Virtual cyclic cellular automata, finite group actions and recursive properties

Abstract

The aim of these notes is three fold. First we introduce virtual cyclic cellular automata and show that the inverse of a reversible (2R+1)-cyclic cellular automaton with periodic boundary conditions is a virtual cyclic cellular automaton. These virtual automata have two special characteristics: they have active and non active cells at specific steps of times and they reflect certain periodicity. We will relate these particularities with a finite cyclic group action on the cellular automaton, and prove that the inverse transition dipolynomial is an invariant dipolynomial under this action. Secondly, we use a recursive estimation of neighbours (REN) algorithm to produce direct examples of virtual cyclic cellular automata, which moreover generalize some of the cellular automata used in applications like collective control or traffic patterns. We also propose a new REN algorithm which allows us to reinterprete a (2R+1)-cyclic cellular automata as a recursive sequence originated from the elementary cellular automaton with base rule 150, and which motivate us to introduce a new notion of a recursive Wolfram number for a (2R+1)-cyclic cellular automaton. Finally we show that this recursive Wolfram number can be computed by the new REN algorithm applied to the base rule 150 and its complementary 105 rule.