Restricted density classification in one dimension

TL;DR

This paper investigates the density classification problem in one-dimensional cellular automata, showing that automata capable of eliminating finite islands in linear time can correctly classify Bernoulli configurations with extreme densities, using a percolation-based proof.

Contribution

It demonstrates that certain cellular automata can almost surely classify Bernoulli configurations near 0 or 1, advancing understanding of density classification limitations.

Findings

Automata with linear-time island washing classify extreme Bernoulli densities correctly.

Percolation arguments are effective in analyzing cellular automaton classification capabilities.

The results connect cellular automaton behavior with percolation theory insights.

Abstract

The density classification task is to determine which of the symbols appearing in an array has the majority. A cellular automaton solving this task is required to converge to a uniform configuration with the majority symbol at each site. It is not known whether a one-dimensional cellular automaton with binary alphabet can classify all Bernoulli random configurations almost surely according to their densities. We show that any cellular automaton that washes out finite islands in linear time classifies all Bernoulli random configurations with parameters close to 0 or 1 almost surely correctly. The proof is a direct application of a "percolation" argument which goes back to Gacs (1986).