Re: [ontolog-forum] Ontology, Analogies and Mapping Disparate Fields

[Ali SH <asaegyn+out@xxxxxxxxx>]

Dear John,

I haven't been at a proper computer until today so I haven't had a chance to respond.

Thanks for the feedback. 

I'm trying to identify the meat of your criticism, but as far as I can tell the discussion has focused on / taken issue with framing and emphasis.

To be clear, you're not objecting to any of the main contributions, which include, but are not limited to:

• two concrete, formal definitions of modular ontology
• which are dependent on the notion of core-hierarchy, 
• which in turn are dependent on recognizing the distinction between conservative and non-conservative extensions, 
• how to arrange said modular ontologies into hierarchies that exhibit desirable properties
• how to define quasi-orderings over such modular hierarchies 
• how to construct an ontology repository that exhibits desirable mathemtical + computational properties
• how to implement semi-automated procedures to decompose/map novel ontologies into such a repository
• formal, detailed, fine-grained specifications for how theories in the lattice may be connected to one another
• formally defining how two ontologies are Similar and Different

And as far as I can tell, it appears your major qualm is that we have not emphasized the link to the Lindebaum Lattice and AGM theory to your preference.  

[JFS] The key generalization that cuts through the muck is something I've

been talking about for years:  The Lindenbaum Lattice for a given
logic and vocabulary.

Hmmm, are not partial orders a generalization of lattices? 

I'd also note that back in 2006 you also wrote [http://ontolog.cim3.net/forum/uos-convene/2006-03/msg00314.html]:

[JFS] Obviously, [the lattice] doesn't buy anything practical since the infinite
lattice is never going to be implemented by anyone.  However, I
have found it a convenient pedagogical tool for talking about and
classifying the operations of combining, extending, and revising
theories.  

We introduce the idea of hierarchy with respect to generic partial orders, and present the meet semi-lattice as a special case (corresponding to the closed hierarchy). A lattice representing all possible axioms in a logic and a non-logical vocabulary is too coarse, as we are interested in making explicit and highlighting particular partial orders over selected sets of axioms. In some cases, these hierarchies will at best be partial orders, in other cases, they correspond to what we call closed hierarchies and these spcial cases could indeed be viewed as sub-lattices of the L-Lattice. 

On Sun, Dec 18, 2011 at 3:14 PM, John F. Sowa <sowa@xxxxxxxxxxx> wrote:

This is an excellent example of how the lattice simplifies and clarifies
the specifications.  The definition is trivial:  your "strongest shared
subtheory" of two theories is their supremum (AKA maximal common
generalization).  That is the ideal definition:  it's precise, formal,
short, and sweet.  According to the theory, the supremum always exists,
but the theory of lattices does *not* say how to compute it.

...
But I'd like to correct a typo
in my previous note to Michael:

The supremum of two or more theories is their minimal common
generalization -- that's what Michael called the "strongest shared
subtheory".  The infinimum is the maximal common specialization.

To be clear, we do observe that closed hierarchies are meet-semilattices (page 8), and while we do not use the word infinimum, we do use the word meet, which is also widely used in lattice theory [1], [1a]. In Definition 13, we again cover the more general case of a partial order in defining the notion of Remainder. The special case where the hierarchy is a lattice is captured in Lemma 4 (page 11).

[JFS] In [my KR book], I extend the three AGM operators with a fourth operator,

which I called analogy in that book.  I later decided to use the term
'relabeling' instead of 'analogy'.  It's narrower and more descriptive.

With regard to the relabeling operator, we draw on work that predates this idea, and in effect define different variations of relabeling, drawing on the well developed literature re interpretations. Specifically, as we note explicitly on page 23, analogy/relabeling as you've used it, could be defined in terms of faithful interpretation (though in 2006 you seem to casually suggest is a synonym for similar to relative interpretations - http://ontolog.cim3.net/forum/uos-convene/2006-03/msg00314.html). And in my thesis, we did explicitly link the work to conceptual metaphors...

