Solution of the Density Classification Problem with Two Cellular Automata Rules

Recently, Land and Belew [Phys. Rev. Lett. 74, 5148 (1995)] have shown that no one-dimensional two-state cellular automaton which classifies binary strings according to their densities of 1's and 0's can be constructed. We show that a pair of elementary rules, namely the ``traffic rule'' 184 and the ``majority rule'' 232, performs the task perfectly. This solution employs the second order phase transition between the freely moving phase and the jammed phase occurring in rule 184. We present exact calculations of the order parameter in this transition using the method of preimage counting.