I have a question that has been bothering me since 2002: what is the full relationship between causal invariance (all causal graphs are isomorphic independent of update order) and global confluence (all branch pairs eventually merge). As is stated many places, confluence is a necessary condition for causal invariance, but can be easily seen to not be sufficient.

I ask since things like CausalInvariantQ seem to be checking confluence instead, and I've heard you say things on livestream like "because this rule contains inverses, it is causal invariant," however from working through some examples it seems that containing inverses only implies confluence. Similarly the Knuth-Benedix completion only produces confluent systems, and not causal invariant ones. On the flipside, from reading Jonathan's paper on GR, it seems clear that you often truly want causal invariance.

I've read all sections of NKS and the WPP introduction several times, and I'm not 100% on this. Does anyone have anyplace that can help clear this up?