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Journal Articles Theoretical Computer Science Year : 2019

Pre-Expansivity in Cellular Automata

A. Gajardo
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V. Nesme
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We introduce the property of pre-expansivity for cellular automata (CA): it is the property of being expansive on asymptotic pairs of configurations (i.e. configurations that differ in only finitely many positions). Pre-expansivity therefore lies between expansivity and pre-injectivity, two important notions of CA theory. We show that there exist one-dimensional positively pre-expansive CAs which are not positively expansive and they can be chosen reversible (while positive expansivity is impossible for reversible CAs). We show however that no bi-dimensional CA which is linear over an Abelian group can be pre-expansive. We also consider the finer notion of k-expansivity (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 and versions

hal-01286018 , version 1 (10-03-2016)
hal-01286018 , version 2 (22-03-2016)
hal-01286018 , version 3 (12-06-2017)
hal-01286018 , version 4 (04-11-2019)


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A. Gajardo, V. Nesme, Guillaume Theyssier. Pre-Expansivity in Cellular Automata. Theoretical Computer Science, 2019, ⟨10.1016/j.tcs.2019.10.034⟩. ⟨hal-01286018v4⟩
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