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Causal invariance definition issue

Posted 1 month ago
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There appears to be a mistake in the definition of causal invariance (Some Quantum Mechanical Properties of the Wolfram Model):

Definition 22. A multiway system is “causal invariant” if and only if the causal networks associated with all paths through the multiway system are (eventually) isomorphic as directed, acyclic graphs.

Consider a pair of string substitution rules A->AB, A->AC with the starting string "A". At each step of the evolution we can choose whether we generate B or C, and the multiway graph for this rule looks like a binary tree. A causal graph for any update order will always look like a single chain, enter image description here, and therefore, all paths are isomorphic as directed, acyclic graphs, and the system must be causally invariant according to the definition. On the other hand, the system is not confluent (Gorard says that confluence is a necessary condition for causal invariance in his presentations), and appears to contradict the whole intuition of a causally invariant system, that was built before that. A definition, corresponding to the intent of causal invariance should look more like this:

"A multiway system is “causal invariant” if and only if the causal networks associated with all paths through the multiway system are (eventually) identical."

It would help though to clarify what does it mean that two causal graphs are "identical" (isomorphism is necessary, but, clearly, not sufficient).

Does this look correct?

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I think you are right, confluence is not a necessary condition for causal invariance. On the same topic, see also this bulletin by Max Piskunov, where he shows that it isn't even a sufficient condition.

Thanks for the bulletin. I was a bit confused by Gorard's technical introduction: https://youtu.be/BV3a0PzNNqE (at 25:11). Does it mean, I should understand causal invariance, as given by the definition? (In other words, there are no mistakes in the definition)

I am confused as well, because even in the paper Algorithmic Causal Sets and the Wolfram Model Gorard states:

global confluence is a necessary (though not sufficient) condition for causal invariance, which can easily be proved by strong induction on the set of updating events.

After having given a definition of causal invariance similar to the one you quoted. And I think that this last paper came out about a week later than Piskunov's bulletin, therefore I must believe that Gorard is still convinced of his statement.

I am sure he has good reasons to believe that global confluence is a necessary condition for causal invariance, but this fact only makes my confusion worse.

In the Wolfram Physics Project, like in any other scientific group, e.g., Bourbaki, there are interesting internal debates among the members concerning the best way to proceed. Respecting the privacy of people involved in those debates by omitting names, I will share the following quote concerning causal invariance (it was not me), which may clarify some of the ideas discussed in this thread (for more information you could also take a look at Wolfram Physics Project on Twitter and follow the members of this project, they are updating their contributions to this project).

By considering only terminating systems, [...] causal invariance does not imply local confluence (which is a different statement than the statement that causal invariance does not imply global confluence), since for terminating systems, local and global confluence are not separable conditions, as a consequence of Newman’s lemma. This is precisely why terminating [Abstract Rewriting Systems] are inappropriate to consider for such analyses [...] and why the definition of causal invariance [...] is an asymptotic statement for [Abstract Rewriting Systems] containing infinite reduction sequences.

So... they are working on a different definition of causal invariance, which does not imply termination? In my post though the system is not terminating, and the causal graphs are isomorphic in an "asymptotic" sense already (I could force-terminated it, if I wanted). Should I consider the system in the post to be causally invariant?

Question:

Should I consider the system in the post to be causally invariant?

Answer (personal opinion): What is important is to be consistent with yourself. After you obtain some interesting results, then you compare your approach with other people's approaches and you decide whether or not to modify your definitions. In new scientific fields, there is always diversity concerning the definitions. When consensus is reached, it is too late: the subject is not new anymore.

I agree with you, the important is to be consistent with yourself. But I still feel like I am missing something, because the example on this post:

  • is not a terminating system
  • is not globally or locally confluent by any definition I know of.
  • is causally invariant by Gorard's definition quoted in the post.

hence, the claim that "causal invariance (by Gorard's definition) implies global confluence in systems containing infinite reduction sequences", seems to be wrong.

The definition of causal invariance originally used by S. Wolfram is the following.

[A] substitution system [...] has the property of causal invariance [if and only if] it gives the same causal network independent of the scheme used to apply its underlying rules.

It is interesting to contrast it with Jonathan's definition.

Definition 22. A multiway system is “causal invariant” if and only if the causal networks associated with all paths through the multiway system are (eventually) isomorphic as directed, acyclic graphs.

Here is a previous discussion about this subject.

By the way, Tommaso Bolognesi has several interesting papers about causality in a framework rather similar to the Wolfram Model. For example: Algorithmic Causal Sets for a Computational Spacetime

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