Mike, and Bill, (01)
FK> This is the same thing as resolving a conflict between two ideas,
> concepts, anything in fact, by moving it one level up in their
> hierarchy. We do that all the time in conflict resolution. (02)
Yes. When you generalize issues about ontology and design by
looking at the underlying theories, you can find many similarities
to other approaches for reconciling theories and ideas. (03)
Conflict resolution is an attempt to get two people to agree to
a common remapping of their personal theories. (04)
AH> There are indeed multiple methods beyond the two you mention... (05)
I agree. I was focusing on two basic methods that apply to theories
of any kind: (06)
1. Generalizing by abstracting away from the details of specific
cases and looking for commonalities. (07)
2. Specializing by taking the union of all the details of all
the specific cases involved. (08)
The operation of specialization/generalization determines a partial
ordering over the set of all possible theories that are expressible
in a given logic. That partial ordering defines a lattice, which
provides a roadmap for showing all possible relationships among
the theories. (09)
AH> We can use a richer vocabulary than simply "generalization" or
> "specialization", utilizing the notion of faithful / partial / weak
> (etc.) interpretations of one theory into another. (010)
I agree. In my 2000 book, I related the lattice to the AGM operators
for theory revision: expansion (which moves to a more specialized
theory), contraction (which moves to a more generalized theory),
and revision (which moves sideways by contractions and expansions). (011)
I also added a fourth operator, which I called *analogy*, but which
I later called *relabeling* because the word 'analogy' has too many
other connotations. The relabeling operator makes jumps across the
lattice by relabeling the names of some or all the functions and
relations. Relabeling is a special case of interpretation. The
more general interpretation can be performed by a combination of
relabeling plus the AGM operators. (012)
All the terms you have used to relate theories can be defined
by sequences of the AGM operators plus relabeling. I have very
strongly endorsed the work that you have been doing with Michael
Grunginger and others. That theoretical work and the tools that
implement it make a very important contribution. (013)
Both approaches are complementary: (014)
1. The lattice is a roadmap that displays the full range of all
possible relationships among all possible theories expressible
with a given logic. (015)
2. The work that you have been developing and implementing provides
the technology for driving along those roads. (016)
AH> Briefly, the semantic mapping procedure specified above constructs
> an image of any target ontology in a repository (satisfying the
> constraints in chap 3), and uses the various types of interpretations
> to generate mappings. While one can fully automate this process in a
> bruteforce manner, with some basic human guidance, this problem can
> be achieved much quicker, and turned from consistency checking into
> sat checking. (017)
Yes. You're developing the automated tools for traversing the map.
The lattice, as a roadmap, wouldn't be of much value without such tools.
They're complementary. (018)
AH> I would note, that while the lattice of theories is useful as
> a theoretic grounding, in practice, joining all what we call
> corehierarchies into one giant lattice gains little benefit. (019)
I agree. I have been preaching modularity for years. I would never
advocate merging theories unless and until there is a strong reason
for doing so. In fact, I have advocated that the large theories
that have already been built, such as Cyc and SUMO, should be
subdivided into smaller reusable modules. The purpose of the
lattice is to show how those modules are related to each other
and to the larger theories. (020)
AH> The definition of a core hierarchy may be found in the FOIS paper. (021)
It's a good paper, and I copied the reference, abstract, and concluding
paragraph at the end of this note. (022)
MB> Can I potentially refer to this note in any papers, notes etc.
> about how we at the EDM Council want to try and deal with
> interoperability with existing standard ontology terms? I'm familiar
> with the ISS paper you presented at SemTech, which covers the Black
> Box approach but I think this summary of how these two approaches
> relate, is useful in itself. Or is there a paper coming which
> will develop this? (023)
I'm glad that you like the approach, and please feel free to cite
the URL of the note in the archives of the ontolog forum. I have
written a lot about the lattice of theories in various papers,
and someday I plan to gather up all the fragments in one place.
But I haven't done that yet. (024)
I also suggest that you cite the publications by other people
working on the SIO project, since they have actually implemented
important aspects. As I said to Ali, all of this work is mutually
reinforcing. (025)
WB> Isn't a third possibility a direct mapping between two theories
> without considering a/the superset? (026)
Yes. But then you have to ask whether the mapping is partial or total: (027)
1. If there is a total mapping of theory A into theory B that
preserves all theorems in A (which I would call a relabeling
and Ali would call an interpretation), then B is a common
specialization of itself with a relabeled version of A. (028)
2. If there is a partial mapping of some subset of A into some
subset of B, then that subset of B is a common generalization
of a relabeled version of A with B. (029)
Some of those mappings can become fairly complex. See the example
in slides 84 to 90 of http://www.jfsowa.com/talks/iss.pdf . (030)
That example shows why I used the term 'analogy', but I switched
to the term 'relabeling' because it is easier to explain to people.
The word 'interpretation', which Ali uses, is common among logicians,
but I despair of trying to teach it to anybody who hasn't spent long
years in becoming indoctrinated in that way of talking. (031)
John
_______________________________________________________________________ (032)
Source: http://stl.mie.utoronto.ca/publications/colorefois.pdf (033)
Ontology Verification with Repositories (034)
Michael GRÜNINGER, Torsten HAHMANN, Ali HASHEMI, Darren ONG (035)
Abstract. In this paper we show how the relationships between first
order ontologies within a repository can be used to support ontology
verification. We discuss the use of representation theorems and
classification theorems to characterize the models of an ontology,
and then show how such results can be obtained from notions such
as relative interpretation. (036)
6. Summary (037)
The concepts and methods discussed in this paper, in particular the
relationships of relative
interpretation together with conservative and nonconservative extension,
can be used
to organize the theories within an ontology repository. We have shown
how to use these
relationships between ontologies to assist us in the characterization of
the models of the
ontologies. In particular, we can use the notion of interpretability to
specify representation
theorems, and use the notion of reducibility of structures to construct
the models
of one ontology from the models of another ontology by exploiting the
relationships
between these ontologies and their modules in the repository. (038)
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