Clandestine Simulations in Cellular Automata

This paper studies two kinds of simulation between cellular automata: simulations based on factor and simulations based on sub-automaton. We show that these two kinds of simulation behave in two opposite ways with respect to the complexity of attractors and factor subshifts. On the one hand, the factor simulation preserves the complexity of limits sets or column factors (the simulator CA must have a higher complexity than the simulated CA). On the other hand, we show that any CA is the sub-automaton of some CA with a simple limit set (NL-recognizable) and the sub-automaton of some CA with a simple column factor (finite type). As a corollary, we get intrinsically universal CA with simple limit sets or simple column factors. Hence we are able to ‘hide’ the simulation power of any CA under simple dynamical indicators.

💡 Research Summary

The paper investigates two distinct notions of simulation between cellular automata (CA): factor‑based simulation and sub‑automaton‑based simulation. A factor simulation exists when there is a surjective, continuous factor map π from the configuration space of a simulator CA B onto that of a simulated CA A such that π∘B = A∘π. Because π projects every trajectory of B onto a trajectory of A, dynamical invariants that are defined via language‑theoretic descriptions—namely the limit set Ω and column factors ℂ—cannot become simpler under a factor simulation. The authors prove that if the simulated CA has a limit set of a certain complexity class (e.g., NL‑recognizable) or column factors of a given class (e.g., shift of finite type), then any CA that factors onto it must belong to at least the same class. In other words, factor simulation preserves or raises complexity; the simulator must be at least as “hard” as the simulated system.

In contrast, a sub‑automaton simulation embeds the state set of the simulated CA as a subset of the simulator’s state set and forces the simulator to behave exactly like the simulated CA on that subset, while allowing arbitrary behavior elsewhere. The authors construct, for every CA T, a simulator CA S whose limit set is NL‑recognizable and another simulator whose column factors are of finite type, yet T appears as a sub‑automaton of S. The construction proceeds by isolating a “core” region that reproduces T’s dynamics and surrounding it with a “clean” region whose evolution forces the global limit set or column factors into a simple language class. Consequently, the complex dynamics of T are hidden inside S, while external observers see only a simple dynamical signature.

From these constructions the paper derives several corollaries. First, intrinsically universal CA—those capable of simulating any other CA—can be built with extremely simple dynamical indicators (simple limit sets or simple column factors). Second, dynamical complexity measures such as limit‑set complexity or column‑factor complexity are insufficient to detect the full computational power of a CA, because powerful simulations can be concealed beneath trivial observable behavior. Third, the results highlight a duality: factor simulations enforce a monotone increase of complexity, whereas sub‑automaton simulations allow arbitrary complexity to be embedded under a façade of simplicity.

The authors discuss the implications for classification theory, suggesting that traditional language‑based invariants must be complemented by other dynamical tools (entropy, measure‑theoretic properties, etc.) to capture simulation capabilities. They also point out potential applications in cryptography and secure computation, where one might wish to embed complex computation within a system that appears simple to an external analyst. Finally, the paper proposes future work on extending the hiding technique to other dynamical invariants and exploring concrete physical or computational models where such concealed simulations could be realized.

Overall, the work provides a nuanced view of simulation in cellular automata, demonstrating that while factor simulations preserve complexity, sub‑automaton simulations can completely mask it, thereby reshaping our understanding of how computational power can be reflected—or hidden—in the observable dynamics of discrete dynamical systems.