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Multidimensional cellular automata: closing property, quasi-expansivity, and (un)decidability issues

Abstract : In this paper we study the dynamics of D-dimensional cellular automata (CA) by considering them as one-dimensional (1D) CA along some direction (slicing constructions). These constructions allow to give the D-dimensional version of important notions as 1D closing property and lift well-known one-dimensional results to the D-dimensional settings. Indeed, like in one-dimensional case, closing D-dimensional CA have jointly dense periodic orbits and biclosing D-dimensional CA are open. By the slicing constructions, we further prove that for the class of closing D-dimensional CA, surjectivity implies surjectivity on spatially periodic configurations (old standing open problem). We also deal with the decidability problem of the D-dimensional closing. By extending the Kariʼs construction from [31] based on tilings, we prove that the two-dimensional, and then D-dimensional, closing property is undecidable. In such a way, we add one further item to the class of dimension sensitive properties, i.e., properties that are decidable in dimension 1 and are undecidable in higher dimensions. It is well-known that there are not positively expansive CA in dimension 2 and higher. As a meaningful replacement, we introduce the notion of quasi-expansivity for D-dimensional CA which shares many global properties (in the D-dimensional settings) with the 1D positive expansivity. We also prove that for quasi-expansive D-dimensional CA the topological entropy (which is an uncomputable property for general CA) has infinite value. In a similar way as quasi-expansivity, the notions of quasi-sensitivity and quasi-almost equicontinuity are introduced and studied.
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Contributor : Enrico Formenti <>
Submitted on : Tuesday, May 10, 2016 - 10:19:49 AM
Last modification on : Friday, January 8, 2021 - 11:22:05 AM

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Alberto Dennunzio, Enrico Formenti, Michael Weiss. Multidimensional cellular automata: closing property, quasi-expansivity, and (un)decidability issues. Theoretical Computer Science, Elsevier, 2014, 516, pp.40-59. ⟨10.1016/j.tcs.2013.11.005⟩. ⟨hal-01313574⟩



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