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

TL;DR

This paper explores topological structures on the set of cellular automata, revealing pathological properties in one topology and establishing continuity and closedness results in another, while also generalizing to measure-induced pseudometrics.

Contribution

It introduces two topologies on cellular automata, analyzes their properties, and extends the framework using shift-invariant measures and Besicovitch distance, providing new insights into automata topology.

Findings

The first topology is neither first-countable nor sequential.

Reversible automata form a closed set in the second topology.

Surjective automata are dense in the first topology.

Abstract

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.