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===Cactus Language===
 
===Cactus Language===
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Table 13 illustrates the ''existential interpretation'' of cactus graphs and cactus expressions by providing English translations for a few of the most basic and commonly occurring forms.
 
Table 13 illustrates the ''existential interpretation'' of cactus graphs and cactus expressions by providing English translations for a few of the most basic and commonly occurring forms.
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The cactus graph and the cactus expression shown here are both described as a ''spike''.
 
The cactus graph and the cactus expression shown here are both described as a ''spike''.
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<pre>
 
<pre>
 
o---------------------------------------o
 
o---------------------------------------o
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The rule of reduction for a lobe is:
 
The rule of reduction for a lobe is:
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<pre>
 
<pre>
 
     x_1  x_2  ...  x_k
 
     x_1  x_2  ...  x_k
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if and only if exactly one of the x_j is a spike.
 
if and only if exactly one of the x_j is a spike.
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In Ref Log, an expression of the form <math>\texttt{((}~ e_1 ~\texttt{),(}~ e_2 ~\texttt{),(}~ \ldots ~\texttt{),(}~ e_k ~\texttt{))}</math>
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expresses the fact that ''exactly one of the <math>e_j\!</math> is true''.  Expressions of this form are called ''universal partition'' expressions, and
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they parse into a type of graph called a ''painted and rooted cactus'' (PARC):
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<pre>
 
<pre>
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In Ref Log, an expression of the form "(( e_1 ),( e_2 ),( ... ),( e_k ))"
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expresses the fact that "exactly one of the e_j is true, for j = 1 to k".
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Expressions of this form are called "universal partition" expressions, and
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they parse into a type of graph called a "painted and rooted cactus" (PARC):
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     e_1  e_2  ...  e_k
 
     e_1  e_2  ...  e_k
 
     o    o          o
 
     o    o          o
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| Just one of the arguments x1, x2, ..., xk  =  ()
 
| Just one of the arguments x1, x2, ..., xk  =  ()
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</pre>
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The interpretation of these operators, read as assertions
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The interpretation of these operators, read as assertions about the values of their listed arguments, is as follows:
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about the values of their listed arguments, is as follows:
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1.  Existential Interpretation:   "Just one of the k argument is false."
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2.  Entitative  Interpretation:   "Not just one of the k arguments is true."
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| Existential Interpretation:
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| Just one of the k argument is false.
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o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
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|-
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| Entitative  Interpretation:
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| Not just one of the k arguments is true.
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<pre>
 
o-------------------o-------------------o-------------------o
 
o-------------------o-------------------o-------------------o
 
|      Graph      |      String      |    Translation    |
 
|      Graph      |      String      |    Translation    |
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|        @        |  ( t ,(r),(s))  |  slices r, s.    |
 
|        @        |  ( t ,(r),(s))  |  slices r, s.    |
 
o-------------------o-------------------o-------------------o
 
o-------------------o-------------------o-------------------o
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</pre>
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o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
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<pre>
 
Table 13.  The Existential Interpretation
 
Table 13.  The Existential Interpretation
 
o-------------------o-------------------o-------------------o
 
o-------------------o-------------------o-------------------o
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|                  |                  |                  |
 
|                  |                  |                  |
 
o-------------------o-------------------o-------------------o
 
o-------------------o-------------------o-------------------o
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</pre>
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o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
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<pre>
 
Table 14.  The Entitative Interpretation
 
Table 14.  The Entitative Interpretation
 
o-------------------o-------------------o-------------------o
 
o-------------------o-------------------o-------------------o
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|                  |                  |                  |
 
|                  |                  |                  |
 
o-------------------o-------------------o-------------------o
 
o-------------------o-------------------o-------------------o
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</pre>
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o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
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<pre>
 
o-----------------o-----------------o-----------------o-----------------o
 
o-----------------o-----------------o-----------------o-----------------o
 
|      Graph      |    String      |  Entitative    |  Existential  |
 
|      Graph      |    String      |  Entitative    |  Existential  |
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o-----------------o-----------------o-----------------o-----------------o
 
o-----------------o-----------------o-----------------o-----------------o
 
</pre>
 
</pre>
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===Differential Logic===
 
===Differential Logic===
Jon Awbrey
12,089

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