Intrinsic Simulations between Stochastic Cellular Automata

The paper proposes a simple formalism for dealing with deterministic, non-deterministic and stochastic cellular automata in a unifying and composable manner. Armed with this formalism, we extend the notion of intrinsic simulation between deterministic cellular automata, to the non-deterministic and stochastic settings. We then provide explicit tools to prove or disprove the existence of such a simulation between two stochastic cellular automata, even though the intrinsic simulation relation is shown to be undecidable in dimension two and higher. The key result behind this is the caracterization of equality of stochastic global maps by the existence of a coupling between the random sources. We then prove that there is a universal non-deterministic cellular automaton, but no universal stochastic cellular automaton. Yet we provide stochastic cellular automata achieving optimal partial universality.

💡 Research Summary

The paper introduces a unified formalism that simultaneously captures deterministic, non‑deterministic, and stochastic cellular automata (CA). By representing each model through a global map—a deterministic function, a set of possible functions, or a probability distribution over functions—the authors obtain a composable framework in which the random source of a stochastic CA is treated as an explicit component. Within this framework they extend the notion of intrinsic simulation (originally defined only for deterministic CA) to the non‑deterministic and stochastic settings. An intrinsic simulation is realized by a simulation morphism consisting of three parts: a spatial encoding that maps blocks of cells of the simulated CA to single cells of the simulator, a temporal encoding that compresses several steps of the simulated CA into one step of the simulator, and a state encoding that translates the alphabet of the simulated CA into that of the simulator.

A central technical contribution is the Coupling Theorem: two stochastic CA implement the same global map if and only if there exists a coupling between their random sources that makes their joint evolution produce identical distributions on configurations. This theorem reduces the problem of checking equality of stochastic global maps to the existence of a suitable joint distribution, a problem that can be tackled with standard coupling or optimal transport techniques. Consequently, the authors provide concrete tools—algorithmic constructions of couplings, analysis of the random source structure, and verification procedures for simulation morphisms—that allow researchers to prove or disprove the existence of an intrinsic simulation between any pair of stochastic CA.

The paper also establishes fundamental limits. By encoding the halting problem of a Turing machine into a two‑dimensional CA, the authors prove that the intrinsic simulation relation is undecidable in dimension two and higher. This result holds for all three classes (deterministic, non‑deterministic, stochastic) and shows that no general algorithm can decide whether a given CA can simulate another.

Regarding universality, the authors construct a universal non‑deterministic CA: a single CA that, via appropriate simulation morphisms, can intrinsically simulate every non‑deterministic CA. The construction mirrors the classic universal deterministic CA but incorporates a nondeterministic transition set that can reproduce any nondeterministic behavior after suitable encoding. In contrast, they prove that no universal stochastic CA exists. The proof hinges on the diversity of stochastic global maps and the coupling requirement: a single stochastic CA cannot provide couplings for all possible random sources, so it cannot simulate every stochastic CA.

Nevertheless, the authors identify a notion of optimal partial universality for stochastic CA. They exhibit specific stochastic CA that are universal for well‑defined subclasses—e.g., all CA whose global maps correspond to a fixed family of Markov chains or to a bounded set of probability distributions. These partially universal models achieve the best possible coverage given the impossibility of full universality.

The paper concludes by outlining future directions: improving the computational complexity of coupling detection, extending partial universality results to richer classes of stochastic dynamics, and exploring connections between stochastic CA and other probabilistic computation models such as probabilistic Turing machines or quantum cellular automata. Overall, the work provides a rigorous foundation for comparing and classifying cellular automata across deterministic, nondeterministic, and stochastic domains, and it supplies practical methods for establishing simulation relationships despite the inherent undecidability in higher dimensions.