Re: R.J. Lipton • Anti-Social Networks
Re: Lou Kauffman • Iterants, Imaginaries, Matrices

Comments I made elsewhere about computer science and (anti-)social networks have a connection with the work in progress on this thread, so it may steal a march to append them here.

Comment 1

I have been interested for a long time now in using graphs to do logic.  For that you need more than positive links — negative relations are more generative than positive relations.  The logical situation is analogous to social networks where people can “unlike” or “¬like” other people, or website networks where the information at one node may contradict the information at another node.  In my pursuits it turns out that particular species of graph-theoretic “cacti” are much more useful than the garden variety trees and unsigned graphs.

Comment 2

For what it’s worth, here is my exposition of “painted cacti” and their application to propositional calculus.

• Cactus Language • Part 1 • Part 2 • Part 3

A painted cactus is a rooted cactus with any number of symbols from a finite alphabet attached to each node.  In their ordinary logical interpretations these symbols (“paints”) stand for boolean variables.

Triangles are interesting in computational contexts because they arise in case-breakdown expressions.  In one of the common interpretations of cactus graphs, a rooted triangular lobe says the values of the two non-root nodes are logically inequivalent.

Resources

• Logic Syllabus
• Logical Graphs • (1) • (2)
• Futures Of Logical Graphs
• Propositional Equation Reasoning Systems
• Differential Propositional Calculus • Part 1 • Part 2

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