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Re: [ontolog-forum] Is there something I missed?

To: "[ontolog-forum]" <ontolog-forum@xxxxxxxxxxxxxxxx>
From: Ed Barkmeyer <edbark@xxxxxxxx>
Date: Fri, 30 Jan 2009 15:33:14 -0500
Message-id: <4983640A.9070806@xxxxxxxx>
Matthew West wrote:
> Dear Mitch
>> - Any (relational) database schema describes a set of relations/an
>> ontology and most are extremely multi-ary.
> [MW] Yes but these are mostly denormalisations to overcome the costs of
> table joins.    (01)

The whole aspect of arity of tables is bound up with the theory of 
"normal forms".  Among other things, a table of 5 columns might still 
only represent a binary relation, and in "4th normal form" that is what 
it represents.  The problem is how many columns it takes to provide the 
key values that identify each of the two participating individuals.  So 
4th normal form requires a table of n columns to express a function F of 
the form:
   F(c\1, ..., c\k) = (c\k+1, ..., c\n)    (02)

So the table representing the total price of each line item of a 
purchase order may have four columns (order#, line#, amount, currency)
where the function "total price" is:
   total_price (order#, line#) = (amount, currency) [4th NF]
   (totalPrice lineItem monetaryAmount)  [binary relation]    (03)

It is only in "5th normal form" that all unary relations are one-column 
tables and all binary relations are two-column tables.  And this is 
accomplished by creating artificial "identifiers" that correspond to 
"co-variant vectors" -- the "compound keys" for the individuals.    (04)

The highly prized "3rd normal form" has the property Matthew observes:
For each column c\i where i > k,
   c\i = F\i(c\1, ..., c\k)
i.e., each data column is a function of _all_ the key columns, but there 
is no requirement for the data columns to have any other relationship to 
each other.  All the functions are different.  Thus the 3rd normal form 
table is the "natural join" of a set of "4th normal form" tables.    (05)

Relational algebra is just yet another mathematical theory that has some 
relationships to knowledge engineering.    (06)

-Ed    (07)

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                FAX: +1 301-975-4694    (08)

"The opinions expressed above do not reflect consensus of NIST,
  and have not been reviewed by any Government authority."    (09)

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