Cellular Automata: Analysis and Applications
by Karl-Peter Hadeler (Author), Johannes Müller (Author)

About the Author
Karl Peter Hadeler, Dr.rer.nat. 1965  (U. of Hamburg), Habilitation 1967 (U. of Hamburg). In 1963/1964 visiting Moscow State University (MGU), 1968/1969 Visiting Associate Professor,U. of Minnesota. 1970 Associate Professor, Technical Department, U. of Erlangen. 1971 Professor of Mathematics, U. of Tübingen. Retired 2005, then 2005-2011 Non-permanent Professor, Arizona State University. Visiting Professor Aarhus, Nijmegen, Georgia Tech, Emory. 2009 John von Neumann Professorship, Technical University of Munich. Member of Center of Excellence (DFG/German NSF). Research interests: Ordinary and partial differential equations (reaction diffusion equations), delay equations, matrix theory, mathematical biology. Since 2011 about ten publications in mathematics.
Johannes Müller studied in Karlsruhe and Tübingen, where he did his habilitation in 2001. After stays in Utrecht and Cologne, he became head of a research group in the Institute for Biomathematics and Biometry in the Helmholtz Center, Munich. Since 2004 he is teaching as a professor at the Technische Universität München. The research interests of Johannes Müller is on the interface of mathematics and life sciences. In particular his research is concerned with the theory of dynamical systems, cellular automata, and stochastic processes respectively their application.

About this book
This book provides an overview of the main approaches used to analyze the dynamics of cellular automata. Cellular automata are an indispensable tool in mathematical modeling. In contrast to classical modeling approaches like partial differential equations, cellular automata are relatively easy to simulate but difficult to analyze. In this book we present a review of approaches and theories that allow the reader to understand the behavior of cellular automata beyond simulations. The first part consists of an introduction to cellular automata on Cayley graphs, and their characterization via the fundamental Cutis-Hedlund-Lyndon theorems in the context of various topological concepts (Cantor, Besicovitch and Weyl topology). The second part focuses on classification results: What classification follows from topological concepts (Hurley classification), Lyapunov stability (Gilman classification), and the theory of formal languages and grammars (Kůrka classification)? These classifications suggest that cellular automata be clustered, similar to the classification of partial differential equations into hyperbolic, parabolic and elliptic equations. This part of the book culminates in the question of whether the properties of cellular automata are decidable. Surjectivity and injectivity are examined, and the seminal Garden of Eden theorems are discussed. In turn, the third part focuses on the analysis of cellular automata that inherit distinct properties, often based on mathematical modeling of biological, physical or chemical systems. Linearity is a concept that allows us to define self-similar limit sets. Models for particle motion show how to bridge the gap between cellular automata and partial differential equations (HPP model and ultradiscrete limit). Pattern formation is related to linear cellular automata, to the Bar-Yam model for the Turing pattern, and Greenberg-Hastings automata for excitable media.

