SUO: Re: Relations And Their Divisitudes
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RATD. Note 16
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JA = Jon Awbrey
TJ = Tom Johnston
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| Table Talk in Translation, Presented as a Two-Column Table |
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| JA: | TJ: |
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| Standard Math Language, | Tom's translation into the language |
| A La Descartes | of relational databases |
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| k-adic relation | table with k columns |
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| k-tuple | row of a table with k columns |
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| relation(s) | table(s) in a relational database |
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| 1-adic projection | result of a relational PROJECT |
| | operation on a table, that |
| | leaves just one column |
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| a k-tuple is defined as | a row of a table with k columns |
| being determined by its | is defined to be determined by the |
| 1-adic projection data, | (ordered) set of relational PROJECT |
| but, | operations, each of which results in |
| k-adic relations in general | a single column instance of the table,|
| general are not determined | but a table with k columns in general |
| by their m-adic projections | is not determined by the (ordered) set|
| for any m < k. | of m PROJECT operations on that table,|
| | for any m > k. |
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TJ: One problem is "determined by", appearing (twice) in Jon's
statement and in my paraphrase. What does it mean?
TJ: Second problem is the purported demonstration of the claim.
I just don't follow it.
Tom,
I think that I have now covered your questions about
(A) determination, and (B) demonstration, except for
the small but crucial matter of the relation between
this weaker form of "projective reducibility" and the
more fundamental form of "compositional reducibility".
But maybe this'd be a good place to pause,
and see if we are still at the same table.
After that I'll turn to the bit about information
being the integral of comprehension and extension.
Jon Awbrey
TJ: My own thoughts about the difference between a row of
a table (a tuple) and the table itself (a relation) is
that the former is part of the extension of the relation,
while the latter (more specifically, the set membership
conditions which define it) represents the intension of
the relation. Next, that one cannot always infer the
intensional rules from the extensional instances because,
at any given moment, the set of all those instances may
not define the boundary conditions of all those rules.
To take a simple example, if one column of the table is
a status-code column, whose domain is the letters A - X
inclusive, it might be that the status-code value of P
is not instantiated in any tuple. Or: a column might be
defined as nullable, although all of the current rows
have a value in that column. In short: that intensional
rules might be inferrable from the full Cartesian Product
of a set of columns (if we knew it was the full Cartesian
Product), but are not inferrable from any subset of the
full Cartesian Product.
TJ: So Jon, can you re-cast your points demonstrating that
last claim of yours listed above, in my language of
working databases? If not, can you give it another
try in your own language?
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