Conservation laws in cellular automata

If is a discrete Abelian group and a finite set, then a cellular automaton (CA) is a continuous map :→ that commutes with all -shifts. If ϕ:→, then, for any a∊, we define Σϕ(a) = ∑_x∊ϕ(a_x) (if finite); ϕ is conserved by if Σϕ is constant under the action of .

We characterize such conservation laws in several ways, deriving both theoretical consequences and practical tests, and provide a method for constructing all one-dimensional CA exhibiting a given conservation law.