Generalized Causal Invariance
Generalized Causal Invariance
Seth J. Chandler 2020
A definition of 2 - 1 causal invariance. Consider all successors of a node. For all non-empty subsets of the immediate successors containing no more than 2 members one can always find a node whose ancestors include all elements of the subset. And this is true for all non-terminal nodes. I believe this is what Wolfram Physics means by causal invariance.
But could we have 3 - 1 causal invariance? Consider all successors of a node. For all non-empty subsets of the immediate successors with no more than 3 members one can always find a node whose ancestors include all elements of the subset. And this is true for all non-terminal nodes.
Or could we have 3-2 causal invariance? Consider all successors of a node. For all non-empty subsets of the immediate successors with no more than 3 members one can always find a subset of size 2 or less who, between them, have ancestors that include all elements of the subset. And this is true for all non-terminal nodes.
And I suppose we could have m - n causal invariance. Consider all successors of a node. For all non-empty subsets of the immediate successors with no more than m members one can always find a subset of nodes of size n or less who, among them, have ancestors that include all elements of the subset. And this is true for all non-terminal nodes.
All graphs should be 1 - 1 causal invariant. Indeed, all graphs should be m-m causal invariant for all m≥1. A graph that is m-n causal invariant should be m-(n+1) causal invariant.
Also, we might want to name the property something else since it’s hard to think of this as causal invariance. Maybe Generalized Confluence?