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Re: [ontolog-forum] N-RELATIONs: Formal Ontology, Semantic Web and Smart

To: ontolog-forum@xxxxxxxxxxxxxxxx
From: "John F. Sowa" <sowa@xxxxxxxxxxx>
Date: Tue, 08 Nov 2011 11:58:17 -0500
Message-id: <4EB95FA9.40805@xxxxxxxxxxx>
I was attending a conference when this thread began, and I didn't
have a chance to comment.  But I'd like to make a few remarks
about the quotation that started it on Friday:    (01)

> Plus RDF doesn't have any *standard* way to tag or represent n-ary
> relations -- we have taken a do-it-yourself attitude[1] -- and thus
> tools cannot predictably recognize n-ary relations as such.    (02)

Before saying anything about any specific formalism or notation,
I'd emphasize the fundamental difference between the subject
matter and the notation used to express it.    (03)

As George Box said, "All models are wrong.  Some are useful."    (04)

We could adapt that principle to say "All notations distort
the structure of the subject.  For various purposes, some are
better or more useful approximations than others."    (05)

As an example, consider the English sentence    (06)

    Sue gives a child a book.    (07)

First point:  this sentence is an approximation to the situation.
If asked, the speaker could add much more detail to explain
the circumstances and the manner of giving and receiving.    (08)

Second, any choice of ontology will make further approximations
that may be quite different from the defaults used in English.
One choice is to represent the verb 'gives' as a triadic relation:    (09)

    (Ex)(Ey)(Child(x) & Book(y) & Gives(Sue,x,y))    (010)

With a notation that uses monadic relations as types or
restrictions on the quantifiers, we could write    (011)

    (Ex:Child)(Ey:Book) Gives(Sue,x,y)    (012)

This uses a triadic relation named Gives.  The issue of how the three
arguments are related to the act of giving is not explicitly shown.
Another representation would use *case relations* or *thematic roles*
of linguistics to show how the participants are related to the action:    (013)

(Ex:Child)(Ey:Book)(Ez:Give)(Agent(z,Sue) & Recipient(z,x) & Theme(z,y))    (014)

This notation treats an action like Give as an entity in the domain
of quantification.  It shows that Sue is the agent of Give, a child
is the recipient, and a book is the theme.    (015)

These three formulas in predicate calculus emphasize different aspects
of the action, but there are some underlying commonalities.  a graph
representation of these formulas can show that.  See the following
diagram:    (016)

    http://www.jfsowa.com/figs/give.gif    (017)

This diagram shows two conceptual graphs for the formulas 2 and 3 above.
The graph on the left has an oval labeled Gives with three arcs.  The
graph on the right has a box labeled Give that is linked to three
dyadic relations labeled Agnt, Rcpt, and Thme.    (018)

Note that both graphs have the same connectivity, but different
numbers and shapes of nodes.  The graph on the left has one oval
(called a relation) that is linked to three boxes (called concepts).    (019)

The graph on the right has 3 relations and 4 concepts.  If you ignore
the difference between boxes and ovals (or circles), the connectivity
is the same:  a node in the middle with three links to chains
consisting of one or more nodes.    (020)

To represent the first formula above, you could take the graph
on the left, remove the type label (Person, Child, or Book) from
each box, and place that label in monadic oval attached to the box.
You would still have a triadic connection in the middle, but with
longer chains attached to it.    (021)

Some observations about this example:    (022)

  1. For mapping to and from natural languages, the version on the
     right with explicit case relations is often convenient.    (023)

  2. For a theorem prover or a relational database, the version
     with a triadic Gives relation may be the most efficient.    (024)

  3. For a mapping to triples, the version on the right may be
     useful *if* the triples can distinguish the types of
     each of the three items.    (025)

  4. For the untyped triples of RDF, more triples are needed
     to show the type labels in the boxes.    (026)

  5. The notation for conceptual graphs uses boxes and ovals
     to distinguish concepts and relations (or square brackets
     and rounded parentheses in the linear CGIF notation).    (027)

  6. I usually say that conceptual graphs are *bipartite*
     graphs with two kinds of nodes and just dyadic arcs.    (028)

  7. However, it is also permissible to say that conceptual
     graphs are *hypergraphs* in which the circles or ovals
     represent arcs that can have 0 or more connections.    (029)

By the way, I avoided the word 'hypergraph' because people who are
not mathematicians find it scary.  They usually feel more comfortable
with graphs that have two kinds of nodes.  But the choice of words
has no effect on the algorithms.    (030)

John    (031)

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