Cellular automata can really solve the parity problem

TL;DR

This paper proves that a single cellular automaton rule can reliably determine the parity of 1s in cyclic binary sequences, resolving previous uncertainties and providing a definitive solution with a full proof.

Contribution

The paper introduces a new fix to the BFO rule and provides a complete proof, confirming that a single-rule cellular automaton can solve the parity problem reliably.

Findings

The new fix corrects the previous BFO rule failure.

A full proof confirms the existence of a single-rule solution.

The solution works for all odd-sized cyclic configurations.

Abstract

Determining properties of an arbitrary binary sequence is a challenging task if only local processing is allowed. Among these properties, the determination of the parity of 1s by distributed consensus has been a recurring endeavour in the context of automata networks. In its most standard formulation, a one-dimensional cellular automaton rule should process any odd-sized cyclic configuration and lead the lattice to converge to the homogeneous fixed point of 0s if the parity of 1s is even and to the homogeneous fixed point of 1s, otherwise. The only known solution to this problem with a single rule was given by Betel, de Oliveira and Flocchini (coined BFO rule after the authors' initials). However, three years later the authors of the BFO rule realised that the rule would fail for some specific configuration and proposed a computationally sound fix, but a proof could not be worked out. Here we provide another fix to the BFO rule along with a full proof, therefore reassuring that a single-rule solution to the problem really does exist.