Solving decision problems by distributed consensus with one-dimensional, binary, radius-$2$ cellular automata over cyclic configurations
Eurico Ruivo, Pedro Paulo Balbi, Kévin Perrot, Marco Montalva-Medel, Eric Goles
TL;DR
The paper addresses whether one-dimensional binary cellular automata with radius $2$ can solve decision problems by distributed consensus on cyclic configurations. It conducts an exhaustive re-evaluation of the entire radius-$2$ rule space for lengths $L$ from 5 to 20, filtering to rules whose basins of attraction are confined to the two fixed points $0^L$ and $1^L$, and identifies 54,928 candidate consensus rules. These are categorized into three classes (A,B,C), with precise proofs establishing consensus behavior for 40,254 rules (about 73.3% of candidates) and detailed analyses of their basin structures; the remaining rules are left as uncharacterised candidates for additional decision problems. The work significantly expands the known landscape of radius-$2$ consensus-capable CA, providing a comprehensive framework for understanding how local rules can induce global consensus and outlining clear directions for extending the analysis to asynchronous updates and broader problem classes. Overall, the findings offer a richer map of how distributed, local computation can realize global decision-making in CA networks, with implications for theory and potential applications in distributed systems.
Abstract
Probing the ability of automata networks to solve decision problems has received a continuous attention in the literature, and specially with the automata reaching the answer by distributed consensus, i.e., their all taking on a same state, out of two. In the case of binary automata networks, regardless of the kind of update employed, the networks should display only two possible attractors, the fixed points $0^L$ and $1^L$, for all cyclic configurations of size $L$. A previous investigation into the space of one-dimensional, binary, radius-2 cellular automata identified a restricted subset of rules as potential solvers of decision problems, but the reported results were incomplete and lacked sufficient detail for replication. To address this gap, we conducted a comprehensive reevaluation of the entire radius-2 rule space, by filtering it with all configuration sizes from 5 to 20, according to their basins of attraction being formed by only the two expected fixed points. A set of over fifty-four thousand potential decision problem solvers were then obtained. Among these, more than forty-five thousand were associated with 3 well-defined decision problems, and precise formal explanations were provided for over forty thousand of them. The remaining candidate rules suggest additional problem classes yet to be fully characterised. Overall, this work substantially extends the understanding of radius-2 cellular automata, offering a more complete picture of their capacity to solve decision problems by consensus.