Derivation and Representation of One-Dimensional, Reversible, Number-Conserving Cellular Automata Rules
Angelo Schranko and Pedro P. B. De Oliveira
Based upon computational experiments, evidence was drawn for the possibility that one-dimensional, q-state, n-neighbour, reversible, number conserving cellular automata rules (RNCCA rules) can be derived by means of compositions of q-state, 2-neighbour, RNCCA rules. Additionally, these experiments also led to empirical compact formulae for the q-ary representation of the rule number of one-dimensional, q-state, n-neighbour, linear RNCCA rules, based upon the q-state, 2-neighbour, RNCCA rules that can be composed to derive them; however, while it was possible to derive formulae that always seem to work for linear RNCCA rules, this is not the case for nonlinear RNCCA rules. A characterisation was also given for linear and nonlinear RNCCA rules, in terms of a feature of their rule table. The spatiotemporal phenomenology of RNCCA rules indicates that the nonlinear variant displays more complex dynamics than the linear kind, but simpler than those of the standard, non-number-conserving reversible rules. Even though the truthful acceptation of our main observations are yet to be grounded by more formal procedures, they are conceptually appealing in that they relate the RNCCA rules in general with the corresponding space for n = 2.
Keywords: One-dimensional cellular automata, reversible rules, number conserving rules, conservative rules, discrete dynamical systems