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

Ville Salo
(University of Turku, Finland)
Ilkka Törmä
(University of Turku, Finland)

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.

In Enrico Formenti: Proceedings 18th international workshop on Cellular Automata and Discrete Complex Systems and 3rd international symposium Journées Automates Cellulaires (AUTOMATA&JAC 2012), La Marana, Corsica, September 19-21, 2012, Electronic Proceedings in Theoretical Computer Science 90, pp. 53–68.
Published: 13th August 2012.

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