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fom

Posts: 1,030
Registered: 12/4/12

fom - 05 - representation based semantics
Posted: Dec 7, 2012 3:59 AM

Plain Text

It is unlikely that the diagram below will post
correctly.

The Euler trail below determines a canonical
order for the 16 columns formed from vertical
and horizontal bars.

More precisely, given an interpretation for
a first connective, a set of instructions
can be given so that traversals of the
Euler trail yield a canonical ordering
always ending with the last three columns
ordered as a truth table interpreting
the complete NOR connective.

The algorithm takes advantage of the
fact that NOR matches LEQ on the
row having the same values in the
range but an opposite value in
the domain.

The canonical enumeration is

<LEQ, OR, DENY, FLIP, NIF, NTRU, AND, NIMP,

T T F F F F T F
F T T F F F F T
T F T T F F F F
F T F T T F F F

XOR, IMP, NAND, TRU, IF, FIX, LET, NOR>

F T F T T T T F
T F T T T T F F
F T T T T F F T
T T T T F F T F

where the truth table formed by the last three columns is

FIX LET | NOR
----------|-----
|
T T | F
T F | F
F F | T
F T | F

and is compatible with the assumed semantics of the initial
member of the sequence,

FIX LET | LEQ
----------|-----
|
T T | T
T F | F
F F | T
F T | F

With this relationship between LEQ as the first connective
of the canonical order and the complete NOR connective as
the last connective of the canonical order, there is no
distinguished truth table configuration, although the
extension of representation independent semantics does
depend on how the elements of the connectivity algebra
are situated relative to an enumeration and the symmetry
of an Euler trail that fixes one particular pair of
of columns have two pairs of distinct symbols as the
starting point of the enumeration.

Once truth-functionality is established, this
extension of semantics using fixed relationships
is ambiguated.

The truth table semantics for all sixteen logical
constants of the affine geometry isolated by the
connectivity algebra is decided.

===========================================================

Step 1:

Select a central vertex

Step 2:

Traverse a central edge to
the other central vertex,
obtaining either

|
-
|
-

or

-
|
_
|

Step 3:

Traverse an edge whose label
differs from the first label
at exactly three loci

Step 4:

Traverse an edge having no
uniquely differentiated locus
such that its label differs
from the first two edge
labels at the locus where
the first two edge labels
match

Step 5:

Traverse an edge having no
uniquely differentiated locus
such that its label differs
from the first two edge
labels at the locus where
the first two edge labels
match

Step 6:

Complete the Euler trail
to have palindromic symmetry
among the last fifteen labels

-
-
|---- - ----|
| - |
| |
\ /
\ /
\ /
\ /
\ /
\ /

*
___/ \___
| __/ \__ -
- -
__ - - __
____/ - | \____
___/ | \___
/ - \
* ---------------------- - ---------------------- *
| \___ | ___/ |
| \____ ____/ |
| \__ - - __/ |
| | - |
| - __ __ | |
| - \___ ___/ - |
| \ / |
| * |
| ___/ \___ |
| / \ |
| _/ \_ |
| / \ |
| / \ |

| | - -
| - | -
- | - |
- - | |

| \ / |
| \_ _/ |
| \ / |
| \___ ___/ |
| \ / |
| * |
| ___/ \___ |
| | __/ \__ - |
| | | |
| __ - - __ |
| ____/ | - \____ |
| ___/ - \___ |
| / | \ |
* ---------------------- | ---------------------- *
\____ - ____/
\____ ____/
\__ | - __/
| |
| __ __ |
- \___ ___/ |
\ /
*

/ \
/ \
/ \
/ \
/ \
/ \
| | |
| | |
|---- | ----|
|