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→Duality : logical and topological: del clutter
==Duality : logical and topological==
==Duality : logical and topological==
There are two types of [[duality (mathematics)|duality]] that have to be kept separately mind in the use of logical graphs, [[De Morgan's laws|logical duality]] and [[topology|topological]] [[dual graph|duality]].
There are two types of [[duality (mathematics)|duality]] that have to be kept separately mind in the use of logical graphs — [[De Morgan's laws|logical duality]] and [[topology|topological]] [[dual graph|duality]].
There is a standard way that graphs of the order that Peirce considered, those embedded in a continuous [[manifold]] like that commonly represented by a plane sheet of paper — with or without the paper bridges that Peirce used to augment its [[genus (mathematics)|topological genus]] — can be represented in linear text as what are called ''[[parsing|parse string]]s'' or ''[[tree traversal|traversal string]]s'' and parsed into ''[[data structure|pointer structure]]s'' in computer memory.
There is a standard way that graphs of the order that Peirce considered, those embedded in a continuous [[manifold]] like that commonly represented by a plane sheet of paper — with or without the paper bridges that Peirce used to augment its [[genus (mathematics)|topological genus]] — can be represented in linear text as what are called ''[[parsing|parse string]]s'' or ''[[tree traversal|traversal string]]s'' and parsed into ''[[data structure|pointer structure]]s'' in computer memory.
Though it's not really there in the most abstract topology of the matter, for all sorts of pragmatic reasons we find ourselves compelled to single out the outermost region of the plane in a distinctive way and to mark it as the ''[[root node]]'' of the corresponding [[dual graph]]. In the present style of Figure the root nodes are marked by horizontal strike-throughs.
Though it's not really there in the most abstract topology of the matter, for all sorts of pragmatic reasons we find ourselves compelled to single out the outermost region of the plane in a distinctive way and to mark it as the ''[[root node]]'' of the corresponding [[dual graph]]. In the present style of Figure the root nodes are marked by horizontal strike-throughs.
Extracting the dual graph from its composite matrix, we get this picture:
Extracting the dual graphs from their composite matrices, we get this picture:
<p>[[Image:Logical_Graph_Figure_5.jpg|center]]</p>
<p>[[Image:Logical_Graph_Figure_5.jpg|center]]</p>
<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 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 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 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>".
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>".