SUO: RE: Reductions Among Relations

["West, Matthew MR SSI-GREA-UK" <Matthew.R.West@is.shell.com>]

Dear Jon,

All you have done below is show that the particular way you have 
chosen to reduce triadic relations to dyadic relations does not
always work. Since the method you have chosen is not the one I
would choose, then I a singularly unimpressed. I does nothing to
prove that there is NO WAY that a triadic relation can be reduced
to a dyadic one without loss of information.

If you apply the method that Pat used, you can always reconstruct
the triadic form, because the triad is a simple join  across the
dyads that it is formed from.

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
=============================================================== 

> -----Original Message-----
> From: Jon Awbrey [mailto:jawbrey@oakland.edu]
> Sent: 21 March 2001 06:11
> To: Arisbe; Complexity Group; SemioCom; Stand Up Ontology
> Cc: Josiah Lee Auspitz; Paul R Chernoff; Lisa Cox; Pat Hayes; John F
> Sowa; West, Matthew MR SSI-GREA-UK
> Subject: Re: Reductions Among Relations
> 
> 
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> 
> RAR Interest Group:
> 
> Last time I gave two cases of 3-adic (triadic) relations
> with projective reductions to 2-adic (dyadic) relations,
> by name, "triadics reducible over projections" (TROP's),
> "triadics reconstructible out of projections" (TROOP's).
> 
> Still, one needs to be very careful and hedgey about saying,
> even in the presence of such cases, that "all is dyadicity".
> I will make some attempt to explain why in the next episode,
> and then I will take up examples of 3-adics that happen to
> be irreducible in this sense, in effect, that are not able
> to be recovered uniquely from their 2-adic projection data.
> Call them "triadics irreducible over projections" (TRIOP's).
> 
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> 
> Note 4
> 
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> 
> Projective Reduction of Triadic Relations:  The Catch
> 
> This scene was initially interjected as a comic relief,
> and so I will spare you the drollery of its repetition.
> The entire pith and substance of it boils down to this
> rather trivial observation, but one whose significance,
> such as it may be, gets overlooked with serious punity,
> and that is the fact that, even though one we have now
> found in the sign relations L(A) and L(B) two examples
> of 3-adic relations which are reconstructible from and
> in this sense reducible to the 2-dimensional data sets
> of their 2-adic projections, we entertain an insidious
> self-deception if we think that we have given the slip
> to 3-adic relations altogether.  I feel sometimes like
> I am straining to call the attentions of fish to water
> by blubbering and glubbering "HOH, HOH, HOH" until the
> sea shall free me, I guess, whenever I try to say this,
> but I guess that I might as well just go ahead and try,
> one more time, while still I have the O^2, and so here
> I go, again, flailing at the logical immersion of this
> entire process of "analysis" (= Greek for washing back)
> in hope of rendering its nigh unto invisible influence
> slightly more opaque to the light of a higher analysis.
> All of which is just my attempt to remind you in a way
> that you will not so soon forget that the very conduct
> of the whole analytic reduction step is an association
> among at least three relations, the analysand plus the
> two or more analytic components, and there it goes yet
> again, this ineluctable triadicity.
> 
> To bring the chase up short, the very idea of reduction,
> when it means reducing one thing to more than one other
> thing, is itself a relation of post-dyadic valence, and
> thus parthenogenetically reproducing itself from itself,
> evokes the polycloven hydra's hoof of Persean 3-adicity.
> 
> Okay, I just thought that you might welcome an intermezzo,
> however scherzo-frenetic it might be, but now it's back
> to the opera that the phantom of the operators wrote.
> 
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> 
> Note 5
> 
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> 
> Projective Reduction of Triadic Relations
> 
> In the story of A and B, it appears to be the case
> that that the triadic relations L(A) and L(B) are
> distinguished from each other, and what's more,
> distinguished from all of the other relations
> in the garden of OSI, for the same O, S, I.
> 
> At least, so says I and my purported proof.
> I am so suspicious of this result myself that
> I will probably not really believe it for a while,
> until I have revisited the problem and the "proof"
> a few times, to see if I can punch any holes in it.
> 
> But let it pass for proven for now,
> and let my feeble faith go for now.
> 
> For the sake of a more balanced account,
> it's time to see if we can dig up any cases
> of "projectively irreducible triadics" (PIT's).
> Any such PIT relation, should we ever fall into one,
> is bound to occasion another, since it is a porismatic
> part of the definition that a triadic relation L is a PIT
> if and only if there exists a distinct triadic relation L'
> such that the dyadic faces of L and L' are indiscernible.
> In this event, then both L and L' fall into the de-genre
> of PIT's together.
> 
> [ Note In Passing:
> |
> | PIT's are the same thing as TRIOP's, but I left in
> | this earlier usage for personal historical reasons,
> | and just in case I someday decide to go back to it.
> ]
> 
> Well, PIT's are not far to find, once you think to look for them --
