Cellular Automata: Analysis and Applications

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This book offers a comprehensive overview of the main approaches for analyzing cellular automata, a crucial tool in mathematical modeling. Unlike classical methods such as partial differential equations, cellular automata are easier to simulate but challenging to analyze. The text reviews various theories that enhance understanding of cellular automata beyond mere simulations. The first section introduces cellular automata on Cayley graphs, characterized through the fundamental Cutis-Hedlund-Lyndon theorems within different topological frameworks (Cantor, Besicovitch, and Weyl topology). The second part delves into classification results derived from topological concepts (Hurley classification), Lyapunov stability (Gilman classification), and formal languages and grammars (Kůrka classification). These classifications suggest a clustering of cellular automata akin to the categorization of partial differential equations into hyperbolic, parabolic, and elliptic types. This section culminates in exploring the decidability of cellular automata properties, examining surjectivity and injectivity, and discussing the Garden of Eden theorems. The third part analyzes cellular automata with distinct properties, often linked to mathematical modeling of biological, physical, or chemical systems. The concept of linearity is utilized to define self-similar limit sets. Models for particle motion illustrate connections between cellular autom