new operators... finally

RE01 Rice Brian T. EM2
Sat, 26 Dec 1998 17:31:51 +0300

i have more to say, of course.  don't take all this too heavily: it's
actually pretty easy stuff, if you just replace my symbols with common-sense
statements.  but then, that's logic, of course.

here's a way to describe what an arrow is within the language itself:
	to represent an arrow by three arrows.  one would be a node.
another would represent the 'tail', leading from the object to which the
original arrow's tail pointed, to the new node.  the remaining arrow would
lead from the new node to the arrow pointed to by the original arrow's head.

this 'abstraction' is probably the simplest first-order abstraction we could
develop (or maybe there is a simpler one).  it basically factors out the
idea that our atoms point to 2 other atoms in an ordered way.  i honestly
can't imagine what to do with it, since it doesn't fit under any system
we've thought about so far, but maybe one of you can see something in it.

here's another 2 new relations.  it's just like the incidence relation R
from the previous post, but this one works in a different ontological
dimension.  these relations are also independent of the nature of the
configuration, and so might be quite useful in reasoning about our
arrow-panes.  we want to state whether a certain arrow's head points to
another certain arrow.  we'll call the relation H, so that 'H(x,y)' states
this fact.  the predicate T should do the same useful thing for tails.  we
could even generalize the notation to an incidence relation P (read "points
to") indexed by the vertex of the arrow, so that 'H(x,y) = P<1>(x,y)' and
'T(x,y) = P<0>(x,y)'. this would allow the easy introduction of
multi-dimensional arrows, but will probably not be used, since it would
change the previously used co-incidence relation into a set of relations
indexed by a tensor (matrix), which would be unduly complicated for now.
this new relation, P, adequately encapsulates the abstraction that i just

(btw, i will refer to the incidence relation of the previous post as a
co-incidence relation to distinguish it from this one from now on).

with this new relation P, we now have a new metaphor for encoding
multi-graphs.  that is, we simply list the 'nodes' of the multi-graph as
well as the arrows between them, and specify the P<0> and P<1> mappings like
a permutation group.

well, this should round out our basic set of first-order relations for the
arrow language.  we've already got something to work with for a first-order
postulation of a new arrow, the relation 'R<>()' (that 0-ary co-incidence
predicate), although it may not be what we need.  tonight, i'm going to
prepare for tomorrow's excursion into how arrow structures can encapsulate
transformations on information systems.  i'm sure some of you have some
ideas about how this will turn out.  i'm going to contact some of the
professors who wrote significant papers in this area in order to discuss
ideas.  perhaps some will listen.  in any case, i would like to know their
stance on my using some of their ideas.