Shifting and Lifting of Cellular Automata

Abstract

We consider the family of all the Cellular Automata (CA) sharing the same local rule but have different memory. This family contains also all the CA with memory m ≤ 0 (one-sided CA) which can act both on A ^ℤ and on A ^ℕ. We study several set theoretical and topological properties for these classes. In particular we investigate if the properties of a given CA are preserved when we consider the CA obtained by changing the memory of the original one (shifting operation). Furthermore we focus our attention to the one-sided CA acting on A ^ℤ starting from the one-sided CA acting on A ^ℕ and having the same local rule (lifting operation). As a particular consequence of these investigations, we prove that the long-standing conjecture [Surjectivity $\Rightarrow$ Density of the Periodic Orbits (DPO)] is equivalent to the conjecture [Topological Mixing $\Rightarrow$ DPO].

This work has been supported by the Interlink/MIUR project “Cellular Automata: Topological Properties, Chaos and Associated Formal Languages”, by the ANR Blanc “Projet Sycomore” and by the PRIN/MIUR project “Formal Languages and Automata: Mathematical and Applicative Aspects”.