Computation of functions on n bits by asynchronous clocking of cellular automata

Abstract

Can a fixed cellular automaton compute different functions on any n-bit inputs, by providing only the n input bits as data and simulate the function only through the clocking scheme (a/synchronicity)? The perhaps surprising answer is: yes! The elementary cellular automaton with Wolfram number 57, as well as several CAs with 4 input cells are capable to compute those bijective functions on n bits that are equivalent to an even permutation on the domain $\{0,\dots, 2^n-1\}$, at least for n ≤ 10. To distinguish between functions, it suffices to vary the temporal order of updating the n cells in a deterministic way. We also characterize the non-bijective functions so computable, where we now need temporal schemes, which are not fully asynchronous. We start with pattern transformations ${\mathbb F}_2^n\ni v\, \mapsto \, w \in{\mathbb F}_2^n$. The thread of all this is a novel paradigm: the algorithm is neither hard-wired (in the CA), nor in the program or data (initial configuration), but only in the temporal order of firing cells. And temporal order is pattern-universal.