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, 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.

💡 Research Summary

The paper investigates whether the two high‑level axioms that characterize classical cellular automata—shift‑invariance (the rule looks the same everywhere) and causality (information cannot travel faster than a fixed bound)—are sufficient to define a robust notion of probabilistic cellular automata (PCA). In deterministic CA, the Curtis‑Lyndon‑Hedlund theorem tells us that any continuous, shift‑invariant map on the space of configurations is exactly a CA. The authors ask whether the probabilistic analogue holds: are non‑signalling, shift‑invariant stochastic maps precisely the class of PCA?

To answer this, they first formalize configurations, random variables, and stochastic maps, and define causality in the probabilistic setting as the absence of influence between arbitrarily distant regions. They then examine several candidate axioms.

1. Non‑signalling alone – They present two counter‑examples. “Parity” is a stochastic map that enforces a global parity constraint: each output bit is uniformly random but the overall parity is always zero. Although it is translation‑invariant and non‑signalling, it creates instantaneous correlations between distant cells, which cannot be generated by any local mechanism. “Magic‑Coins” is another construction where two far‑apart coin tosses are perfectly correlated while remaining independent of any inputs. Both demonstrate that non‑signalling does not forbid the spontaneous creation of long‑range correlations, so it is insufficient to capture the physical intuition behind PCA.

2. Non‑correlating (no spontaneous correlations) – The authors strengthen the requirement: after one time step, any two disjoint infinite regions must remain statistically independent (their joint state must factor as a tensor product). Parity satisfies this stronger condition, but a new construction called “GenNLBox” shows that even non‑correlating, shift‑invariant, non‑signalling maps can be non‑local. GenNLBox is a generalization of the well‑known non‑local box (NL‑Box) from quantum foundations; it violates Bell inequalities maximally, yet it does not create new correlations beyond those already present in the input. Hence, non‑correlating alone still does not characterize PCA.

3. Common‑cause and screening‑off principles – Inspired by Reichenbach’s principle of common cause, the authors explore whether imposing a screening‑off condition (conditioning on a common cause renders distant variables independent) would close the gap. They formalize this principle for stochastic maps over configurations and test it against the previously introduced counter‑examples. Unfortunately, GenNLBox again satisfies the common‑cause condition while remaining non‑local, showing that even these more sophisticated causality notions fail to rule out unimplementable stochastic dynamics.

The paper also critiques the existing “Standard‑PCA” definition, which models a PCA as a deterministic CA followed by a homogeneous, independent noise step applied to each cell. They point out three major shortcomings: (i) it does not capture all reasonable stochastic maps (Parity is excluded), (ii) the class is not closed under composition (the product of two Standard‑PCA need not be a Standard‑PCA), and (iii) it lacks a principled, high‑level justification.

In summary, the authors conclude that the straightforward translation of the deterministic CA axioms to the probabilistic realm does not work. Neither non‑signalling, nor non‑correlating, nor even the common‑cause principle suffices to single out a physically meaningful class of PCA. The paper thus calls for a deeper, perhaps entirely new, set of axioms that simultaneously guarantee shift‑invariance, a robust notion of causality, compositional closure, and an operational (local) implementation. Future work is suggested to develop such a framework, possibly by integrating insights from quantum information theory, stochastic processes, and the foundations of causality.