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It is easy to see the relationship between the parenthetical expressions of Peirce's logical graphs, that somewhat clippedly picture the ordered containments of their formal contents, and the associated [[dual graph]]s, that constitute the species of [[rooted tree]]s here to be described.
It is easy to see the relationship between the parenthetical expressions of Peirce's logical graphs, that somewhat clippedly picture the ordered containments of their formal contents, and the associated [[dual graph]]s, that constitute the species of [[rooted tree]]s here to be described.
In the case of our last example, a moment's contemplation of the following picture will lead us to see that we can get the corresponding parenthesis string by starting at the root of the tree, climbing up the left side of the tree until we reach the top, then climbing back down the right side of the tree until we return to the root, all the while reading off the symbols, in this particular case either "(" or ")", that we happen to encounter in our travels.
In the case of our last example, a moment's contemplation of the following picture will lead us to see that we can get the corresponding parenthesis string by starting at the root of the tree, climbing up the left side of the tree until we reach the top, then climbing back down the right side of the tree until we return to the root, all the while reading off the symbols, in this case either "<math>(\!</math>" or "<math>)\!</math>", that we happen to encounter in our travels.
<p>[[Image:Logical_Graph_Figure_6.jpg|center]]</p>
<p>[[Image:Logical_Graph_Figure_6.jpg|center]]</p>
This ritual is called ''[[tree traversal|traversing]]'' the tree, and the string read off is often called the ''[[traversal string]]'' of the tree. The reverse ritual, that passes from the string to the tree, is called ''[[parsing]]'' the string, and the tree constructed is often called the ''[[parse tree|parse graph]]'' of the string. The speakers thereof tend to be a bit loose in this language, often using ''[[parse string]]'' to mean the string that gets parsed into the associated graph.
This ritual is called ''[[tree traversal|traversing]]'' the tree, and the string read off is often called the ''[[traversal string]]'' of the tree. The reverse ritual, that passes from the string to the tree, is called ''[[parsing]]'' the string, and the tree constructed is often called the ''[[parse tree|parse graph]]'' of the string. The speakers thereof tend to be a bit loose in this language, often using ''[[parse string]]'' to mean the string that gets parsed into the associated graph.
We have now treated in some detail various forms of the axiom that is formulated in string form as "(( )) = ". For the sake of comparison, let's record the planar and dual forms of the axiom that is formulated in string form as "( )( ) = ( )".
We have now treated in some detail various forms of the axiom or initial equation that is formulated in string form as " <math>((~))~=</math> ". For the sake of comparison, let's record the planar and dual forms of the axiom that is formulated in string form as "<math>(~)(~)~=~(~)</math>".
First the planar form:
First the plane-embedded maps:
<pre>
<p>[[Image:Logical_Graph_Figure_7.jpg|center]]</p>
</pre>
Next the planar and dual forms superimposed:
Next the plane maps and their dual trees superimposed:
<pre>
<p>[[Image:Logical_Graph_Figure_8.jpg|center]]</p>
</pre>
Finally the dual form by itself:
Finally the dual trees by themselves:
<pre>
<p>[[Image:Logical_Graph_Figure_9.jpg|center]]</p>
And here are the parse trees with their traversal strings indicated:
<p>[[Image:Logical_Graph_Figure_10.jpg|center]]</p>
</pre>
We have at this point enough material to begin thinking about the forms of [[analogy]], [[iconicity]], [[metaphor]], [[morphism]], whatever you want to call it, that are pertinent to the use of logical graphs in their various logical interpretations, for instance, those that Peirce described as ''[[entitative graph]]s'' and ''[[existential graph]]s''.
We have at this point enough material to begin thinking about the forms of [[analogy]], [[iconicity]], [[metaphor]], [[morphism]], whatever you want to call it, that are pertinent to the use of logical graphs in their various logical interpretations, for instance, those that Peirce described as ''[[entitative graph]]s'' and ''[[existential graph]]s''.