Effect of Boundary Conditions on Cellular Automata that Classify Density

Abstract

The properties of two-state nearest-neighbour cellular automata (CA) that are capable of density classification are discussed. It is shown that these CA actually conserve the total density, rather than merely classifying it. This is also the criterion for any CA simulation of DFT. The effect of boundary and periodicity conditions upon the evolution of such CA are elaborated by considering linear and cyclic lattices and boundary conditions consistent with the conservation criterion. In a bounded linear lattice, it is possible to achieve a configuration with a single 01 and no 10 domain wall, or vice versa, but this is not possible in a cyclic lattice, where these two domain walls have to appear in pairs. This determines the final stable state of the automaton.