> indeed, the landscape of "formal or mathematical existence" (FOME)
> is both figuratively and litterally rife with them!
> 
> What follows is the account of a couple,
> that I will dub "L^0" and "L^1".
> 
> But first, even though the question of projective reduction
> has to do with 3-adic relations as a general class, and is
> thus independent of their potential use as sign relations,
> it behooves us to consider the bearing of these reduction
> properties on the topics of interest to us for the sake
> of communication and representation via sign relations.
> 
> [ Notation:
> |
> | Let any of the locutions, L c XxYxZ, L on XxYxZ, L sub XxYxZ,
> | substitute for the peculiar style of "in-line" or "in-passing"
> | reference to subsethood that has become idiomatic in mathematics,
> | and that would otherwise use the symbol that has been customary
> | since the time of Peano to denote "contained in" or "subset of".
> ]
> 
> Most likely, any triadic relation L on XxYxZ that is imposed on
> the arbitrary domains X, Y, Z could find use as a sign relation,
> provided that it embodies any constraint at all, in other words,
> so long as it forms a proper subset L of the entire space XxYxZ.
> But these sorts of uses of triadic relations are not guaranteed
> to capture or constitute any natural examples of sign relations.
> 
> In order to show what a projectively irreducible 3-adic relation
> looks like, I now present a pair of 3-adic relations that have the
> same 2-adic projections, and thus cannot be distinguished from each
> other on the basis of this data alone.  As it happens, these examples
> of triadic relations can be discussed independently of sign relational
> concerns, but structures of their basic ilk are frequently 
> found arising
> in signal-theoretic applications, and they are no doubt 
> keenly associated
> with questions of redundant coding and therefore of reliable 
> communication.
> 
> Projectively Irreducible Triadic Relations, or
> Triadic Relations Irreducible Over Projections:
> 
> Consider the triadic relations L^0 and L^1
> that are specified in the following set-up:
> 
> | B = {0, 1}, with the "+" signifying addition mod 2,
> | analogous to the "exclusive-or" operation in logic.
> |
> | B^k = {<x<1>, ..., x<k>> : x<j> in B for j = 1 to k}.
> 
> In what follows, the space XxYxZ is isomorphic to BxBxB = B^3.
> For lack of a good isomorphism symbol, I will often resort to
> writing things like "XxYxZ iso BxBxB" or even "XxYxZ = B^3".
> 
> | Relation L^0
> |
> | L^0 = {<x, y, z> in B^3 : x + y + z = 0}.
> |
> | L^0 has the following four triples
> | of the form <x, y, z> in B^3:
> |
> |    <0, 0, 0>
> |    <0, 1, 1>
> |    <1, 0, 1>
> |    <1, 1, 0>
> 
> | Relation L^1
> |
> | L^1 = {<x, y, z> in B^3 : x + y + z = 1}.
> |
> | L^1 has the following four triples
> | of the form <x, y, z> in B^3:
> |
> |    <0, 0, 1>
> |    <0, 1, 0>
> |    <1, 0, 0>
> |    <1, 1, 1>
> 
> Those are the relations,
> here are the projections:
> 
> Taking the dyadic projections of L^0
> we obtain the following set of data:
> 
> | (L^0)<XY> has these four triples
> | of the form <x, y> in XxY:
> | 
> |    <0, 0>
> |    <0, 1>
> |    <1, 0>
> |    <1, 1>
> 
> | (L^0)<XZ> has these four triples
> | of the form <x, z> in XxZ:
> | 
> |    <0, 0>
> |    <0, 1>
> |    <1, 1>
> |    <1, 0>
> 
> | (L^0)<YZ> has these four triples
> | of the form <y, z> in YxZ:
> | 
> |    <0, 0>
> |    <1, 1>
> |    <0, 1>
> |    <1, 0>
> 
> Taking the dyadic projections of L^1
> we obtain the following set of data:
> 
> | (L^1)<XY> has these four triples
> | of the form <x, y> in XxY:
> | 
> |    <0, 0>
> |    <0, 1>
> |    <1, 0>
> |    <1, 1>
> 
> | (L^1)<XZ> has these four triples
> | of the form <x, z> in XxZ:
> | 
> |    <0, 1>
> |    <0, 0>
> |    <1, 0>
> |    <1, 1>
> 
> | (L^1)<YZ> has these four triples
> | of the form <y, z> in YxZ:
> | 
> |    <0, 1>
> |    <1, 0>
> |    <0, 0>
> |    <1, 1>
> 
> Now, for ease of verifying the data, I have written
> these sets of pairs in the order that they fell out
> on being projected from the given triadic relations.
> But, of course, as sets, their order is irrelevant,
> and it is simply a matter of a tedious check to
> see that both L^0 and L^1 have exactly the same
> projections on each of the corresponding planes.
> 
> To summarize:
>  
> The relations L^0, L^1 sub B^3 are defined by the following equations,
> with algebraic operations taking place as in the "Galois Field" GF(2),
> that is, with 1 + 1 = 0.
> 
> 1.  The triple <x, y, z> in B^3 belongs to L^0  iff  x + y + z = 0.
>     L^0 is the set of even-parity bit-vectors, with  x + y = z.
> 
> 2.  The triple <x, y, z> in B^3 belongs to L^1  iff  x + y + z = 1.
>     L^1 is the set of odd-parity bit-vectors,  with  x + y = z + 1.
> 
> The corresponding projections of L^0 and L^1 are identical.
> In fact, all six projections, taken at the level of logical
> abstraction, constitute precisely the same dyadic relation,
> isomorphic to the whole of BxB and expressible by means of
> the universal constant proposition 1 : BxB -> B.  In sum:
> 
>    (L^0)<XY>  =  (L^1)<XY>  =  1<XY>  =  BxB  =  B^2,
>    (L^0)<XZ>  =  (L^1)<XZ>  =  1<XZ>  =  BxB  =  B^2,
>    (L^0)<YZ>  =  (L^1)<YZ>  =  1<YZ>  =  BxB  =  B^2.
> 
> Therefore, L^0 and L^1 constitute examples of
> "projectively irreducible triadic relations",
> "triadic relations irreducible on projections".
> 
> Since I think that it is always
> best to leave them wanting more,
> that really is enough for today.
> 
> Many Regards,
> 
> Jon Awbrey
> 
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