Solving two-dimensional density classification problem with two probabilistic cellular automata

TL;DR

This paper introduces a probabilistic approach using two stochastic cellular automata rules to solve the two-dimensional density classification problem, overcoming limitations of deterministic rules in higher dimensions.

Contribution

It proposes a novel probabilistic method combining two rules to achieve accurate density classification in two dimensions, where deterministic solutions are lacking.

Findings

Probabilistic rules enable correct classification with probability approaching 1.

The approach generalizes the density classification task to two dimensions.

Stochastic rules mimic agent behaviors for effective classification.

Abstract

The density classification problem is one of the simplest yet non-trivial computing tasks which seem to be ideally suitable for cellular automata (CA). Unfortunately, there exists no one-dimensional two-state CA which classifies binary strings according to their densities. If, however, in place of simple cells one uses agents which change their behaviour from one rule to another after a fixed number of iterations, the classification can be performed by the traffic rule 184 and the majority rule 232. This two-rule solution cannot be easily generalized to two (or higher) dimensions, because it critically depends on a kinetic phase transition occurring in the rule 184. No rule exhibiting analogous transition is known in two dimensions, most likely because no such rule exists. We propose, therefore, to approach this problem form a slightly different angle, namely by introducing a stochastic component into each of the two rules. If one precedes each iteration of rule 184 by the stochastic "lane changing rule," and each iteration of rule 232 by the stochastic "crowd avoidance" rule, in the limit of infinitely many iterations the classification can be performed correctly with probability 1. This solution can be described either in the language of CA, or using the paradigm of agents which move and proliferate on the 2D lattice, following probabilistic rules.