Changes

MyWikiBiz, Author Your Legacy — Sunday October 26, 2025
Jump to navigationJump to search

Directory:Jon Awbrey/Papers/Cactus Language (view source)

Revision as of 15:58, 27 May 2009

245 bytes added ,  15:58, 27 May 2009
→‎Cactus Language: markup
Line 2,544: Line 2,544:  
===Cactus Language===
 
===Cactus Language===
   −
'''Note.''' Need to find the original context of this fragment.
+
{| align="center" cellpadding="6" width="90%"
 +
| align="right" | '''Note.''' Need to find the original context of this fragment.
 +
|}
    
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.
Line 2,556: Line 2,558:  
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''.
   −
{| align="center" cellspacing="6" width="90%"
+
{| align="center" cellpadding="6" width="90%"
−
| align="center" |
+
|
 
<pre>
 
<pre>
 
o---------------------------------------o
 
o---------------------------------------o
Line 2,573: Line 2,575:  
The rule of reduction for a lobe is:
 
The rule of reduction for a lobe is:
   −
{| align="center" cellspacing="6" width="90%"
+
{| align="center" cellpadding="6" width="90%"
−
| align="center" |
+
|
 
<pre>
 
<pre>
 
     x_1  x_2  ...  x_k
 
     x_1  x_2  ...  x_k
Line 2,592: Line 2,594:  
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.
    +
In Ref Log, an expression of the form <math>\texttt{((}~ e_1 ~\texttt{),(}~ e_2 ~\texttt{),(}~ \ldots ~\texttt{),(}~ e_k ~\texttt{))}</math>
 +
expresses the fact that ''exactly one of the <math>e_j\!</math> is true''.  Expressions of this form are called ''universal partition'' expressions, and
 +
they parse into a type of graph called a ''painted and rooted cactus'' (PARC):
 +
 +
{| align="center" cellpadding="6" width="90%"
 +
|
 
<pre>
 
<pre>
−
In Ref Log, an expression of the form "(( e_1 ),( e_2 ),( ... ),( e_k ))"
  −
expresses the fact that "exactly one of the e_j is true, for j = 1 to k".
  −
Expressions of this form are called "universal partition" expressions, and
  −
they parse into a type of graph called a "painted and rooted cactus" (PARC):
  −
   
     e_1  e_2  ...  e_k
 
     e_1  e_2  ...  e_k
 
     o    o          o
 
     o    o          o
Line 2,618: Line 2,621:  
|
 
|
 
| Just one of the arguments x1, x2, ..., xk  =  ()
 
| Just one of the arguments x1, x2, ..., xk  =  ()
 +
</pre>
 +
|}
   −
The interpretation of these operators, read as assertions
+
The interpretation of these operators, read as assertions about the values of their listed arguments, is as follows:
−
about the values of their listed arguments, is as follows:
     −
1.  Existential Interpretation:   "Just one of the k argument is false."
+
{| align="center" cellpadding="6" width="90%"
−
2.  Entitative  Interpretation:   "Not just one of the k arguments is true."
+
| Existential Interpretation:
−
 
+
| Just one of the k argument is false.
−
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
+
|-
 +
| Entitative  Interpretation:
 +
| Not just one of the k arguments is true.
 +
|}
    +
{| align="center" cellpadding="6" width="90%"
 +
|
 +
<pre>
 
o-------------------o-------------------o-------------------o
 
o-------------------o-------------------o-------------------o
 
|      Graph      |      String      |    Translation    |
 
|      Graph      |      String      |    Translation    |
Line 2,705: Line 2,715:  
|        @        |  ( t ,(r),(s))  |  slices r, s.    |
 
|        @        |  ( t ,(r),(s))  |  slices r, s.    |
 
o-------------------o-------------------o-------------------o
 
o-------------------o-------------------o-------------------o
 +
</pre>
 +
|}
   −
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
+
{| align="center" cellpadding="6" width="90%"
−
 
+
|
 +
<pre>
 
Table 13.  The Existential Interpretation
 
Table 13.  The Existential Interpretation
 
o-------------------o-------------------o-------------------o
 
o-------------------o-------------------o-------------------o
Line 2,802: Line 2,815:  
|                  |                  |                  |
 
|                  |                  |                  |
 
o-------------------o-------------------o-------------------o
 
o-------------------o-------------------o-------------------o
 +
</pre>
 +
|}
   −
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
+
{| align="center" cellpadding="6" width="90%"
−
 
+
|
 +
<pre>
 
Table 14.  The Entitative Interpretation
 
Table 14.  The Entitative Interpretation
 
o-------------------o-------------------o-------------------o
 
o-------------------o-------------------o-------------------o
Line 2,900: Line 2,916:  
|                  |                  |                  |
 
|                  |                  |                  |
 
o-------------------o-------------------o-------------------o
 
o-------------------o-------------------o-------------------o
 +
</pre>
 +
|}
   −
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
+
{| align="center" cellpadding="6" width="90%"
−
 
+
|
 +
<pre>
 
o-----------------o-----------------o-----------------o-----------------o
 
o-----------------o-----------------o-----------------o-----------------o
 
|      Graph      |    String      |  Entitative    |  Existential  |
 
|      Graph      |    String      |  Entitative    |  Existential  |
Line 2,988: Line 3,007:  
o-----------------o-----------------o-----------------o-----------------o
 
o-----------------o-----------------o-----------------o-----------------o
 
</pre>
 
</pre>
 +
|}
    
===Differential Logic===
 
===Differential Logic===
Jon Awbrey
12,122

edits

Retrieved from "https://mywikibiz.com/Special:MobileDiff/87199"

Navigation menu