Block Invariance in Elementary Cellular Automata
Eric Goles, Marco Montalva-Medel, Henning Mortveit and Salvador Ramirez-Flandes
Consider an elementary cellular automaton (ECA) under periodic boundary conditions. Given an arbitrary partition of the set of vertices we consider the block updating, i.e. the automaton’s local function is applied from the first to the last set of the partition such that vertices belonging to the same set are updated synchronously. The automaton is said block-invariant if the set of periodic configurations is independent of the choice of the block updating. When the sets of the partition are singletons we have the sequential updating: vertices are updated one by one following a permutation π. In  the authors analyzed the π- invariance of the 28 = 256 possible ECA rules (or the 88 non-redundant rules subset). Their main result was that for all n > 3, exactly 41 of these non-redundant rules are π-invariant. In this paper we determine the subset of these 41 rules that are block invariant. More precisely, for all n > 3, exactly 15 of these rules are block invariant. Moreover, we deduce that block invariance also implies that the attractor structure itself is independent of the choice of the block update.
Keywords: Elementary cellular automata, block updates, periodic points, block invariance.