Computing by Temporal Order: Asynchronous Cellular Automata

Our concern is the behaviour of the elementary cellular automata with state set {0,1} over the cell set Z/nZ (one-dimensional finite wrap-around case), under all possible temporal rules (asynchronicity). Over the torus Z/nZ (n≤ 10),we will see that the ECA with Wolfram update rule 57 maps any v ∈ F2 to any w ∈ F2, varying the temporal rule. We furthermore show that all even (element of the alternating group) bijective functions on the set F2∼= {0, . . . ,2n−1}, can be computed by ECA-57, by iterating it a sufficient number of times with varying temporal rules, at least for n ≤ 10. We characterize the non-bijective functions computable by asynchronous rules. The thread of all this is a novel paradigm: The algorithm is neither hard-wired (in the ECA), nor in the program or data (initial configuration), but in the temporal order of updating cells, and temporal order is pattern-universal.