Applying causality principles to the axiomatization of probabilistic cellular automata

TL;DR

This paper explores whether the axiomatic principles of causality and shift-invariance in cellular automata extend to probabilistic cases, aiming to formalize their global evolution through bottom-up operational descriptions.

Contribution

It investigates the applicability of causality principles to probabilistic cellular automata, extending axiomatic characterizations beyond classical and quantum cases.

Findings

Causality and shift-invariance imply operational descriptions in classical and quantum CA.

Probabilistic CA require additional conditions for similar axiomatization.

The study advances understanding of causal structures in stochastic cellular automata.

Abstract

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, reversible and quantum cases, 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. Keywords: Characterization, noise, Markov process, stochastic Einstein locality, screening-off, common cause principle, non-signalling, Multi-party non-local box.