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Directory:Jon Awbrey/Papers/Propositional Equation Reasoning Systems (view source)

Revision as of 13:30, 14 August 2009

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→‎Variations on a theme of transitivity: format table
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To study the differences between these two versions of transitivity within what is locally a familiar context, let's view the propositional forms involved as if they were elementary cellular automaton rules, resulting in the following Table.
 
To study the differences between these two versions of transitivity within what is locally a familiar context, let's view the propositional forms involved as if they were elementary cellular automaton rules, resulting in the following Table.
   −
{| align="center" cellpadding="10" style="text-align:center; width:90%"
+
<br>
 +
 
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{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
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|+ <math>\text{Table 43.}~~\text{Composite and Compiled Order Relations}</math>
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|- style="background:#f0f0ff"
 
|
 
|
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<pre>
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<p><math>\mathcal{L}_1</math></p>
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Table 43.  Composite and Compiled Order Relations
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<p><math>\text{Decimal}</math></p>
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o---------o------------o-----------------o----------------o-------------o
+
|
−
| L_1    | L_2        | L_3            | L_4            | L_5        |
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<p><math>\mathcal{L}_2</math></p>
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|        |            |                |                |            |
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<p><math>\text{Binary}</math></p>
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| Decimal | Binary     | Vector         | Cactus         | Order       |
+
|
−
o---------o------------o-----------------o----------------o-------------o
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<p><math>\mathcal{L}_3</math></p>
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|         |         p : 1 1 1 1 0 0 0 0 |               |             |
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<p><math>\text{Vector}</math></p>
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|         |         q : 1 1 0 0 1 1 0 0 |               |             |
+
|
−
|         |         r : 1 0 1 0 1 0 1 0 |               |             |
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<p><math>\mathcal{L}_4</math></p>
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o---------o------------o-----------------o----------------o-------------o
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<p><math>\text{Cactus}</math></p>
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|         |           |                |                |            |
+
|
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| q_207  | q_11001111 | 1 1 0 0 1 1 1 1 | (p  (q))      | p =< q      |
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<p><math>\mathcal{L}_5</math></p>
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|        |            |                |                |            |
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<p><math>\text{Order}</math></p>
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| q_187  | q_10111011 | 1 0 1 1 1 0 1 1 |      (q  (r)) | q =< r      |
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|- style="background:#f0f0ff"
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|        |            |                |                |            |
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| &nbsp;
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| q_175  | q_10101111 | 1 0 1 0 1 1 1 1 | (p       (r)) | p =< r     |
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| align="right" | <math>p\colon\!</math>
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|        |            |                |                |            |
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| <math>1~1~1~1~0~0~0~0</math>
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| q_139  | q_10001011 | 1 0 0 0 1 0 1 1 | (p (q))(q (r)) | p =< q =< r |
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| &nbsp;
−
|        |            |                |                |            |
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| &nbsp;
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o---------o------------o-----------------o----------------o-------------o
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|- style="background:#f0f0ff"
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</pre>
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| &nbsp;
−
| (43)
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| align="right" | <math>q\colon\!</math>
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| <math>1~1~0~0~1~1~0~0</math>
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| &nbsp;
 +
| &nbsp;
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|- style="background:#f0f0ff"
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| &nbsp;
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| align="right" | <math>r\colon\!</math>
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| <math>1~0~1~0~1~0~1~0</math>
 +
| &nbsp;
 +
| &nbsp;
 +
|-
 +
|
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<math>\begin{matrix}
 +
q_{207}
 +
\\[4pt]
 +
q_{187}
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\\[4pt]
 +
q_{175}
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\\[4pt]
 +
q_{139}
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\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
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q_{11001111}
 +
\\[4pt]
 +
q_{10111011}
 +
\\[4pt]
 +
q_{10101111}
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\\[4pt]
 +
q_{10001011}
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\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
1~1~0~0~1~1~1~1
 +
\\[4pt]
 +
1~0~1~1~1~0~1~1
 +
\\[4pt]
 +
1~0~1~0~1~1~1~1
 +
\\[4pt]
 +
1~0~0~0~1~0~1~1
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\end{matrix}</math>
 +
|
 +
<math>\begin{matrix}
 +
\texttt{(} p \texttt{~(} q \texttt{))}
 +
\\[4pt]
 +
\texttt{(} q \texttt{~(} r \texttt{))}
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\\[4pt]
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\texttt{(} p \texttt{~(} r \texttt{))}
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\\[4pt]
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\texttt{(} p \texttt{~(} q \texttt{))~(} q \texttt{~(} r \texttt{))}
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\end{matrix}</math>
 +
|
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<math>\begin{matrix}
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p \le q
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\\[4pt]
 +
q \le r
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\\[4pt]
 +
p \le r
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\\[4pt]
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p \le q \le r
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\end{matrix}</math>
 
|}
 
|}
 +
 +
<br>
    
Taking up another angle of incidence by way of extra perspective, let us now reflect on the venn diagrams of our four propositions.
 
Taking up another angle of incidence by way of extra perspective, let us now reflect on the venn diagrams of our four propositions.
Jon Awbrey
12,080

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