The Garden of Eden Theorem over Generalized Cellular Automata

The Garden of Eden theorem is a fundamental result in the theory of cellular automata, which establishes a necessary and sufficient condition for the surjectivity of a cellular automaton with a finite alphabet over an amenable group. Specifically, the theorem states that such an automaton is surjective if and only if it is pre-injective, where pre-injectivity requires that any two almost equal configurations with the same image under the automaton must be equal. This paper focuses on establishing the Garden of Eden theorem over a \(\varphi \)-cellular automaton by demonstrating both Moore theorem and Myhill theorem over \(\varphi \)-cellular automata are true. These results have significant implications for the theoretical framework of the Garden of Eden theorem and its applicability across diverse groups or altered versions of the same group. Overall, this paper provides a more comprehensive study of \(\varphi \)-cellular automata and extends the Garden of Eden theorem to a broader class of automata.