SUO: RE: Re: Exposition
Dear Jon,
> ¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤
>
> Strand 3. Embedded Tables & Extended Relations
>
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>
> JA: There were a few issues about "embedded", "indexed", "keyed",
> or "ordered" relational tables that were initially raised in
> the present frame, and so I will try to address them now, at
> least until the running accumulation of bits grows too large.
>
> JA: This filiation is offspring to your comment that,
> with regard to your putative method of reduction:
>
> | MW: Where I come from this method would
> | be informally known as reification.
>
> JA: Having seen more of your elaboration of what you are
> actually doing,
> I begin to see this more as a matter of two kinds of
> embedding that
> you are carrying out on the given relation, and one of
> these brands
> of embedding, extension, or extenuation does bear a
> relation to the
> operation that is classically known as "hypostatic
> abstraction", or
> else as "reification". But all of these operations act
> to increase
> the arity of the relation, so I really do not see how
> they help the
> case that you want to establish. I have sorted out
> these questions
> for separate discussion in this note.
>
> JA: Concreteness in the beginning is hardly misplaced,
> so let us consider a discreetly simple example of
> a relational database, just to pick one at random
> from off the grainy strand beneath our feet, here:
>
> JA: Relation L(&) = {<x, y, z> in B^3 : x & y = z}.
>
> o-----o-----o---------o
> | x | y | z = x&y |
> o-----o-----o---------o
> | 0 | 0 | 0 |
> | 0 | 1 | 0 |
> | 1 | 0 | 0 |
> | 1 | 1 | 1 |
> o-----o-----o---------o
>
> JA: There are many different ways to embed a k-adic relational table
> in a more spacious and sumptious table, that is, one that enjoys
> either more rows or more columns or both, and with whatever form
> of extenutaion is chosen to extend the relation thus represented
> either to larger samples of its initial cartesian product domain,
> or else to relations of augmented audacity, arete, or valoration.
>
> JA: But you have already instanced for me the two most common types
> of embedding or extension, and so it is meet that we take these
> up first. Let us just contemplate their basic forms all in one
> place, and leave off discussing the details to later in the day.
>
> JA: Extension 1. Abstract Hypostasy, Reflective Entitlement.
>
> o---------o-----o-----o---------o
> | t | x | y | z = x&y |
> o---------o-----o-----o---------o
> | "L(&)" | 0 | 0 | 0 |
> | "L(&)" | 0 | 1 | 0 |
> | "L(&)" | 1 | 0 | 0 |
> | "L(&)" | 1 | 1 | 1 |
> o---------o-----o-----o---------o
MW: I don't follow this.
>
> JA: Extension 2. Indexing, Keying, Ordering, Sequencing.
>
> o---------o-----o-----o---------o
> | n | x | y | z = x&y |
> o---------o-----o-----o---------o
> | 1 | 0 | 0 | 0 |
> | 2 | 0 | 1 | 0 |
> | 3 | 1 | 0 | 0 |
> | 4 | 1 | 1 | 1 |
> o---------o-----o-----o---------o
>
> JA: Some of the names, of course, are not yet standard.
>
> MW: In the interests of clarity, this can be seen as an
> intermediate step in the transformation I am making.
>
> MW: Having introduced the index (the reification step) ...
>
> Gasp!
>
> MW: then each of the components of each row is related to that index.
> This gives you a basis for creating a relation for each
> column, so:
>
> MW: n = {1,2,3,4}
>
> x = {<1,0>, <2,0>, <3,1>, <4,1>}
>
> y = {<1,0>, <2,1>, <3,0>, <4,1>}
>
> z = {<1,0>, <2,0>, <3,0>, <4,1>}
>
> MW: I fail to see why you think x, y, z are not relations.
> I have swopped 1 tradic relation for 3 dyadic relations
> plus a set.
>
> Unhuh, and are these three 2-adic relations
> just hanging out there, blowing in the wind,
> as if they were to wash up on the shores of
> your certainty, each one a detached message,
> all dispersed in welters of many-splintered
> brands and colors of bottles, that the wine
> darks seas of the world give up to you, and
> you, on receipt of this gift, how would you
> even know that each message had any bearing
> on the meaning of the others, if not by way
> of some clue that you can persieve or purse
> seine or just plain abdeuce to be contained
> therein, read between the letters and lines?
MW: Not at all, they are linked by the objects
they contain (though it would be sensible not
to use 1,2,3,4 as the index as you have done
since at least 1 is also used to represent
a different object in the matrix).
> And are these characters, in their pedigree,
> any kind'o'kin to the gang'o'four you hinge
> them on, or do they hang but separately, as
> nothing but instances of one dummy variable
> after'n'other?
MW: Very colourful, but you lost me.
>
> MW: It is relatively straight forward to do the
> transformation in each direction, so they
> are equivalent.
>
> Not so.
>
> You must remember this:
> A set is still a set,
> as the index goes by.
>
> Consider an "indexed 3-adic relation",
> which is really this 4-adic relation:
>
> o---------o-----o-----o---------o
> | n | x | y | z = x&y |
> o---------o-----o-----o---------o
> | 1 | 1 | 1 | 1 |
> | 2 | 1 | 0 | 0 |
> | 3 | 0 | 1 | 0 |
> | 4 | 0 | 0 | 0 |
> o---------o-----o-----o---------o
>
> This is a distinct 4-adic relation that might
> well arise in any number of applications from
> what is fundamentally the same 3-adic relation.
> Indeed, there are 4! = 24 distinct such 4-adics
> that may easily arise in one setting or another
> out of the same 3-adic relation that we had in
> the beginning.
MW: I don't follow this move. To me 1,2,3,4 are (or
should be) unique object identifiers in the Universe
of Discourse (1 isn't here so there is a problem).
> So the mapping is not 1 to 1,
> but 1 to 24, and this does not make for any
> kind of invertible transformation, nor any
> kind of equivalence between formal objects.
>
> You have come to the very threshold of teridentity, my friend.
MW: Save me, save me!!!!
>
> | You tell him to come in sit down
> | But something makes you turn around
> | The door is open you can't close your shelter
> | You try the handle of the road
> | It opens do not be afraid
> | It's you my love, you who are the stranger
> | It is you my love, you who are the stranger.
> |
> | Leonard Cohen, 'The Stranger Song'
>
> Jon Awbrey
>
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>
Regards
Matthew
===============================================================
Matthew West http://www.matthew-west.org.uk/
Principal Consultant Shell Visiting Professor
Operations & Asset Management The Keyworth Institute
Shell Services International The University of Leeds
http://www.shellservices.com/ http://www.keyworth.leeds.ac.uk/
H3229, Shell Centre, London, SE1 7NA, UK.
Tel: +44 207 934 4490 Fax: 7929 Mobile: +44 7796 336538
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