On Cellular Automata
Tawfiq Hamed, Mohammad Saleh
TL;DR
The paper generalizes cellular automata by modeling the universe as a group and introducing $\\phi$-cellular automata to bridge CA across group homomorphisms. It develops a prodiscrete uniform framework, proves a generalized Uniform Curtis-Hedlund theorem for $\\phi$-CA, and extends the theory to linear alphabets, decomposition via homomorphism factorization, and covering-map constructions on circulant graphs. Key contributions include precise $\\phi$-CA definitions, uniform-continuity characterizations, a decomposition theorem, a linear CA extension, and a covering-map framework that yields injective $\\phi$-CA for circulant covers. These results deepen the algebraic and topological foundations of CA on group-based and circulant-graph structures and open avenues for studying more general covers and dynamical properties in graph-driven CA systems.
Abstract
Cellular automata are a fundamental computational model with applications in mathematics, computer science, and physics. In this work, we explore the study of cellular automata to cases where the universe is a group, introducing the concept of \( φ\)-cellular automata. We establish new theoretical results, including a generalized Uniform Curtis-Hedlund Theorem and linear \( φ\)-cellular automata. Additionally, we define the covering map for \( φ\)-cellular automata and investigate its properties. Specifically, we derive results for quotient covers when the universe of the automaton is a circulant graph. This work contributes to the algebraic and topological understanding of cellular automata, paving the way for future exploration of different types of covers and their applications to broader classes of graphs and dynamical systems.