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Solvable Cellular Automata

Methods and Applications

  • Book
  • © 2023

Overview

Authors:
  1. Henryk Fukś

    1. Department of Mathematics and Statistics, Brock University, St. Catharines, Canada

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  • Addresses solvability of cellular automata with many results and applications
  • Provides an appendix with solution formulas, both deterministic and probabilistic, for over 60 cellular automata
  • Provides a quick overview table which are solvable in both deterministic and probabilistic sense

Part of the book series: Understanding Complex Systems (UCS)

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About this book

The main focus of the book is solvability of cellular automata, that is, expressing the state of a given cell after a given number of steps by an explicit formula.  The author considers solutions of two types of initial value problems for cellular automata, the deterministic one and the probabilistic one. In the first chapter the basic concepts of cellular automata theory are introduced. Deterministic initial value problem is introduced next and solutions for selected simple rules are also presented. In the following chapters various techniques for solving the deterministic problem are introduced, using elementary CA rules of increasing complexity as examples. The second part of the book introduces the concept of probability measure in the context of cellular automata and the probabilistic initial value problem for both deterministic and probabilistic rules. The book is amply illustrated with examples and applications such as the density classification problem, phase transitions in traffic models or the diffusion of innovations model. In the appendix, solution formulae (both deterministic and probabilistic) for over 60 elementary cellular automata rules are listed. Ruelle-Frobenius-Perron equations for all 88 minimal elementary cellular automata are also provided. 

Keywords

Table of contents (12 chapters)

  1. Front Matter

    Pages i-xix
    Download chapter PDF

  2. Deterministic Cellular Automata

    • Henryk Fukś
    Pages 1-13

  3. Deterministic Initial Value Problem

    • Henryk Fukś
    Pages 15-26

  4. Multiplicative and Additive Rules

    • Henryk Fukś
    Pages 27-36

  5. More Complex Rules

    • Henryk Fukś
    Pages 37-54

  6. Exploiting Rule Identities

    • Henryk Fukś
    Pages 55-61

  7. Rules with Additive Invariants

    • Henryk Fukś
    Pages 63-91

  8. Construction of the Probability Measures

    • Henryk Fukś
    Pages 93-111

  9. Probabilistic Solutions

    • Henryk Fukś
    Pages 113-152

  10. Probabilistic Cellular Automata

    • Henryk Fukś
    Pages 153-170

  11. Applications

    • Henryk Fukś
    Pages 171-201

  12. Approximate Methods

    • Henryk Fukś
    Pages 203-223

  13. Beyond Solvable Elementary Rules

    • Henryk Fukś
    Pages 225-235

  14. Back Matter

    Pages 237-296
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Authors and Affiliations

  • Department of Mathematics and Statistics, Brock University, St. Catharines, Canada

    Henryk Fukś

About the author

Henryk Fukś  is a Professor at the Department of Mathematics and Statistic of Brock University in St. Catharines, Canada. He received his Ph.D. at the University of Illinois at Chicago in 1997, and afterwards held the NSERC postdoctoral fellowship at the University of Guelph and the Fields Institute for Research in Mathematical Sciences in Toronto. He published over 70 research papers, mostly in the field of cellular automata. In addition to cellular automata, his research interests include complex systems, cooperative phenomena, phase transitions, mathematical modelling as well as history of mathematics.

Bibliographic Information

  • Book Title: Solvable Cellular Automata

  • Book Subtitle: Methods and Applications

  • Authors: Henryk Fukś

  • Series Title: Understanding Complex Systems

  • DOI: https://doi.org/10.1007/978-3-031-38700-5

  • Publisher: Springer Cham

  • eBook Packages: Physics and Astronomy, Physics and Astronomy (R0)

  • Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023

  • Hardcover ISBN: 978-3-031-38699-2Published: 10 October 2023

  • Softcover ISBN: 978-3-031-38702-9Due: 10 November 2023

  • eBook ISBN: 978-3-031-38700-5Published: 09 October 2023

  • Series ISSN: 1860-0832

  • Series E-ISSN: 1860-0840

  • Edition Number: 1

  • Number of Pages: XIX, 296

  • Number of Illustrations: 13 b/w illustrations, 28 illustrations in colour

  • Topics: Complex Systems, Theory of Computation, Complexity, Mathematical Physics

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