Stochastic Cellular Automata Solutions to the Density Classification Problem

Abstract

In the density classification problem, a binary cellular automaton should decide whether an initial configuration contains more 0s or 1s. The answer is given when all cells of the CA agree on a given state (0 or 1). This problem is known for having no exact solution in the case of binary deterministic one-dimensional CA. We investigate how randomness in CA may help us solve the problem. We analyse the behaviour of stochastic CA rules that perform the density classification task. We show that describing stochastic rules as a ''blend'' of deterministic rules allows us to derive quantitative results on the classification time and the classification time of previously studied rules. We introduce a new rule whose effect is to spread defects and to wash them out. This stochastic rule solves the problem with an arbitrary precision, that is, its quality of classification can be made arbitrarily high, though at the price of a longer time to converge. We experimentally demonstrate that this rule exhibits good scaling properties and that it attains qualities of classification never reached so far.