The Garden of Eden theorem: old and new
Tullio Ceccherini-Silberstein, Michel Coornaert
Abstract
We review topics in the theory of cellular automata and dynamical systems that are related to the Moore-Myhill Garden of Eden theorem.
Table of Contents
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The Garden of Eden theorem: old and new
Abstract
We review topics in the theory of cellular automata and dynamical systems that are related to the Moore-Myhill Garden of Eden theorem.
Paper Structure
This paper contains 39 sections, 32 theorems, 148 equations.
Table of Contents
- Introduction
- Configuration spaces and shifts
- Notation
- Configurations spaces
- Group actions
- Shifts
- Patterns
- Subshifts
- Cellular automata
- Definition
- The Curtis-Hedlund-Lyndon theorem
- Operations on cellular automata
- Surjectivity of cellular automata, GOE configurations, and GOE patterns
- Pre-injectivity of cellular automata and mutually erasable patterns
- The Garden of Eden theorem for $\mathbb{Z}^d$
- ...and 24 more sections
Key Result
Theorem 3.8
Let $G$ be a countable group and let $A,B$ be finite sets. Let $\tau \colon X \to Y$ be a map from a subshift $X \subset A^G$ into a subshift $Y \subset B^G$. Then the following conditions are equivalent:
Theorems & Definitions (111)
- Example 2.1
- Example 2.2
- Example 2.3
- Example 2.4
- Example 2.5
- Remark 2.6
- Definition 3.1
- Example 3.2
- Example 3.3: Majority vote
- Remark 3.4
- ...and 101 more