Cellular automata can really solve the parity problem

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 proposed solution to this problem with a single rule was given more than 10 years ago (and coined BFO rule after the authors’ initials). However, three years later its authors realised that the rule would fail for a specific configuration and proposed a computationally sound fix, but a proof could not be worked out. Here we provide a fix to that failing rule along with a full proof, therefore reassuring that a single-rule solution to the problem really does exist.