Re: SUO: Reducibility Among Relations
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>I am going to string together a cleaned-up composite version of
>an earlier thread on the topic of reducibility among relations --
>viewed another way, on the extent to which it is possible to
>construct relations between complex relations and simpler
>relations. The aim here, once we get past questions of
>what is reducible in what way and what not in no way,
>is to develop concrete and fairly general methods
>for analyzing the structures of those relations
>that are indeed amenable to a useful analysis --
>and here I probably ought to emphasize that
>I am talking about the structure of each
>relation in itself, at least, to the
>extent that it presents itself in
>extensional form, and not just
>the syntax of this or that
>relational expression.
>
>By way of a lightly diverting overture, let's begin
>with an examplar of a "degenerate triadic relation",
>a particular version of the "between relation", but
>let us make it as simple as we possibly can and not
>attempt to analyze even that much of a case in full
>or final detail, but leave something for the finale.
>
>Let B = {0, 1}.
>
>Let the relation named "Rise<2>"
>such that Rise<2> c B^2 = B x B,
>be exactly this set of 2-tuples:
>
>| Rise<2> = {<0, 0>,
>| <0, 1>,
>| <1, 1>}
>
>Let the relation named "Rise<3>"
>such that Rise<3> c B^3 = BxBxB,
>be exactly this set of 3-tuples:
>
>| Rise<3> = {<0, 0, 0>,
>| <0, 0, 1>,
>| <0, 1, 1>,
>| <1, 1, 1>}
>
>Then Rise<3> is a "degenerate 3-adic relation"
>because it can be expressed as the conjunction
>of a couple of 2-adic relations, specifically:
>
>Rise<3><x, y, z> iff [Rise<2><x, y> and Rise<2><y, z>].
>But wait just a minute! You read me clearly to say already --
>and I know that you believed me! -- that no 3-adic relation
>can be decomposed into any 2-adic relations, so what in the
>heck is going on!? Well, "decomposed" implies the converse
>of "composition", which has to mean "relational composition"
>in the present context,
Why does it *have to mean* that?(There are other notions of
decomposition.) And what exactly *is* that? I suspect that what you
are leading towards (and as usual without actually getting there) is
some kind of relational algebra, where relations are things and there
are operations which compose new relations from old ones. If my
suspicions are correct, then PLEASE don't take us on any more of
these roundabout tours through gardens of trivial examples rendered
in ASCII art, but cut to the chase. Tell us the operations of the
relational algebra you propose we should take seriously. It can
probably be done in a page or so. There are many possibilities out
there, and you may have some new ones; but we can't make any useful
progress until we know which one you are talking about.
>Okay, there is a lot more to say, even about such a simple example,
>but I have a feeling that this much is just about enough for today.
No, it wasnt anywhere near enough. It was just enough to be
irritating and just not quite enough to convey any actual content. If
exasperation were music, Jon, you would have perfect pitch.
Pat Hayes
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