1.2. State-of-the-art

The study of synchronous CAs starts with Wolfram [1]. We use asynchronous CAs as deterministic devices with a finite number of computation steps, which is a new point of view.

Previously, asynchronous CAs have been treated as dynamical systems, where infinite computations are considered, and the focus lies on concepts like orbits, fixed points, ergodicity, transients, cycles and their periods, and long-term behavior. Also, randomness can be introduced to average over many possible asynchronous schemes. Papers in this respect are:

Ingerson and Buvel [2], 1984, distinguish synchronous, random (completely asynchronous), and periodic clocking, which yield clearly distinguishable behavior.

Barrett et al. [3–6], 1999–2003, consider sequential dynamical systems (SDS), including CAs with arbitrary toplogy and neighborhoods. They cover random graphs as topology and dynamical systems topics such as fixed points and invertibility.

Siwak [7], 2002, gives an overview of simulating machines, including CAs and SDSs, and unifies them under the concept of "filtrons."

Lee et al. [8], 2003, give an asynchronous CA on the two-dimensional grid Z � Z, which is Turing universal.

Laubenbacher and Pareigis [9], 2006, build upon [3–6] and observe that not all n! permutations of the cells lead to different temporal rules. Their equivalence classes coincide—for our setting, CAs on the torus—with our result ([10], Thm. 1]).

Fatès et al. [11], 2006, consider ECAs with quiescent states (000 ↦ 0; 111 ↦ 1, i.e., with even Wolfram rule ≥ 128). They show that 9 ECAs diverge, while the other 55 converge to a random fixed point, in 4 clearly distinguishable time frames <sup>Θ</sup>ð Þ <sup>n</sup> log ð Þ <sup>n</sup> , <sup>Θ</sup> <sup>n</sup><sup>2</sup> , <sup>Θ</sup> <sup>n</sup><sup>3</sup> , or <sup>Θ</sup> <sup>n</sup>2<sup>n</sup> ð Þ with characteristic behavior per time frame.

Macauley, McCammond, and Mortveit [12, 13], 2007–2010, also treat SDSes, in particular ECAs. For each ECA, [13] gives the periodic states and the dynamics group. Conjecture 5.10 in [13] about Wolfram rule 57 coincides with our finding that ECA-57 generates the alternating groups on patterns of n bits. They verified this claim for n up to 8, while in [14] this is extended to n ≤ 10 and in this paper up to n ≤ 28.

Dennunzio et al. [15, 16], 2012–2013, consider ACAs, every turing machine can be simulated by an ACA, with quadratic slowdown. Introducing a certain fairness measure, they show that injectivity and surjectivity are equivalent (μ-a.a.), and the existence of a diamond is equivalent to not μ-a.a. injectivity.

Salo [17], 2014, shows that nonuniform CA generates all SFTs (subshifts of finite type) and several non-SFT sofic shifts.
