One-dimensional cellular automata with a unique active transition

2024-11-06

arXiv.org

AI Summary

This work investigates the dynamical behavior of one-dimensional cellular automata governed by a single active transition rule—i.e., whose local function deviates from the identity map at exactly one pattern $ p $. Using symbolic dynamics, topological dynamical systems theory, and combinatorial pattern analysis, we establish the first complete classification framework for this class. We prove that the global dynamics must be either idempotent or strictly almost equicontinuous, and that idempotence holds if and only if $ p $ contains a nontrivial translationally symmetric subpattern. This result precisely characterizes how a minimal structural constraint on the local rule—namely, a single-point deviation—fully determines the essential nature of the global dynamics. It provides the first rigorous characterization and decidability criterion for the “local perturbation → global phase transition” mechanism in discrete dynamical systems.

Technology Category

• Multiagent Systems: Agent-Based Simulation and Emergent Behavior
• Cognitive Modeling & Cognitive Systems: Symbolic Representations
• Reasoning under Uncertainty: Sequential Decision Making

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
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

A one-dimensional cellular automaton $ au : A^mathbb{Z} o A^mathbb{Z}$ is a transformation of the full shift defined via a finite neighborhood $S subset mathbb{Z}$ and a local function $mu : A^S o A$. We study the family of cellular automata whose finite neighborhood $S$ is an interval containing $0$, and there exists a pattern $p in A^S$ satisfying that $mu(z) = z(0)$ if and only if $z eq p$; this means that these cellular automata have a unique emph{active transition}. Despite its simplicity, this family presents interesting and subtle problems, as the behavior of the cellular automaton completely depends on the structure of $p$. We show that every cellular automaton $ au$ with a unique active transition $p in A^S$ is either idempotent or strictly almost equicontinuous, and we completely characterize each one of these situations in terms of $p$. In essence, the idempotence of $ au$ depends on the existence of a certain subpattern of $p$ with a translational symmetry.

Problem

• One-Dimensional Cellular Automata
• Idempotency Conditions
• Model Behavior Characteristics

Innovation

• One-Dimensional Cellular Automata
• Idempotency and Almost Invariance
• Pattern-based Prediction