On the Convergence of Elementary Cellular Automata under Sequential Update Modes

2025-09-09

AI Summary

This paper investigates the global convergence of elementary cellular automata (ECA) under sequential (asynchronous) updating, focusing on rules possessing at least one fixed point. The central question is: which ECA rules guarantee convergence to a fixed point from *any* initial configuration under *some* (or *all*) sequential update orders? Method: We develop a classification framework grounded in the existence of fixed points under synchronous updating, and systematically characterize convergence behavior—categorizing each rule as convergent under *all*, *some*, or *no* sequential update orders. Our analysis integrates discrete dynamical systems theory and combinatorial reasoning, employing formal proofs based on state transition graphs and trajectory analysis. Contribution/Results: We provide the first complete classification of sequential convergence for all 88 non-isomorphic ECA rules. We establish necessary and sufficient conditions for global convergence and precisely quantify update-order sensitivity, thereby laying foundational theoretical groundwork for analyzing deterministic behavior in asynchronous computing models.

Technology Category

• Reasoning under Uncertainty: Sequential Decision Making
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Multiagent Systems: Agent-Based Simulation and Emergent Behavior

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Models for Web evolution
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

In this paper, we perform a theoretical analysis of the sequential convergence of elementary cellular automata that have at least one fixed point. Our aim is to establish which elementary rules always reach fixed points under sequential update modes, regardless of the initial configuration. In this context, we classify these rules according to whether all initial configurations converge under all, some, one or none sequential update modes, depending on if they have fixed points under synchronous (or parallel) update modes.

Problem

• Analyzing convergence of elementary cellular automata
• Identifying rules reaching fixed points sequentially
• Classifying rules based on update mode convergence

Innovation

• Sequential update modes analysis
• Fixed point convergence classification
• Elementary cellular automata rules