John F. Sowa wrote:
> As George Box said, "All models are wrong. Some are useful."
> (01)
For the record, Box wrote this in 1979, which is within the memory of
the IT world. But mathematical models predate the IT world, and the
same aphorism was penned by W. Edwards Deming in 1947. (I think we
could call that 'prior art'.) (02)
I am indebted to another colleague for the aphorism of Manfred Eigen,
which I have found to be even more useful: (03)
"A theory has only the alternative of being right or wrong. A
model has a third possibility: it may be right but irrelevant." (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)
We could. But it seems to me fairer to say, as semanticists do, that
the only way to transfer concepts and information from person to person
is to express them, and no simple means of expression conveys all of the
concept that the speaker possesses. Even the comprehension of a single
sentence requires all the trappings of membership in a 'speech community'. (06)
In the same way, formal grammatical structures are accompanied by
explicit hierarchical interpretation rules, and most importantly by
semantic (non-grammatical) interpretation of the actual terms that
appear in the grammatical slots. For n-ary relations, the issue is only
whether the grammar of the language in question has some specific
structure and interpretation rules that capture the basic intent of the
n-ary relation. The problem stated in the original email is that RDF
and RDFS do not have any such specific structure and interpretation rule. (07)
Now, with respect to the terms that fill the slots, it is often the case
that a relationship that the speaker thinks of as n-ary actually has a
conceptual factorization that the speaker may not have perceived. That
is, some 'n-ary relations' lose no information by being factored into
binary relations. Compare the following:
John gives the book to Mary
John meets Mary in New York. (08)
In the first case the English prepositional phrase "to Mary" is somehow
intrinsic to the intent of the verb "gives". That is, "gives" is
conceptually ternary. By comparison, the prepositional phrase "in New
York" is extrinsic to the intent of the verb "meets". There is nothing
really ternary about "meets ... in". The second example could be
phrased: The situation "John meets Mary" occurs in New York. And that
phrasing involves 3 binary relationships: person meets person;
situation is described by proposition; situation occurs in location.
(John's examples demonstrate these two approaches.) That is, there is
nothing artificial about converting the second example to binary
relations, but there is something artificial in doing that to the first
example. (09)
In my experience, there are intrinsically ternary and quatenary verb
concepts. English grammar supports them with the use of prepositional
phrases, but listeners use those prepositional phrases in comprehending
the verb concept. And for that matter, there are also binary verb
concepts that are expressed in English using intransitive verbs and
prepositional phrases: John goes to the store. But many English
prepositional phrases are purely adverbial, and are understood by
listeners as additional details that are not central to the verb
concept. English grammar does not make this distinction; technically it
treats all prepositional phrases as adverbial (or adjectival). The
listener learns these interpretation rules on a verb-specific basis.
The problem is that we can't do that with formal languages. (010)
(Other languages use different technical strategies for dealing with
multiple roles relative to a verb, and some of those structures have
only the 'additional role' interpretation. I haven't seen that
technique used in formal languages, but it is clear that it could be.) (011)
My point is that we must first distinguish between n-ary database
relations, which may represent compound statements or relations with
adverbial modifiers, and conceptually n-ary relations. Then we must
provide grammatical elements for compound statements and adverbial
phrases, as distinct from n-ary relations. Finally, we need some
grammatical structure that is used to represent truly n-ary relations
and nothing else. SQL is a bad example, because it merges n-ary
relations, compound statements and adverbial modifiers into one
grammatical structure. CLIF is a bad example, because it doesn't have a
way to express adverbs, either. English is a bad example, because it
doesn't have an unambiguous grammatical representation of n-ary
relations. If I understand correctly, John's CGL has unique syntax for
adverbial phrases and unique syntax for n-ary relations. So we have a
model for what might be done as an RDF extension, in much the same way
that CGL is a CLIF extension. But we would still have to get the target
community to agree to understand the chosen conventions. (012)
Oh, and one more thing: you have to get journeyman knowledge engineers
to understand the difference. I am reminded of an observation by Sjir
Nijssen that, if you give modelers different constructs for similar
concepts, half of them will use the wrong one half the time. (013)
-Ed (014)
> As an example, consider the English sentence
>
> Sue gives a child a book.
>
> 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.
> (015)
Exactly. The content of the expression (the sentence) is by no means
all of the speaker's conceptualization of the situation. (And that says
nothing whatever about the relationship between the speaker's
conceptualization and the actual situation, which we may argue
philosophically may be beyond our mental grasp.) (016)
> Second, any choice of ontology will make further approximations
> that may be quite different from the defaults used in English.
> (017)
That is, any ontological language will have a formal grammar with a
strict interpretation. But that says nothing about the meaning of
'Sue', 'gives', 'child', or 'book', each of which is in the formal
grammar only a 'term' with some set of matching character sequences.
99% of the intent is carried by those terms, and not by the grammatical
construct. (018)
N-ary relations is definitely a grammatical issue, not a conceptual
one. John describes two grammatical structures that nominally carry the
same basic intent, but they require different interpretation conventions
to achieve that information transfer. (019)
> One choice is to represent the verb 'gives' as a triadic relation:
>
> (Ex)(Ey)(Child(x) & Book(y) & Gives(Sue,x,y))
>
> With a notation that uses monadic relations as types or
> restrictions on the quantifiers, we could write
>
> (Ex:Child)(Ey:Book) Gives(Sue,x,y)
>
> 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:
>
> (Ex:Child)(Ey:Book)(Ez:Give)(Agent(z,Sue) & Recipient(z,x) & Theme(z,y))
>
> 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.
>
> 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:
>
> http://www.jfsowa.com/figs/give.gif
>
> 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.
>
> 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).
>
> 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.
>
> 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.
>
> Some observations about this example:
>
> 1. For mapping to and from natural languages, the version on the
> right with explicit case relations is often convenient.
>
> 2. For a theorem prover or a relational database, the version
> with a triadic Gives relation may be the most efficient.
>
> 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.
>
> 4. For the untyped triples of RDF, more triples are needed
> to show the type labels in the boxes.
>
> 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).
>
> 6. I usually say that conceptual graphs are *bipartite*
> graphs with two kinds of nodes and just dyadic arcs.
>
> 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.
>
> 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.
>
> John
>
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> (020)
--
Edward J. Barkmeyer Email: edbark@xxxxxxxx
National Institute of Standards & Technology
Manufacturing Systems Integration Division
100 Bureau Drive, Stop 8263 Tel: +1 301-975-3528
Gaithersburg, MD 20899-8263 Cel: +1 240-672-5800 (021)
"The opinions expressed above do not reflect consensus of NIST,
and have not been reviewed by any Government authority." (022)
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