Re: Zeroth Order Ontology

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ZOO.  Note 11

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Still exploring the space L_2 of rooted trees in a somewhat aimless fashion,
let's look at how we might apply the arithmetic initials I_1 and I_2 to the
task of evaluating of yet another example.

o-----------------------------------------------------------o
| Example E_2                                               |
o-----------------------------------------------------------o
|                                                           |
|     o       o   o   o                                     |
|      \     /     \ /                                      |
|     o o   o o   o o o                                     |
|      \|   |/     \|/                                      |
|       o   o       o                                       |
|        \ /        |                                       |
|         o---------o                                       |
|         |                                                 |
|         |                                                 |
|         @                                                 |
|                                                           |
o=============================< I_2.  Refold (()) >=========o
|                                                           |
|             o   o   o                                     |
|            /     \ /                                      |
|     o     o o   o o o                                     |
|      \    |/     \|/                                      |
|       o   o       o                                       |
|        \ /        |                                       |
|         o---------o                                       |
|         |                                                 |
|         |                                                 |
|         @                                                 |
|                                                           |
o=============================< I_2.  Refold (()) >=========o
|                                                           |
|                 o   o                                     |
|                  \ /                                      |
|     o       o   o o o                                     |
|      \     /     \|/                                      |
|       o   o       o                                       |
|        \ /        |                                       |
|         o---------o                                       |
|         |                                                 |
|         |                                                 |
|         @                                                 |
|                                                           |
o=============================< I_1.  Condense ()() >=======o
|                                                           |
|                   o                                       |
|                   |                                       |
|     o       o   o o o                                     |
|      \     /     \|/                                      |
|       o   o       o                                       |
|        \ /        |                                       |
|         o---------o                                       |
|         |                                                 |
|         |                                                 |
|         @                                                 |
|                                                           |
o=============================< I_2.  Refold (()) >=========o
|                                                           |
|     o       o   o   o                                     |
|      \     /     \ /                                      |
|       o   o       o                                       |
|        \ /        |                                       |
|         o---------o                                       |
|         |                                                 |
|         |                                                 |
|         @                                                 |
|                                                           |
o=============================< I_2.  Refold (()) >=========o
|                                                           |
|             o   o   o                                     |
|            /     \ /                                      |
|           o       o                                       |
|          /        |                                       |
|         o---------o                                       |
|         |                                                 |
|         |                                                 |
|         @                                                 |
|                                                           |
o=============================< I_2.  Refold (()) >=========o
|                                                           |
|                 o   o                                     |
|                  \ /                                      |
|                   o                                       |
|                   |                                       |
|         o---------o                                       |
|         |                                                 |
|         |                                                 |
|         @                                                 |
|                                                           |
o=============================< I_1.  Condense ()() >=======o
|                                                           |
|                   o                                       |
|                   |                                       |
|                   o                                       |
|                   |                                       |
|         o---------o                                       |
|         |                                                 |
|         |                                                 |
|         @                                                 |
|                                                           |
o=============================< I_2.  Refold (()) >=========o
|                                                           |
|         o---------o                                       |
|         |                                                 |
|         |                                                 |
|         @                                                 |
|                                                           |
o=============================< I_2.  Refold (()) >=========o
|                                                           |
|         @                                                 |
|                                                           |
o=============================< QEI >=======================o

In this way, one discovers the formal equation recorded below:

o-----------------------------------------------------------o
| Equation E_2                                              |
o-----------------------------------------------------------o
|                                                           |
|     o       o   o   o                                     |
|      \     /     \ /                                      |
|     o o   o o   o o o                                     |
|      \|   |/     \|/                                      |
|       o   o       o                                       |
|        \ /        |                                       |
|         o---------o                                       |
|         |                                                 |
|         |                                                 |
|         @                             =         @         |
|                                                           |
o-----------------------------------------------------------o
| ( (()(()))((())()) ( (()(()())()) ))  =                   |
o-----------------------------------------------------------o

Using the square bracket notation for a "formal equivalence class" (FEC),
one says that "( (()(()))((())()) ( (()(()())()) ))" is in the FEC [!e!]
of the empty string or in the FEC [" "] of the blank character.  We have
the result that  ["( (()(()))((())()) ( (()(()())()) ))"] = [" "] = [""].

Jon Awbrey

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