Linear reversible second-order cellular automata and their first-order matrix equivalents

Abstract

Linear or one-dimensional reversible second-order cellular automata, exemplified by three cases named as RCA1–3, are introduced. Displays of their evolution in discrete time steps, , from their simplest initial states and on the basis of updating rules in modulo 2 arithmetic, are presented. In these, shaded and unshaded squares denote cells whose cell variables are equal to one and zero respectively. This paper is devoted to finding general formulas for, and explicit numerical evaluations of, the weights N(t) of the states or configurations of RCA1–3, i.e. the total number of shaded cells in tth line of their displays. This is achieved by means of the replacement of RCA1–3 by the equivalent linear first-order matrix automata MCA1–3, for which the cell variables are matrices, instead of just numbers () as for RCA1–3. MCA1–3 are tractable because it has been possible to generalize to them the heavy duty methods already well-developed for ordinary first-order cellular automata like those of Wolfram's Rules 90 and 150. While the automata MCA1–3 are thought to be of genuine interest in their own right, with untapped further mathematical potential, their treatment has been applied here to expediting derivation of a large body of general and explicit results for N(t) for RCA1–3. Amongst explicit results obtained are formulas also for each of RCA1–3 for the total weight of the configurations of the first times, .