Shifting and Lifting of Cellular Automata

  • Authors:

  • Luigi Acerbi;Alberto Dennunzio;Enrico Formenti

  • Affiliations:

  • Università degli Studi di Milano---Bicocca, Dipartimento di Informatica, Sistemistica e Comunicazione, Via Bicocca degli Arcimboldi 8, 20126 Milano, Italy;Università degli Studi di Milano---Bicocca, Dipartimento di Informatica, Sistemistica e Comunicazione, Via Bicocca degli Arcimboldi 8, 20126 Milano, Italy;Université de Nice-Sophia Antipolis, Laboratoire I3S, 2000 Route des Colles, 06903 Sophia Antipolis, France

  • Venue:
  • CiE '07 Proceedings of the 3rd conference on Computability in Europe: Computation and Logic in the Real World
  • Year:
  • 2007

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Abstract

We consider the family of all the Cellular Automata (CA) sharingthe same local rule but have different memory. This family containsalso all the CA with memory m≤ 0 (one-sided CA) whichcan act both on Aℤand on Aℕ. We study several set theoretical andtopological properties for these classes. In particular weinvestigate if the properties of a given CA are preserved when weconsider the CA obtained by changing the memory of the original one(shifting operation). Furthermore we focus our attention to theone-sided CA acting on Aℕstarting fromthe one-sided CA acting on Aℤand havingthe same local rule (lifting operation). As a particularconsequence of these investigations, we prove that thelong-standing conjecture [Surjectivity $\Rightarrow$ Density of thePeriodic Orbits (DPO)] is equivalent to the conjecture [TopologicalMixing $\Rightarrow$ DPO].