Non-trivial automata networks do exist that solve the global majority problem with the local majority rule

TL;DR

This paper investigates whether automata networks using only local majority rules can solve the global majority problem, identifying non-trivial cases where this is possible and explaining the underlying reasons.

Contribution

It demonstrates the existence of non-trivial automata networks that solve the global majority problem using local rules, expanding understanding of computational capabilities of such systems.

Findings

Non-trivial automata networks can solve the global majority problem.

Conditions under which local majority rule leads to correct global consensus.

Insights into why certain automata networks succeed in solving the problem.

Abstract

The global majority problem, often referred to as the Density Classification Task, is a classical benchmark in the context of probing the computational capabilities of automata networks. It poses the simple yet challenging problem of determining, by totally local means, whether an arbitrary initial configuration of binary states can evolve to a final, homogeneous global configuration that reflects the initial global majority. Although it is known that in the specific case of cellular automata with periodic boundaries no rule is able to solve the problem, in other formulations solutions are known and, in others, the problem is still open. Aligned with the latter, here we explore the possibility of solving the problem with automata networks, operating only with the local majority rule, with a focus on identifying non-trivial cases where it can be solved and explaining why they do so.