A multiway system that generates causal graphs which are all isomorphic as acyclic digraphs is said to exhibit causal invariance, and the causal graph itself is also said to be causally invariant. Essentially, causal invariance means that no matter which evolution is chosen for a system, the history is the same in the sense that the same events occur and they have the same causal relationships. The figures above illustrate two nontrivial substitution systems that exhibit the same causal graphs independent of the order in which the rules are applied (Wolfram 2002, pp. 500-501).
Whenever two rule hypotheses overlap in an evolution, the corresponding system is not causally invariant. Hence, the simplest way to search for causal invariance
is to use rules whose hypotheses can never overlap except trivially. An overlap can
involve one or two strings. For example, 
does not have any overlaps. However, 
can overlap as 
, and the set of strings 
can overlap as 
.
A mobile automaton simulated by a string substitution system is an example of a causally invariant network in which rule hypotheses overlap, as long as the initial condition contains only a single active cell.
