Intrinsic Simulations between Stochastic Cellular Automata

TL;DR

This paper introduces a formal framework for intrinsic simulation in stochastic cellular automata, extending previous concepts to non-deterministic and stochastic cases, and explores their universality properties.

Contribution

It develops a unifying formalism for stochastic cellular automata and extends intrinsic simulation notions, providing tools to analyze their universality and simulation capabilities.

Findings

Existence of a universal non-deterministic cellular automaton.

No universal stochastic cellular automaton exists.

Stochastic cellular automata can achieve partial universality.

Abstract

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.