Elementary Cellular Automata with Minimal Memory and Random Number Generation

Cellular automata (CA) are discrete, spatially explicit extended dynamic systems. CA systems are composed of adjacent cells or sites arranged as a regular lattice, which evolve in discrete time steps. Each cell is characterized by an internal state whose value belongs to a finite set. The updating of these states is made simultaneously according to a common local transition rule involving a neighborhood of each cell. Thus, if si HTL is taken to denote the value of cell i at time step T, the site values evolve by iterating the mapping si HT+1L  fJ:sj HTL>, j œ iN, with i standing for the set of cells in the neighborhood of cell i. Here we will consider the simplest scenario, that of elementary CA [1], that is, one-dimensional CA with two possible state values Hs œ 80, 1<L, and rules operating on nearest neighbors: si  fJsi-1 HTL , si, si+1 HTL N. Elementary rules are characterized by a sequence