SUO: RE: Reductions Among Relations
Dear Jon,
You took as an example:
| x is between a and b
|
| if and only if
|
| a < x and x < b
which you analysed down to some more general elements, but without
eliminating the triadicity. You can of course eliminate triadic
relations (different from eliminating triadicity) if you look at it
this way:
There is a number space n
the upper bound of n is b
the lower bound of n is a
x is in n.
Note I have 3 dyadic relations to replace 1 triadic relation.
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: 24 March 2001 05:00
> To: Complexity Group; SemioCom; Stand Up Ontology
> Cc: Josiah Lee Auspitz; Paul R Chernoff; Lisa Cox; Mark
> Pierre Richards;
> John F Sowa; West, Matthew MR SSI-GREA-UK
> Subject: Re: Reductions Among Relations
>
>
> ¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤
>
> RAR^N To Go Troops:
>
> Starting from the simplest notions of Rise and Fall,
> I may easily have chosen to leave it as an exercise
> for the reader to discover suitable generalizations,
> say from Rise<k> and Fall<k> for k of order 2 and 3,
> to the slightly more general case in which k is any
> natural number, that is, finite, integral, positive.
> But that is far too easy a calisthenic, and no kind
> of a work-out to offer our band of fearless readers,
> and so the writer picks up the gage that he himself
> throws down, and for his health runs the easy track!
>
> Let B = {0, 1}.
>
> Let the relation Rise<k> c B^k
> be defined in the following way:
>
> | Rise<k><x<1>, ..., x<k>>
> |
> | iff
> |
> | Rise<2><x<1>, x<2>> and Rise<k-1><x<2>, ..., x<k>>
>
> Let the relation Fall<k> c B^k
> be defined in the following way:
>
> | Fall<k><x<1>, ..., x<k>>
> |
> | iff
> |
> | Fall<2><x<1>, x<2>> and Fall<k-1><x<2>, ..., x<k>>
>
> But let me now leave off, for the time being,
> from the temptation to go any further in the
> direction of increasing k than I ever really
> intended to, on beyond 2 or 3 or thereabouts,
> for that is not the aim of the present study.
>
> ¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤
>
> Note 8
>
> ¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤
>
> In this note I revisit the "Between" relation on reals,
> and then I rework it as a discrete and finite analogue
> of its transcendantal self, as a Between relation on B.
> Ultimately, I want to use this construction as working
> material to illustrate a method of defining relational
> compositions in terms of projections. So let us begin.
>
> Last time I defined Rise and Fall relations on B^k.
> Working polymorphously, as some people like to say,
> let us go ahead and define the analogous relations
> over the real domain R, not even bothering to make
> new names, but merely expecting the reader to find
> the aptest sense for a given context of discussion.
>
> Let R be the set of real numbers.
>
> Let the relation named "Rise<2>"
> such that Rise<2> c R^2 = R x R,
> be defined in the following way:
>
> | Rise<2><x, y>
> |
> | iff
> |
> | [x = y] or [x < y]
>
> Let the relation named "Fall<2>"
> such that Fall<2> c R^2 = R x R,
> be defined in the following way:
>
> | Fall<2><x, y>
> |
> | iff
> |
> | [x > y] or [x = y]
>
> There are clearly a number of redundancies
> between the definitions of these relations,
> but I prefer the symmetry of this approach.
>
> The next pair of definitions will be otiose, too,
> if viewed in the light of the comprehensive case
> that follows after, but let us go gently for now.
>
> Let the relation named "Rise<3>"
> such that Rise<3> c R^3 = RxRxR,
> be defined in the following way:
>
> | Rise<3><x, y, z>
> |
> | iff
> |
> | Rise<2><x, y> and Rise<2><y, z>
>
> Let the relation named "Fall<3>"
> such that Fall<3> c R^3 = RxRxR,
> be defined in the following way:
>
> | Fall<3><x, y, z>
> |
> | iff
> |
> | Fall<2><x, y> and Fall<2><y, z>
>
> Then Rise<3> and Fall<3> are "degenerate 3-adic relations"
> insofar as each of them bears expression as a conjunction
> whose conjuncts are expressions of 2-adic relations alone.
