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Sensitivity and Topological Mixing are Undecidable for Reversible One-dimensional Cellular Automata
Ville Lukkarila

It is shown by a reduction from the reversible Turing machine halting problem that sensitivity is undecidable even for reversible one-dimensional cellular automata. With a few additional constructions, the undecidability of topological mixing and the undecidability of topological transitivity follow. Furthermore, sets of topologically mixing cellular automata and non-sensitive cellular automata are recursively inseparable. It follows that Devaney’s chaos and Knudsen’s chaos are undecidable dynamical properties.

Keywords: Cellular automata, Chaos, Sensitivity to initial conditions, Topological mixing, Topological transitivity, Undecidability

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