ONT 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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