2. Notations: ECA and update rules

1. Introduction

1.1. What is etherealware?

186 From Natural to Artificial Intelligence - Algorithms and Applications

which the CA's cells are updated.

1.2. State-of-the-art

Turing universal.

Computation takes place in dedicated hardware or on general-purpose hardware by dedicated software. Different functionality requires either changing the hardware (think ASIC, FPGA) or changing the software running on it. Firmware is an intermediate concept, where the hardware

Etherealware is the first way to use fixed hardware (certain cellular automata (CA) in this case), run fixed software (the same update rule for all cells, for all time, for all purposes), and still deliver diversity in the resulting function: by changing only the clocking scheme, the order in

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),

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

Siwak [7], 2002, gives an overview of simulating machines, including CAs and SDSs, and

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

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,

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

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

is modified by microprogramming a CPU or personalizing an FPGA.

and periodic clocking, which yield clearly distinguishable behavior.

dynamical systems topics such as fixed points and invertibility.

unifies them under the concept of "filtrons."

CAs on the torus—with our result ([10], Thm. 1]).

characteristic behavior per time frame.

Here, continuing the work in [10, 14], we again employ CAs as computing devices, whose work comes to an end, when the pattern transformation or function evaluation has been obtained. Also, the clocking, the temporal update rule, is completely deterministic and replaces the usual ways of representing an algorithm, either in software (initial data) or in hardware (choice of ECA and connecting graph). Thus, the algorithm resides exclusively in the clocking scheme. We therefore call functions computable in this way as "clocking-computable functions." The main additional contribution of this paper is the introduction of unfair clocking schemes.
