Conservation Laws in Cellular Automata

If X is a discrete abelian group and A a finite set, then a cellular automaton (CA) is a continuous map F : A −→ A that commutes with all X-shifts. If φ : A −→ R, then, for any a ∈ A, we define Σφ(a) = ∑ x∈X φ(ax) (if finite); φ is conserved by F if Σφ is constant under the action of F. We characterize such conservation laws in several ways, deriving both theoretical consequences and practical tests, and provide a method for constructing all onedimensional CA exhibiting a given conservation law. If A is a finite set (with discrete topology), and X an arbitrary indexing set, then A (the space of all functions X 7→ A) is compact and totally disconnected in the Tychonoff topology. If (X,+,O) is a discrete abelian group, then X acts on itself by translation; this induces a shift action of X on A: if a = [ax|x∈X] ∈ A , and u ∈ X, then σ(a) = [bx|x∈X], where bx = a(x+u). A cellular automaton (CA) is a continuous map F : A−→ A which commutes with all shifts. The Curtis-Hedlund-Lyndon Theorem [1] says that F is a CA if and only if there is some finite B ⊂ X (a “neighbourhood of the identity”) and a local map f : A −→ A so that, for all a ∈ A and x ∈ X, F(a)x = f ( a|B+x ) . Here, for any W ⊂ X, we define a|W = [aw|w∈W] ∈ A . For example, if X = Z and B = [−B...B], then for any z ∈ Z, a|B+z = [az−B, . . . , az+B]. Without loss of generality, we assume B is symmetric, in the sense that (b ∈ B) ⇐⇒ (−b ∈ B). We assume X is abelian only for expositional simplicity; all results extend easily to nonabelian X.