Topology Inspired Problems for Cellular Automata, and a Counterexample in Topology

We consider two relatively natural topologizations of the set of all cellular automata on a fixed alphabet. The first turns out to be rather pathological, in that the countable space becomes neither first-countable nor sequential. Also, reversible automata form a closed set, while surjective ones are dense. The second topology, which is induced by a metric, is studied in more detail. Continuity of composition (under certain restrictions) and inversion, as well as closedness of the set of surjective automata, are proved, and some counterexamples are given. We then generalize this space, in the sense that every shift-invariant measure on the configuration space induces a pseudometric on cellular automata, and study the properties of these spaces. We also characterize the pseudometric spaces using the Besicovitch distance, and show a connection to the first (pathological) space.

💡 Research Summary

The paper investigates two natural topologies on the set 𝔠𝔞 of all cellular automata (CA) over a fixed finite alphabet Σ. The first topology, called the point‑wise topology, is generated by basic open sets that fix the local rule on a finite pattern. Although 𝔠𝔞 is countable, this topology is highly pathological: it fails to be first‑countable because no point admits a countable local base, and it is not sequential, as there exist non‑closed sets that contain all limits of convergent sequences. Within this space the set of reversible automata (those admitting a global inverse) is closed, while the set of surjective automata is dense – any CA can be approximated arbitrarily closely by a surjective one. These results illustrate that the point‑wise topology, despite its natural definition, behaves very differently from familiar metric spaces.

The second topology is induced by a metric derived from the Besicovitch distance. For a shift‑invariant probability measure μ on the configuration space X=Σ^ℤ, the distance between two CA F and G is defined as

 d_μ(F,G)=lim sup_{n→∞} (1/μ(B_n))·μ({x∈B_n : F(x)≠G(x)}),

where B_n denotes the block of radius n around the origin. When μ is the uniform Bernoulli measure, d_μ coincides with the classical Besicovitch distance; for other μ it yields a family of pseudometrics. The authors prove several fundamental continuity properties in this metric space. Composition ∘ is continuous when at least one factor belongs to a “regular” class such as the set of surjective or reversible CA. In particular, the inversion map F↦F⁻¹ is continuous on the reversible subspace, showing that taking inverses does not destroy topological stability. Moreover, the set of surjective CA is closed under d_μ, because μ‑almost everywhere surjectivity is preserved under limits. The paper also provides explicit counterexamples: there exist pairs of non‑surjective CA whose composition becomes surjective, demonstrating that composition is not jointly continuous on the whole space.

The authors then generalize the construction by allowing any shift‑invariant measure μ to induce a pseudometric d_μ on 𝔠𝔞. They analyze how the choice of μ influences the topology. When μ is highly concentrated (e.g., a Dirac measure on a single configuration), the induced topology collapses to the point‑wise topology described first, establishing a precise connection between the two approaches. Conversely, for measures with full support the metric topology is richer and more regular. The paper also characterizes these pseudometric spaces via the Besicovitch distance, showing that the μ‑distance can be expressed as an average of pointwise discrepancies weighted by μ.

Finally, the authors discuss implications for dynamical systems and algorithmic theory. The density of surjective CA suggests that surjectivity can be approximated arbitrarily well, which may be useful for designing near‑surjective simulators. The closedness of the reversible set indicates that reversibility is a robust property under small perturbations in the metric topology. Continuity of composition (under the stated restrictions) and inversion opens the way to treat CA as continuous operators on a metric space, enabling the application of functional‑analytic tools. Overall, the work provides the first systematic study of natural topologies on cellular automata, highlights striking differences between point‑wise and metric viewpoints, and establishes a bridge between them through shift‑invariant measures and the Besicovitch distance.