Exploring one-dimensional, binary, radius-2 cellular automata, over cyclic configurations, in terms of their ability to solve decision problems by distributed consensus

2025-10-01

AI Summary

This study investigates the capacity of one-dimensional binary radius-2 cyclic cellular automata (CAs) to solve decision problems via distributed consensus. Method: We introduce a novel framework combining attractor basin enumeration with cyclic configuration evolution analysis. For all cycle lengths from 5 to 20, we exhaustively traverse initial configurations, perform state-space exploration, and detect fixed points to fully reconstruct the radius-2 rule space. Contribution/Results: We identify over 54,000 rules capable of decision-making; for more than 40,000, we provide precise formal characterizations. Over 45,000 rules are verified to stably solve three well-defined decision problems. Crucially, we discover multiple previously uncharacterized problem classes, significantly expanding the known boundaries of distributed consensus capabilities in radius-2 CAs. Our results demonstrate the broad feasibility of achieving global consensus through local interactions in small-scale cyclic configurations.

Technology Category

• Multiagent Systems: Distributed Problem Solving
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Reasoning under Uncertainty: Causality

Application Category

• Web Mining and Content Analysis: Models for Web evolution
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements

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.

Problem

• Investigating binary radius-2 cellular automata solving decision problems
• Identifying rules achieving consensus through two fixed-point attractors
• Characterizing automata networks that reach distributed binary consensus

Innovation

• Comprehensive reevaluation of radius-2 cellular automata rules
• Filtered rules using configuration sizes from 5 to 20
• Identified over fifty-four thousand potential decision solvers