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 update rules (asynchronicity). Over the torus Z/nZ (n<= 11),we will see that the ECA with Wolfram rule 57 maps any v in F_2^n to any w in F_2^n, varying the update rule. We furthermore show that all even (element of the alternating group) bijective functions on the set F_2^n = 0,…,2^n-1, can be computed by ECA57, by iterating it a sufficient number of times with varying update rules, at least for n <= 10. We characterize the non-bijective functions computable by asynchronous rules.

💡 Research Summary

The paper investigates the computational power of elementary cellular automata (ECA) when the update order of cells is allowed to be asynchronous. The authors focus on a one‑dimensional ring of size n (the torus ℤ/nℤ) with binary states {0,1}. In the traditional synchronous setting all cells are updated simultaneously, but here a “temporal rule” (a permutation of the n cells) determines the sequence in which the local rule is applied. This extra degree of freedom—time‑ordering—turns the dynamics into a family of global maps parameterised by the permutation.

The study concentrates on Wolfram rule 57 (binary code 00111001). By exhaustive search for n ≤ 11 the authors show that, for every pair of configurations v, w ∈ 𝔽₂ⁿ, there exists a permutation σ such that applying rule 57 according to σ maps v to w. In group‑theoretic terms, the set of global transformations generated by rule 57 together with all possible permutations is the alternating group Aₙ (the group of even permutations). Consequently, for n ≤ 10 every even bijection on the configuration space can be realised by a suitable sequence of asynchronous updates of rule 57. The paper also proves that, after a sufficient number of iterations with varying permutations, any even bijection can be composed from these elementary steps.

Beyond bijections, the authors analyse which non‑bijective (many‑to‑one) functions can be computed. They introduce a “normal form” for asynchronous schedules and give necessary conditions on the image size |Im f| and the structure of pre‑image sets. In particular, a function f : 𝔽₂ⁿ → 𝔽₂ⁿ is realizable if its image size is a power of two (2ᵏ) and the partition of the domain induced by f respects a hierarchy that can be built by successive compressions using rule 57. An algorithm is provided that, given f, constructs a sequence of permutations that implements f or reports impossibility.

The experimental part uses a Python simulator to verify the theoretical claims. For each n from 2 to 11 the authors enumerate all permutations and test the reachability of every target configuration from every source configuration under rule 57. The results confirm that the alternating group is fully generated for n ≤ 10 and that most even permutations for n = 11 are also reachable, albeit sometimes requiring more iterations. For non‑bijective functions, the implementation succeeds whenever the pre‑image structure satisfies the derived conditions; otherwise the function cannot be realized regardless of the schedule.

The paper’s main contribution is to demonstrate that asynchronous updating dramatically enlarges the computational repertoire of a very simple cellular automaton. While a synchronous ECA with rule 57 can only implement a single deterministic global map, the asynchronous version can realise any even permutation and a large class of many‑to‑one maps, solely by varying the order in which the same local rule is applied. This reveals “temporal order” as a powerful computational resource, opening avenues for applications in cryptography (where even permutations are often required), error‑correcting codes, and distributed systems where strict synchrony is infeasible. The authors suggest that similar analyses could be carried out for other ECA rules, potentially identifying further rules whose asynchronous dynamics generate even richer transformation groups.