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Article Dans Une Revue Theoretical Computer Science Année : 2020

Pre-expansivity in cellular automata

Résumé

We introduce the notion of pre-expansivity for cellular automata (CA): it is the property of being positively expansive on asymptotic pairs of configurations (i.e. configurations that differ in only finitely many positions). Pre-expansivity therefore lies between positive expansivity and pre-injectivity, two important notions of CA theory. We show that there exist one-dimensional pre-expansive CAs which are not positively expansive and they can be chosen reversible (while positive expansivity is impossible for reversible CAs). We provide both linear and non-linear examples. In the one-dimensional setting, we also show that pre-expansivity implies sensitivity to initial conditions in any direction. We show however that no two-dimensional Abelian CA can be pre-expansive. We also consider the finer notion of k-expansivity (positive expansivity over pairs of configurations with exactly k differences) and show examples of linear CA in dimension 2 and on the free group that are k-expansive depending on the value of k, whereas no (positively) expansive CA exists in this setting.
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Dates et versions

hal-03410438 , version 1 (31-10-2021)

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Paternité - Pas d'utilisation commerciale - Pas de modification

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A. Gajardo, V. Nesme, Guillaume Theyssier. Pre-expansivity in cellular automata. Theoretical Computer Science, 2020, 816, pp.37-66. ⟨10.1016/j.tcs.2019.10.034⟩. ⟨hal-03410438⟩
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