Cellular Automata on Probability Measures

2025-03-19

AI Summary

Traditional cellular automata (CA) assume deterministic cell states, limiting their ability to model complex systems with intrinsic uncertainty. To address this, we propose Cellular Automata on Measure spaces (CAM), where each cell’s state is defined as a probability measure—enabling a unified framework for representing spatially heterogeneous uncertainty. CAM generalizes CA to arbitrary probability measure spaces, subsuming classical cases such as Bernoulli measures, and rigorously establishes its mathematical foundations by integrating measure theory, symbolic dynamics, and stochastic graph modeling. Theoretical analysis uncovers novel connections between CAM and symbolic dynamics, particularly through measure-preserving transformations and shift-invariant measures. Numerical experiments demonstrate CAM’s effectiveness in modeling the evolution of stochastic graphs, capturing both structural randomness and temporal dynamics. By embedding uncertainty directly into the state space, CAM substantially extends the expressive power of classical CA and provides a principled new paradigm for dynamic representation of uncertain complex systems.

Technology Category

• Reasoning under Uncertainty: Uncertainty Representations
• Multiagent Systems: Multiagent Systems under Uncertainty
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

Classical Cellular Automata (CCAs) are a powerful computational framework widely used to model complex systems driven by local interactions. Their simplicity lies in the use of a finite set of states and a uniform local rule, yet this simplicity leads to rich and diverse dynamical behaviors. CCAs have found applications in numerous scientific fields, including quantum computing, biology, social sciences, and cryptography. However, traditional CCAs assume complete certainty in the state of all cells, which limits their ability to model systems with inherent uncertainty. This paper introduces a novel generalization of CCAs, termed Cellular Automata on Measures (CAMs), which extends the classical framework to incorporate probabilistic uncertainty. In this setting, the state of each cell is described by a probability measure, and the local rule operates on configurations of such measures. This generalization encompasses the traditional Bernoulli measure framework of CCAs and enables the study of more complex systems, including those with spatially varying probabilities. We provide a rigorous mathematical foundation for CAMs, demonstrate their applicability through concrete examples, and explore their potential to model the dynamics of random graphs. Additionally, we establish connections between CAMs and symbolic dynamics, presenting new avenues for research in random graph theory. This study lays the groundwork for future exploration of CAMs, offering a flexible and robust framework for modeling uncertainty in cellular automata and opening new directions for both theoretical analysis and practical applications.

Problem

• Extends Cellular Automata to model probabilistic uncertainty
• Introduces Cellular Automata on Measures (CAMs) framework
• Enables modeling of systems with spatially varying probabilities

Innovation

• Extends Cellular Automata with probabilistic uncertainty
• Introduces Cellular Automata on Measures (CAMs)
• Models systems with spatially varying probabilities