[Paper Review] The Garden of Eden theorem: old and new

Tullio Ceccherini‐Silberstein, Michel Coornaert|arXiv (Cornell University)|Jul 27, 2017

Cellular Automata and Applications32 references3 citations

TL;DR

This paper provides a comprehensive survey of the Garden of Eden theorem and its generalizations in cellular automata and dynamical systems. It establishes that for amenable groups, surjectivity of a cellular automaton is equivalent to pre-injectivity—extending the classical Moore-Myhill theorem beyond ℤᵈ to all amenable groups, with key results on entropy, homoclinicity, and algebraic mean dimension.

ABSTRACT

We review topics in the theory of cellular automata and dynamical systems that are related to the Moore-Myhill Garden of Eden theorem.

Motivation & Objective

• To unify and extend the classical Garden of Eden theorem beyond ℤᵈ to general amenable groups.
• To clarify the role of amenability in determining whether surjectivity implies pre-injectivity in cellular automata.
• To generalize the theorem to subshifts, algebraic dynamical systems, and hyperbolic dynamical systems.
• To explore the interplay between entropy, mean dimension, and the existence of Garden of Eden configurations.
• To provide a self-contained reference linking cellular automata, group theory, and dynamical systems through the lens of the Garden of Eden theorem.

Proposed method

• Uses Følner nets and entropy-based arguments to generalize the Garden of Eden theorem to countable amenable groups.
• Applies the Curtis-Hedlund-Lyndon theorem to characterize cellular automata via continuity and equivariance.
• Introduces algebraic mean dimension as a key invariant for algebraic cellular automata over amenable groups.
• Employs the notion of stable spaces with bounded propagation and dense holonomy in Gromov’s generalization to simplicial graphs.
• Leverages homoclinicity and strong irreducibility to extend results to subshifts and expansive systems.
• Utilizes the concept of pre-injectivity and mutually erasable patterns to characterize surjectivity.

Experimental results

Research questions

• RQ1 For which groups does the Garden of Eden theorem—surjectivity if and only if pre-injectivity—hold?
• RQ2 How does entropy and mean dimension relate to the existence of Garden of Eden configurations in cellular automata?
• RQ3 Can the Garden of Eden theorem be extended to algebraic cellular automata and principal algebraic dynamical systems?
• RQ4 What is the role of amenability in the failure or validity of the Garden of Eden theorem for nonamenable groups?
• RQ5 To what extent can Gromov’s generalization of the theorem be applied to stable spaces over amenable simplicial graphs?

Key findings

• The Garden of Eden theorem holds for all amenable groups, and fails for nonamenable groups, as shown by Bartholdi and Kielak.
• For amenable groups, surjectivity of a cellular automaton is equivalent to pre-injectivity, generalizing the original Moore-Myhill result on ℤᵈ.
• In algebraic cellular automata over amenable groups, surjectivity is equivalent to (* careful) pre-injectivity, as formalized via algebraic mean dimension.
• Gromov’s generalization extends the theorem to stable spaces of bounded propagation on amenable simplicial graphs with bounded degree and dense pseudogroups of partial isometries.
• The theorem holds for strongly irreducible, finite-type subshifts with equal entropy under maps of bounded propagation and dense holonomy.
• The class of groups satisfying the Garden of Eden theorem is precisely the class of amenable groups, resolving a long-standing question posed by Schupp.