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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