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

  • Authors:

  • Alberto Dennunzio;Enrico Formenti;Michael Weiss

  • Affiliations:

  • Universití degli studi di Milano-Bicocca, Dipartimento di Informatica Sistemistica e Comunicazione, viale Sarca 336, 20126 Milano, Italy;Université Nice-Sophia Antipolis, Laboratoire I3S, 2000 Route des Colles, 06903 Sophia Antipolis, France;Université Nice-Sophia Antipolis, Laboratoire I3S, 2000 Route des Colles, 06903 Sophia Antipolis, France

  • Venue:
  • Theoretical Computer Science
  • Year:
  • 2014

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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 Kari@?s 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.