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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.
List of contents
Deterministic cellular automata.- Deterministic initial value problem.- Multiplicative and additive rules.- More complex rules.- Exploiting rule identities.- Rules with additive invariants.- Construction of probability measures.- Probabilistic solutions.- Probabilistic cellular automata.- Applications.- Local structure approximation.- Non-solvable rules.- Appendices.
About the author
Henryk Fuk¿ 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.
Report
The book is carefully written with thorough explanations, and could be used as a resource for calculations associated with elementary cellular automata or self-study of an interesting explicit viewpoint of cellular automata. (Thomas B. Ward, zbMATH 1552.37001, 2025)
