[Paper Review] A family of weakly universal cellular automata in the hyperbolic plane with two states

Maurice Margenstern|arXiv (Cornell University)|Feb 8, 2012

Cellular Automata and Applications13 references5 citations

TL;DR

This paper constructs a family of weakly universal cellular automata on hyperbolic plane grids {p,3} for p ≥ 13, using a planar railway model with only two states. The construction achieves universality via a rotation-invariant rule set that simulates a two-register machine, with the set of evolving cells forming a planar graph, marking a minimal two-state solution in the hyperbolic plane.

ABSTRACT

In this paper, we construct a family of weakly universal rotation invariant cellular automaton for all grids $\{p,3\}$ of the hyperbolic plane for $p\geq 13$. The scheme is general for $p\geq 17$ and for $13\leq p<17$, we give such a cellular automaton for $p=13$, which is enough. Also, an important property of this family is that the set of cells of the cellular automaton which are subject to changes is actually a planar set. The problem for $p<13$ for a truly planar construction is still open. The best result, for $p=7$, is four states and was obtained by the same author.

Motivation & Objective

• To construct a weakly universal cellular automaton in the hyperbolic plane with only two states, achieving universality in a truly planar configuration.
• To extend previous results on universality in hyperbolic cellular automata by reducing the number of states from four to two while preserving planarity.
• To resolve the open problem of planar two-state universality in hyperbolic grids, specifically for {p,3} tilings with p ≥ 13.
• To provide a uniform rule set for p ≥ 17 and a specific construction for p = 13, completing the family for p ≥ 13.
• To demonstrate that the set of evolving cells forms a planar graph with infinitely many cycles, ensuring spatial coherence and avoiding 3D embeddings.

Proposed method

• The railway model is employed to simulate a two-register machine, using tracks, switches (fixed, flip-flop, memory), and signal propagation to encode computation.
• A novel planar crossing mechanism using roundabouts is introduced to replace 3D bridges, enabling two-state universality in the hyperbolic plane.
• The automaton uses a rotation-invariant rule set derived from context-sensitive state transitions, with states encoded as B (black) and W (white).
• Signal propagation is managed via particles moving along tracks, with flash signals from switches used to update memory states and control logic.
• The construction relies on a finite initial configuration that is ultimately periodic along two rays, with infinite support only in the active computation paths.
• Rules are systematically derived for key components (e.g., Z1, D1, H, K) to ensure correct signal transmission and state transitions during computation.

Experimental results

Research questions

• RQ1 Can a weakly universal cellular automaton be constructed in the hyperbolic plane with only two states while maintaining a truly planar structure?
• RQ2 What is the minimal number of states required for universality in a planar hyperbolic cellular automaton, and can this be achieved below four states?
• RQ3 How can signal crossings be implemented in a planar hyperbolic grid without resorting to 3D embeddings or additional states?
• RQ4 For which values of p is a two-state weakly universal cellular automaton possible in the {p,3} tiling of the hyperbolic plane?
• RQ5 Can a uniform rule set be defined for all {p,3} tilings with p ≥ 17, ensuring consistent universality?

Key findings

• A weakly universal cellular automaton with only two states is constructed for all {p,3} tilings of the hyperbolic plane with p ≥ 13.
• The set of cells that change state during computation forms a planar graph with infinitely many cycles, confirming the construction’s spatial planarity.
• For p ≥ 17, a uniform set of rotation-invariant rules is provided, enabling a single rule set to work across all such tilings.
• A specific construction is given for p = 13, which is sufficient to complete the family for p ≥ 13, despite the lack of uniformity at lower p.
• The railway model is successfully adapted to the hyperbolic plane with a novel planar crossing mechanism using roundabouts, avoiding 3D embeddings.
• The construction is verified via a computer program that checks the consistency of the rule set, ensuring correct signal propagation and state transitions.