Re: Zeroth Order Ontology

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ZOO.  Note 8

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Generators and Relations.

Last time we presented a quotient structure L/[//] = <[/|//]> in terms
of generators and relations, where the generators proliferate elements
of the formal language or "space" in question and where the relations
incorporate those elements under the severed headings of a partition
or "formal equivalence relation" (FER), in effect, collapsing some
of the distinctions that the generators created between elements.

Let's now pursue this paradigm into the forest of rooted trees, the
qualities of "finite" and "free" being the defaults in this context.

Let L be the formal language of rooted trees, considered in parallel with the
corresponding traversal strings, the formal language of paired-up parentheses.
L happens to be, in a very precise mathematical sense, the "archetype" of all
context-free languages, known in algebra, combinatorics, and formal language
theory as the "Dyck language on two symbols":

Cf.  http://www.combinatorics.org/Volume_3/Html/v3i1f1.html

Let L' = L/{I_1, I_2} be the quotient structure on L that
results from taking the arithmetic initials I_1 and I_2
as the "coincidental" relations.

o-----------------------------------------------------------o
|                                                           |
|                 o   o                 o                   |
|                  \ /                  |                   |
|                   @         =         @                   |
|                                                           |
o-----------------------------------------------------------o
|                                                           |
|                ( ) ( )      =        ( )                  |
|                                                           |
o-----------------------------------------------------------o
| Axiom I_1.   Distract <---- | ----> Condense              |
o-----------------------------------------------------------o

o-----------------------------------------------------------o
|                                                           |
|                   o                                       |
|                   |                                       |
|                   o                                       |
|                   |                                       |
|                   @         =         @                   |
|                                                           |
o-----------------------------------------------------------o
|                                                           |
|                 (( ))       =                             |
|                                                           |
o-----------------------------------------------------------o
| Axiom I_2.     Unfold <---- | ----> Refold                |
o-----------------------------------------------------------o

Exercise for the reader.  Try to see why every string or tree in L
is equivalent to exactly one of the following two strings or trees:

The empty expression "" that traverses a rooted node:

          @

The spike expression "()" that traverses a rooted edge:

          o
          |
          @

I'm pretty sure that's true, anyway.

Jon Awbrey

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