Cellular Automata on Probability Measures

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Main Authors:	Formenti, Enrico, Hafiz, Faizal, Kunze, Amelia, La Torre, Davide
Format:	Preprint
Published:	2025
Subjects:	Cellular Automata and Lattice Gases Information Theory
Online Access:	Acceder al recurso
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author	Formenti, Enrico Hafiz, Faizal Kunze, Amelia La Torre, Davide
author_facet	Formenti, Enrico Hafiz, Faizal Kunze, Amelia La Torre, Davide
contents	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.
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publishDate	2025
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spellingShingle	Cellular Automata on Probability Measures Formenti, Enrico Hafiz, Faizal Kunze, Amelia La Torre, Davide Cellular Automata and Lattice Gases Information Theory 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.
title	Cellular Automata on Probability Measures
topic	Cellular Automata and Lattice Gases Information Theory
url	https://arxiv.org/abs/2503.15086

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