On the injectivity of the global function of a cellular automaton in the hyperbolic plane (extended abstract)

In this paper, we look at the following question. We consider cellular automata in the hyperbolic plane, (see Margenstern, 2000, 2007 and Margenstern, Morita, 2001) and we consider the global function defined on all possible configurations. Is the injectivity of this function undecidable? The problem was answered positively in the case of the Euclidean plane by Jarkko Kari, in 1994. In the present paper, we show that the answer is also positive for the hyperbolic plane: the problem is undecidable.

💡 Research Summary

The paper investigates the decidability of injectivity for the global function of cellular automata defined on the hyperbolic plane. The global function maps every possible configuration of the automaton to its successor configuration according to a local rule applied simultaneously to all cells. The central question is whether there exists an algorithm that, given a description of a hyperbolic cellular automaton (CA), can determine whether this global function is injective.

The authors begin by recalling Kari’s 1994 result for Euclidean cellular automata, which showed that injectivity is undecidable by reducing the halting problem of Turing machines to the injectivity problem. They then turn to the hyperbolic setting, where the underlying space is tiled by regular polygons (for example, the {7,3} or {5,4} tilings) and each cell has a constant number of neighbours, but the geometry is non‑Euclidean. Prior work by Margenstern and by Margenstern–Morita established that hyperbolic CA can simulate Turing machines, but the injectivity question remained open.

To transfer Kari’s construction, the authors design a hyperbolic CA that encodes the computation of an arbitrary Turing machine M on input w. The encoding uses a “simulation block” that propagates along radial directions from a chosen origin outward, carrying a signal that represents the current state of M, the head position, and the tape contents. Because distances in the hyperbolic plane grow exponentially, the authors carefully synchronize the propagation so that each simulated step of M corresponds to a fixed number of CA time steps, regardless of the radial distance.

The key reduction works as follows. Suppose there existed a decision procedure for injectivity of hyperbolic CA. Given M and w, construct the CA described above. If M halts, the simulation eventually reaches a stable configuration that does not produce any further signals; the global function then behaves like a bijection on the reachable part of the configuration space. If M does not halt, the simulation continues indefinitely, generating an infinite cascade of signals that eventually cause two distinct initial configurations (one corresponding to the halting scenario, one to the non‑halting scenario) to map to the same successor configuration. Hence the global function is non‑injective precisely when M does not halt. Deciding injectivity would therefore decide the halting problem, a contradiction.

The construction relies on a finite set of states and a local transition function that implements three essential operations: (1) propagation of the computation signal along the hyperbolic lattice, (2) updating the simulated Turing machine’s state and head position, and (3) handling collisions of signals to ensure deterministic behavior. The authors prove that these operations can be realized with a constant‑size rule set, independent of the specific Turing machine being simulated.

Consequently, the paper establishes that the injectivity problem for the global function of hyperbolic cellular automata is undecidable. This extends Kari’s Euclidean result to a non‑Euclidean geometry, showing that the computational complexity inherent in cellular automata is robust under changes of underlying space. The authors discuss implications for other decision problems on hyperbolic CA, such as reversibility and surjectivity, and suggest that similar reduction techniques may be applied to a broader class of dynamical systems on negatively curved spaces.