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Connectivity constraints on the rule (Wolfram Stephen graphs)

Posted 5 months ago

Hi, I am analyzing chapter 3.2 in the book "A Class of Models with the Potential to Represent Fundamental Physics" by Wolfram Stephen and I am having trouble understanding left connectivity constraint on rule. I would be grateful if someone could show this by example of some rule. Dominik.

Posted 2 months ago

Since no one else has responded, I will give this a try. I think left connectivity means nothing more nor less than that every element on the left-hand side is connected, either directly or indirectly, by links or pathway to every other element on the left-hand side. So for examples: (a,b),(b,c) is connected because the two pairs share the element b in common, but (a,b),(c,d) is not connected because the two pairs do not share any elements in common. (a,b),(c,d),(c,b) is connected because the third pair shares the element b in common with the first pair and the element c in common with the second pair. However, (a,b),(c,d),(b,e) is not connected, because the two connected pairs (a,b),(b,e) are not connected to (c,d), because the second pair shares no elements in common with the first or third pairs.

More helpful sections of Wolfram's book, A Project to Find the Fundamental Theory of Physics, include Sections 2.9 Connectedness and Section 3.19 Rules Involving Disconnected Pieces. Section 3.2 is a bit dense if one simply starts there. Wolfram states (p. 92), "But for many purposes we will want to impose connectivity constraints on the rule." I assume here that Wolfram means "physics purposes." It would be an odd physics rule that allowed, for instance, a particle in the Milky Way galaxy to interact, immediately and directly, with a particle in the Andromeda galaxy. Unless they are quantum entangled somehow, in which case the quantum rules would somehow have to establish the connection. Or completely new laws of physics.

POSTED BY: Carl Lundgren
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