A family of weakly universal cellular automata in the hyperbolic plane with two states

TL;DR

This paper introduces a family of weakly universal cellular automata with two states in the hyperbolic plane, applicable to various grids, with a focus on planarity and universality for different p-values.

Contribution

It constructs weakly universal, rotation-invariant cellular automata with two states for all grids p,3 in the hyperbolic plane, expanding the known universality results.

Findings

Applicable to all grids p,3 with p geq 13

The set of changing cells forms a planar set

Established universality for p=13 and general p geq 17

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.