In this case, the paper's design choice was to emphasize and build on the work in [2] and [3] which was (I think) the first work to formally introduce interpreting one theory in terms of another. One reason for this design choice is that it immediately connects to the concrete and successful projects at [4], [5] and is able to accommodate (weaker?) forms of relabeling - specifically partial, relative and faithful interpretations. In this light, analogy is the special case where the interpretation is faithful (plus a couple of other nuances, though one can't be sure based on your previous writing).

While I can appreciate you wanting a greater emphasis on the role of the AGM operators and the Lindebaum Lattice (they were indeed mentioned), the group opted to use the well established language of interpretations to formalize the various relationships in terms of hierarchies, modularity, reducibility, translations and mappings.  I just quickly reviewed the definitions, lemmata and theorems in the paper, and I think only two (generously, maybe 3) might have been affected by framing the results primarily in terms of a lattice and not a partial order. Of course,  in so doing, we would have also lost our more general notion of hierarchy...

The place where the AGM postulates are most relevant might be in the specification of the semantic mapping procedures. Please note that the first procedure, Finding Reducible Theories, is predicated on the general notion of interpretability, hence analogy alone would not suffice. Of course, the actual procedures themselves are quite clear on their own and draw on the various flavours of interpretation which are nowhere to be found in AGM lit.

That said, it might still be possible to provide a high level account of the procedures in terms of the AGM operators.

If one were to be charitable and extend the connection to AGM, one might consider the entire repository as undergoing a "belief" revision via the addition of novel ontologies. Following this analogy, one possible way of thinking about some of the presented procedures (especially those in section 7) is that they can be viewed as implementing a version of AGM belief revision to the entire repository. Of course, backing up this claim with actual proofs and theorems would have eaten up a few more pages and would have still required the introduction of interpretations. Similarly, I suppose one might also conceptualize a three way semantic mapping as a sort of assisted, referenced belief revision, though again, it would have required quite a few more pages to formally, precisely back up these claims.  As Michael noted in off-line discussions, both these conceptualizations are extending the AGM framework to repositories as an analogy to belief revision. Perhaps we should have also highlighted the analogous connection to Schank's work on memory [6] and formally captured his intuition as well? :P

In contrast, in the mathematical logic of [2] and [3], the vocabulary needed to specify our intuitions are already formally specified and suffice for our purposes.

When we begin deploying reasoning services over COLORE, the AGM postulates may play a more prominent role in helping ontologists design or pick their axioms. Of course, if someone else believes that the analogy between a repository and belief revision merits a paper, they are more than welcome to write up their thoughts and back up their claims.

All in all, I appreciate that you have emphasized these connections, though we opted to use the terminology introduced in [2] and [3] and defined hierarchies via the more general partial orders. As far as I can tell, your criticism has focused not on any of our contributions or results, but style / framing and what you consider "muck" - though I submit that if you ever get around to formally capturing your intuitions at the level of detail (with all the nuances) as in this paper, you will find the "muck" to be unavoidable. In any event, I look forward to reading your upcoming Principles of Logic and Ontology textbook .

Best,
Ali

[1] Birkhoff, G. Lattice Theory, 3rd ed. Providence, RI: Amer. Math. Soc., 1967.

[1a] Grätzer, G. Lattice Theory: First Concepts and Distributive Lattices. San Francisco, CA: W. H. Freeman, 1971

[2] Burstall, R.M. and Goguen, J.A. (1977) Putting theories together to make specifications. International Joint Conference on Artificial Intelligence 1977, pp. 1045-1058.

[3] Enderton, H. (1972) Mathematical Introduction to Logic, Academic Press.

[4] Farmer,W. M. (2000) An Infrastructure for Intertheory Reasoning. Proc. of the Seventeenth Int. Conference on Automated Deduction (CADE-17), LNCS 1831, pp. 115–131.

[5] http://imps.mcmaster.ca/
[6] Schank, Roger. Tell Me A Story: A new look at real and artificial memory. Scribners, 1990.

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