Applying Causality Principles to the Axiomatization of Probabilistic Cellular Automata

Cellular automata (CA) consist of an array of identical cells, each of which may take one of a finite number of possible states. The entire array evolves in discrete time steps by iterating a global evolution G. Further, this global evolution G is required to be shift-invariant (it acts the same everywhere) and causal (information cannot be transmitted faster than some fixed number of cells per time step). At least in the classical [13], reversible [17] and quantum cases [1], these two top-down axiomatic conditions are sufficient to entail more bottom-up, operational descriptions of G. We investigate whether the same is true in the probabilistic case.