>
> Just in order to complete the development
> of this thought, let us then finish it so:
>
> Let the relation Rise<k> c R^k
> be defined in the following way:
>
> | Rise<k><x<1>, ..., x<k>>
> |
> | iff
> |
> | Rise<2><x<1>, x<2>> and Rise<k-1><x<2>, ..., x<k>>
>
> Let the relation Fall<k> c R^k
> be defined in the following way:
>
> | Fall<k><x<1>, ..., x<k>>
> |
> | iff
> |
> | Fall<2><x<1>, x<2>> and Fall<k-1><x<2>, ..., x<k>>
>
> If there was a point to writing out this last step,
> I think that it may well have been how easy it was
> not to write, not literally to "write" at all, but
> simply to "cut and paste" the definitions from the
> boolean case, and then but to change the parameter
> B into the parameter R at a mere one place in each.
>
> Let us then push on, in a retrograde way, returning to the orbit
> of those very first relations that got us into the midst of this
> quandary in the first place, to wit, the relations in medias res,
> the relations of betwixt and between and all of their sundry kin.
> But let us this time place the paltry special relations on which
> we fixed the first time around back within the setting of a much
> broader and a much more systematically examined context, that is,
> an extended family of related relations or variations on a theme.
>
> I hope that you will be able to recall that cousin of
> the Between relation that we took up here once before,
> and that you will be able to recognize its characters,
> even if I now disguise it under a new name and partly
> dissemble them under a new manner of parameterization.
> Where you might take the name "IO" to mean "in order",
> it is the relation defined on three real numbers thus:
>
> Let the relation named "IO<213>"
> such that IO<213> c R^3 = RxRxR,
> be defined in the following way:
>
> | IO<213><x, a, b> iff [a < x < b],
> |
> | equivalently,
> |
> | IO<213><x, a, b> iff [a < x] and [x < b].
>
> Corresponding to the 3-adic relation IO<213> c R^3 = RxRxR,
> there is a "proposition", a function io<213> : R^3 -> B,
> that I will describe, until a better name comes along,
> as the "relation map" that is "dual to" the relation.
> It is also known as the "indicator" of that relation.
>
> Consider the boolean analogue or the logical variant of IO,
> with real domains : R now replaced by boolean domains : B.
>
> The boolean analogue of the ordering "<" is the implication "=>",
> so the logical variant of the relation IO<213> is given this way:
>
> Let the relation named "IO<213>"
> such that IO<213> c B^3 = BxBxB,
> be defined in the following way:
>
> | IO<213><x, a, b>
> |
> | iff
> |
> | [a => x] and [x => b]
>
> When it does not risk any confusion,
> one can express this also like this:
>
> | IO<213><x, a, b>
> |
> | iff
> |
> | a => x => b
>
> Corresponding to the 3-adic relation IO<213> c B^3 = BxBxB,
> there is a "proposition", a function io<213> : B^3 -> B,
> that I will describe, until a better name comes along,
> as the "relation map" that is "dual to" the relation.
> It is also known as the "indicator" of that relation.
>
> At this point I want to try and get away with a bit
> of additional flexibility in the syntax that I use,
> reusing some of the same names for what are distinct
> but closely related types of mathematical objects.
>
> In particular, I would like to have the license
> to speak a bit more loosely about these objects,
> to ignore the distinction between "relations" of
> the form Q c X<1> x ... x X<k> and "relation maps"
> of the form q : X<1> x ... x X<k> -> B, and even
> on sundry informal occasions to use the very same
> names for them -- The Horror! -- hoping to let the
> context determine the appropriate type of object,
> except where it may be necessary to maintain this
> distinction in order to avoid risking confusion.
>
> In order to keep track of all of the players -- not to
> mention all of the refs! --
> it may help to re-introduce a diagram that I have used many
> times before, as a
> kind of a play-book or programme, to sort out the burgeoning
> teams of objects
> and the cryptic arrays of signs that we need to follow
> throughout the course
> of this rather extended into overtime game:
>
> --------------------------------///---------------------------
> -----------
> Objective Framework (OF) Interpretive Framework (IF)
> --------------------------------///---------------------------
> -----------
> Formal or Mathematical Objects Formal or Mathematical
> Signs & Texts
> --------------------------------///---------------------------
> -----------
>
> Propositions Expressions
> (Logical) (Propositional)
> * *
> | |
> | |
> * *
> / \ / \
> / \ / \
> * * * *
> Sets Maps Set Names Map Names
> (Geometric) (Functional) (Geometric) (Functional)
>
> --------------------------------///---------------------------
> -----------
> B^k B^k -> B "IO<213>" "io<213>"
> R^k R^k -> B "IO<213>" "io<213>"
> X^k X^k -> B "Q" "q"
> --------------------------------///---------------------------
> -----------
>
> Matthew West wrote to Adam Pease:
>
> | I think there are two key issues here.
