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

Revision as of 03:57, 28 May 2009

1,255 bytes added ,  03:57, 28 May 2009
→‎Notes Found in a Cactus Patch: markup + rewrite
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<pre>
 
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o---------------------------------------o
 
o---------------------------------------o
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</pre>
 
</pre>
 
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</pre>
 
</pre>
 
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<pre>
 
o-------------------o-------------------o-------------------o
 
o-------------------o-------------------o-------------------o
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<pre>
 
Table 13.  The Existential Interpretation
 
Table 13.  The Existential Interpretation
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<pre>
 
<pre>
 
Table 14.  The Entitative Interpretation
 
Table 14.  The Entitative Interpretation
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<pre>
 
o-----------------o-----------------o-----------------o-----------------o
 
o-----------------o-----------------o-----------------o-----------------o
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Start with a proposition of the form <math>x ~\operatorname{and}~ y,</math> which is graphed as two labels attached to a root node:
 
Start with a proposition of the form <math>x ~\operatorname{and}~ y,</math> which is graphed as two labels attached to a root node:
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<pre>
 
o---------------------------------------o
 
o---------------------------------------o
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In this style of graphical representation, the value <math>\operatorname{true}</math> looks like a blank label and the value <math>\operatorname{false}</math> looks like an edge.
 
In this style of graphical representation, the value <math>\operatorname{true}</math> looks like a blank label and the value <math>\operatorname{false}</math> looks like an edge.
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<pre>
 
o---------------------------------------o
 
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<pre>
 
<pre>
 
o---------------------------------------o
 
o---------------------------------------o
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Don't think about it &mdash; just compute:
 
Don't think about it &mdash; just compute:
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<pre>
 
o---------------------------------------o
 
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To make future graphs easier to draw in ASCII, I will use devices like '''<code>@=@=@</code>''' and '''<code>o=o=o</code>''' to identify several nodes into one, as in this next redrawing:
 
To make future graphs easier to draw in ASCII, I will use devices like '''<code>@=@=@</code>''' and '''<code>o=o=o</code>''' to identify several nodes into one, as in this next redrawing:
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o---------------------------------------o
 
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However you draw it, these expressions follow because the expression <math>x + dx,\!</math> where the plus sign indicates addition in <math>\mathbb{B},</math> that is, addition modulo 2, and thus corresponds to the exclusive disjunction operation in logic, parses to a graph of the following form:
 
However you draw it, these expressions follow because the expression <math>x + dx,\!</math> where the plus sign indicates addition in <math>\mathbb{B},</math> that is, addition modulo 2, and thus corresponds to the exclusive disjunction operation in logic, parses to a graph of the following form:
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<pre>
 
o---------------------------------------o
 
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Next question:  What is the difference between the value of the proposition <math>xy\!</math> "over there" and the value of the proposition <math>xy\!</math> where you are, all expressed as general formula, of course?  Here 'tis:
 
Next question:  What is the difference between the value of the proposition <math>xy\!</math> "over there" and the value of the proposition <math>xy\!</math> where you are, all expressed as general formula, of course?  Here 'tis:
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o---------------------------------------o
 
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Last question, for now:  What is the value of this expression from your current standpoint, that is, evaluated at the point where <math>xy\!</math> is true?  Well, substituting <math>1\!</math> for <math>x\!</math> and <math>1\!</math> for <math>y\!</math> in the graph amounts to the same thing as erasing those labels:
 
Last question, for now:  What is the value of this expression from your current standpoint, that is, evaluated at the point where <math>xy\!</math> is true?  Well, substituting <math>1\!</math> for <math>x\!</math> and <math>1\!</math> for <math>y\!</math> in the graph amounts to the same thing as erasing those labels:
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o---------------------------------------o
 
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And this is equivalent to the following graph:
 
And this is equivalent to the following graph:
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<pre>
 
<pre>
 
o---------------------------------------o
 
o---------------------------------------o
Jon Awbrey
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