The rule “BA” → “AB” is presented in Section 5.9 of the Physics Project book (“The Significance of Causal Invariance”) as being causal invariant. The instances of its causal graph shown therein for 5 time steps and initial condition “BBBAA” is given as example, showing that all 5 instances are isomorphic to each other.
I’ve run the example with the MultiwaySystem function (ResourceFunction[“MultiwaySystem”][{“BA” → “AB”}, “BBBAA”, 5, “CausalGraphInstances”]) and the function output matched the book content.
However:
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Running the system for only 4 time steps, not all instances are isomorphic to each other, since in this case there 2 types of graphs (among the 5 instances) which are clearly not isomorphic to each other.
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In the subsequent example in the section – that changes the initial condition to “BBBBAAAA” – the MultiwaySystem function couldn’t generate the instance causal graph shown in the book, regardless of the number of time steps I tried (from 4 to 10). Actually, for all these time steps different graph types are generated which precludes the existence of isomorphism among them all.
After further experimentations with the MultiwaySystem function, I have a hypothesis that might explain the observations I made: the function may not have a bug after all, but the book would be lacking critically important information about when the individual causal graphs should be isomorphic to each other.
Here are the facts:
- Considering rule BA => AB and initial condition BBBAA:
– For t = 4: Not all graph instances are isomorphic to each other, since there are 2 types of graphs.
– For t = 5: All 5 graph instances are isomorphic to each other and to those of Section 5.9.
– From t = 6: All graph instances are isomorphic to each other and to those for t = 5.
– Convergence of the evolution and causal graphs (i.e., their reaching a fixed point) only happens at t = 6.
- Considering the same rule BA => AB, but now initial condition BBBAA:
– Convergence of the evolution and causal graphs (i.e., their reaching a fixed point) only happens at t = 16.
– For t = 13: Not all graph instances are isomorphic to each other.
– For t = 14: All 24024 graph instances are isomorphic to each other.
– For t = 15: All 24024 graph instances are isomorphic to each other.
– For t = 16: All individual 24024 causal graph instances are isomorphic to the three ones shown in Section 5.9.
Conclusion:
It seems that isomorphism among the causal graph instances can only be guaranteed from the time step (and onwards) the evolution (and, therefore, its associated causal graph) has reached a fixed point. However, isomorphism may happen before the convergence.
So, the conditions by which isomorphism happens are not discussed in the sections related to causal graphs (especially Section 5.9).