> |
> | 1. When you use a larger relation as a primitive,
> | you will have more types of relation. Whereas if
> | you analyse them down to more fundamental relations,
> | you may have more relations, but there will be fewer
> | types of relation. This is usually an advantage.
> | Indeed we have found that there is reuse at the
> | individual relation level too, which means an
> | effective reduction in repetition (with the
> | opportunity for inconsistency that gives).
> |
> | 2. When you analyse to more fundamental concepts
> | you give yourself the opportunity to be able to make
> | inferences that arise from teh more fundamental level.
> |
> | In my view it is OK to have more complex relations provided
> | they are rigourously defined in terms of underlying concepts.
>
> ¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤
>
> What Matthew says here makes a lot of sense to me.
>
> By way of illustrating the application of these heuristic principles
> in the case of Adam Pease's Between relation, we might consider the
> following analysis:
>
> Given a =/= b, I will pull the usual trick and say,
> "without loss of generality" (WOLOG), assume a < b.
>
> Then:
>
> | x is between a and b
> |
> | if and only if
> |
> | a < x and x < b
>
> Now, if we are operating under what are likely to be
> the default assumptions, the intended context, or the
> standard interpretation of these statements, then the
> sentence "x < y" invokes a dyadic relation LT c R x R,
> where R = Reals, and this in turn corresponds to the
> proposition lt : RxR -> B, where B = {0, 1} as usual.
>
> This allows us to articulate the sentence "x is between a and b"
> as invoking a certain 3-adic relation "1st between 2nd and 3rd",
> or let us call it "IO<213>" for short, such that IO<213> c R^3,
> and this in turn corresponds to the proposition io<213> : R^3 -> B.
>
> Last but not least, recall that conjunction is just a function
> that maps two propositions into a proposition, in other words,
> a function of the form 'and' : (X -> B) x (X -> B) -> (X -> B).
>
> Now here's a good question: What is X?
> To be specific, is it R^2 or is it R^3?
>
> We appear to have a little bit of a problem
> matching up the divers pieces of the puzzle.
>
> What this requires is a little gimmick that I call a "tacit
> extension".
> In this case, we need a way of taking the function lt : R^2 -> B and
> using it to construct two different functions of type : R^3 -> B.
>
> Thus we need a couple of functions on functions, both having
> this form:
>
> te : (R<1> x R<2> -> B) -> (R<1'> x R<2'> x R<3'> -> B),
>
> Here we must now distinguish particular copies of R
> in the cartesian products RxR = R^2 and RxRxR = R^3.
>
> | Tacit extension 21 is defined by: (te<21>(f))(x, a, b) = f(a, x).
> |
> | Tacit extension 13 is defined by: (te<13>(f))(x, a, b) = f(x, b).
>
> Applying these tacit extensions to lt we get:
>
> | (te<21>(lt))(x, a, b) = lt<21>(x, a, b) = lt(a, x)
> |
> | (te<13>(lt))(x, a, b) = lt<13>(x, a, b) = lt(x, b)
>
> In effect, the tacit extension of the "less than" map,
> given three arguments, just compares the proper pair
> of them, with a "don't care" condition on the other.
>
> Putting all of these pieces together, we have this picture:
>
> lt(a, x) . . lt(x, b)
> | |
> | |
> te<21> o o te<13>
> | |
> | |
> lt<21>(x, a, b) . . lt<13>(x, a, b)
> \ /
> \ /
> \ /
> [and]
> |
> |
> |
> .
> io<213>(x, a, b)
>
> This gives a functional decomposition of the proposition io<213>
> making use of two different applications of the proposition lt,
> a couple of tacit extensions te<21> and te<13>, and the reusable
> function 'and' that connects the tacit extensions of lt together.
>
> What is the use of all of this? -- especially in view of the fact
> that we did not reduce the maximum arity of the relations involved?
>
> In sum, we started with a 3-adic relation map io<213> : R^3 -> B,
> getting a pair of 2-adic relation maps of the form lt : R^2 -> B,
> plus two tacit extensions of the form te : (R^2 -> B) ->
> (R^3 -> B),
> and a 3-adic relation of the shape 'and' : (R^3 -> B)^2 ->
> (R^3 -> B).
>
> Well, the advantage, over the long haul, comes precisely from the fact
> that the seemingly more complex pieces that we reached in the analysis
> are pieces that are highly reusable in situation after situation, and
> so it is worth the overhead, eventually, that it takes to understand
> these recurring components as thoroughly as possible.
>
> That brings an end to these Legends of the Fall,
> And so I wish a gentle night to you one and all.
>
> Jon Awbrey
>
> ¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤
>