Table of contents
1 Introduction 1
    1. 1 Discreteness 1
    1. 2 The Game of Life 3
    1. 3 Contact Automata 4
    1. 4 Some Wolfram Automata 6
    1. 5 Greenberg-Hastings Automata 10
    1. 6 Langton’s Ant and Life Without Death 11
    1. 7 A Nice Little Automaton 12
    1. 8 History and Applications 13
    1. 9 Outline of This Work 13
2 Cellular Automata: Basic Definitions 19
    2. 1 The Grid 19
        2. 1. 1 Abelian or Regular Grids 20
        2. 1. 2 Non-Abelian Grids 22
    2. 2 The Neighborhood 26
    2. 3 Elementary State and the Global State 28
    2. 4 The Local and the Global Function 30
    2. 5 Excursion: The Growth Function of a Cayley Graph 33
3 Cantor Topology of Cellular Automata 37
    3. 1 Prelude: Cantor Sets and Cantor Spaces 38
        3. 1. 1 The Classical Mid-Third Cantor Set 38
        3. 1. 2 Cantor Spaces 42
    3. 2 Cantor Metric for Cellular Automata 45
    3. 3 The Curtis-Hedlund-Lyndon Theorem 48
    3. 4 Spatial Structure and Simplifications 52
        3. 4. 1 Examples: Structures That Are Not Cellular Automata 57
        3. 4. 2 Simplification of the State Space 60
        3. 4. 3 Simplification of the Neighborhood 61
        3. 4. 4 Simplification of the Grid 62
    3. 5 Cellular Automata and Continuous Maps on Cantor Spaces 65
        3. 5. 1 Bijective Maps 66
        3. 5. 2 General Maps: The Universal Cellular Automaton 67
4 Besicovitch and Weyl Topologies 75
    4. 1 Definition of the Besicovitch and Weyl Space 75
    4. 2 Topological Properties 81
        4. 2. 1 Besicovitch Spaces 82
        4. 2. 2 Weyl Spaces 90
    4. 3 Cellular Automata on Besicovitch and Weyl Spaces 98
    4. 4 A CHL Theorem for Besicovitch and Weyl Spaces 103
5 Attractors 111
    5. 1 Dynamical Systems, ω -Limit Sets and Attractors  112
        5. 1. 1 Dynamical Systems 112
        5. 1. 2 ω -Limit Sets and Attractors  114
    5. 2 Structure of Attractors: Finite Grids 119
    5. 3 Intersection of Attractors and Quasi-Attractors 119
    5. 4 Conleys Decomposition Theorem, Attractors, and Chains 125
    5. 5 Bernoulli Measure on Cellular Automata 132
    5. 6 Structure of Attractors—Infinite Grids: Hurley Classification 140
6 Chaos and Lyapunov Stability 155
    6. 1 Topological Chaos 155
    6. 2 Permuting Cellular Automata 159
        6. 2. 1 Surjective Cellular Automata 160
        6. 2. 2 Topological Transitivity 165
        6. 2. 3 Denseness of Periodic Points 166
    6. 3 Lyapunov Stability and Gilman Classification 169
        6. 3. 1 Class Gilman 1 171
        6. 3. 2 Class Gilman 2 173
        6. 3. 3 Class Gilman 3 174
        6. 3. 4 Class Gilman 4 175
7 Language Classification of Kůrka 179
    7. 1 Grammar 179
    7. 2 Finite Automata 181
    7. 3 Finite Automata and Regular Languages 184
    7. 4 Cellular Automata and Language: Kůrka Classification 186
        7. 4. 1 Class Kůrka 1 190
        7. 4. 2 Class Kůrka 2 191
        7. 4. 3 Kůrka 3 194
8 Turing Machines, Tiles, and Computability 197
    8. 1 Turing Machines 197
    8. 2 Universal Turing Machine 201
    8. 3 Computational Universality of Cellular Automata 205
    8. 4 Undecidable Problems 207
        8. 4. 1 The Paradox of Epimenides 207
        8. 4. 2 Russel’s Paradox 208
        8. 4. 3 Richard’s Paradox 209
        8. 4. 4 The Word Problem 210
        8. 4. 5 The Halting Problem 211
        8. 4. 6 The Immortality Problem 213
        8.4.7 Non-computability of ω -Limit Sets for Cellular Automata  213
    8. 5 Tiles 214
        8. 5. 1 Definitions and Examples 214
        8. 5. 2 Tessellations of Free Groups 217
        8. 5. 3 Aperiodic Tessellations on Z 2  220
        8. 5. 4 Undecidability of the Domino Problem in
        8. 5. 5 Undecidability of the Finite Domino Problem in
        8. 5. 6 Group or Graph 246
        8. 5. 7 Domino Problem and Monadic Second Order Logic 249
9 Surjectivity and Injectivity of Global Maps 253
    9. 1 The Garden of Eden 254
    9. 2 Algorithms for One-Dimensional Cellular Automata 262
        9. 2. 1 Stationary Points 262
        9. 2. 2 Surjectivity 267
        9. 2. 3 Injectivity and Bijectivity 275
    9. 3 Undecidability Higher Dimensional Cellular Automata 279
        9. 3. 1 Stationary Points 280
        9. 3. 2 Surjectivity 280
10 Linear Cellular Automata 287
    10. 1 Representation of Linear Cellular Automata 287
    10. 2 Surjectivity, Injectivity and Bijectivity 292
    10. 3 Fractal Sets and Linear Cellular Automata 296
        10. 3. 1 Introductory Example and the Fermat Property 297
        10. 3. 2 Limit Sets of Linear Cellular Automata 300
        10. 3. 3 Iterated Function Systems 307
        10. 3. 4 Matrix Substitution Systems 314
        10. 3. 5 Cellular Automata and Matrix Substitution Systems 324
11 Particle Motion 335
    11. 1 Particle Motion: Formal Approach 335
        11. 1. 1 Modelling Diffusion by Continuous Models 336
        11. 1. 2 Naive Cellular Automata Models for Diffusion 338
    11. 2 From PDE to Cellular Automata: Ultradiscrete Limit 340
        11. 2. 1 Heat Equation 342
        11. 2. 2 The Burgers Equation 343
        11. 2. 3 Ultradiscrete Limit and Burgers Equation 350
    11. 3 Microscopic Models for Diffusion 354
        11. 3. 1 Straight Movement 355
        11. 3. 2 Lattice Gas Cellular Automata 360
12 Pattern Formation 377
    12. 1 Fractal Mollusc Patterns 377
    12. 2 Turing Pattern 378
        12. 2. 1 Turing-Pattern in Partial Differential Equations 378
        12. 2. 2 Excursion: Hopfield Nets 380
        12. 2. 3 Bar-Yam-Model for Turing Pattern 384
    12. 3 Greenberg-Hastings Model for Excitable Media 386
        12. 3. 1 Definitions 387
        12. 3. 2 The Winding Number 391
        12. 3. 3 The Potential 396
    12. 3. 4 Survival of Configurations 400
13 Applications in Various Areas 405
    13. 1 Sandpile Automata and Self-Organized Criticality 405
    13. 2 Epidemiology 411
        13. 2. 1 Mean Field Approximation 411
        13. 2. 2 SIRS Model and Mean Field Approximation 413
        13. 2. 3 Polynomial Growth: Clustering of Contact Networks 416
    13. 3 Evolution 418
        13. 3. 1 Evolution 419
        13. 3. 2 Spatial Model 420
        13. 3. 3 Heuristic Analysis 423
A Basic Mathematical Tools 427
    A.1 Basic Definitions from Topology 427
    A.2 Basic Algebraic Theory 433
        A.2.1 Group Theory 433
        A.2.2 Ring Theory 438
        A.2.3 Fields 441
    A.3 Basic Measure Theory 442
References 455
Index 463

Series: Springer Monographs in Mathematics
Hardcover: 467 pages
Publisher: Springer; 1st ed. 2017 edition (May 28, 2017)
Language: English
ISBN-10: 3319530429
ISBN-13: 978-3319530420

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