arrow-structure syntax and semantics

RE01 Rice Brian T. EM2
Thu, 10 Dec 1998 14:47:30 +0800

[the benefits of nodes being built from arrows]
here's something else.  let's say we've made a graph describing the
relationships between several objects.  this graph consists of arrows and
nodes, which are themselves unlinked to any particular structure.  now,
let's say that we want to 'draw a metaphor' between the relationship denoted
by this graph and some other relationship which is among arrows that _do_
belong to structures.  this second relationship type is the one, of course,
about which we would be primarily concerned, since it is the mechanism by
which we enrich our existing arrow structures.  the metaphor could basically
consist of a mapping from the nodes in the first diagram to the arrows where
the 'second' diagram would be placed.  i'm not sure where to take this idea,
since it is embryonic.  i could say that all 'shapes' would be attached to
nodes in this system, and that node instantiation would provide
distinguishability for diagrams' nodes.  on the other hand, this method
could simply serve the purpose of deleting nodes as concepts in the system,
by simply drawing metaphors from shapes which already point to arrows which
'participate' in arrow structures to obviate the redundancy of the
node-containing structure.  on the other hand, this second idea could exist
"at a lower level" with a "higher level" displaying a "parent shape"
connected to nodes whence the "daughter shapes" would derive by metaphor.
it seems more intuitive this way while 'saving space' at the 'lower level',
however it also strays from the Self paradigm of prototyping objects, which
is quite elegant, though not intuitive in the case of arrow structures.

[the benefits of infinitary encodings]
i remember reading an abstract computer science book recently on the
relationship between language theory and computation theory, and it went
into detail about the construction of finite-state machines for languages.
I also remember a statement about some Turing machines being inconstructible
for certain given first-order languages, as well as the size of certain
state-machine constructions becoming infinitary (even uncountable) for
certain lucrative languages.  after all, does the Church-Turing thesis prove
the undecideability of standard logic by _finite_-state machines?
the point is that i _do_ have a basis for this "infinitary" obsession,
strange (and esoteric) though it seems.  state-machines, for example, are
_easily_ described by arrow diagrams, and i assume that it is not too far
off that we will have actually (ok, virtually) implemented infinitary arrow