An upper bound on the number of states for a strongly universal hyperbolic cellular automaton on the pentagrid

TL;DR

This paper establishes an upper bound of 9 states for a rotation-invariant hyperbolic cellular automaton on the pentagrid with an undecidable halting problem, advancing understanding of automaton complexity in hyperbolic spaces.

Contribution

It provides a concrete upper bound of 9 states for such automata, extending previous work and demonstrating undecidability in this context.

Findings

Existence of a 9-state hyperbolic cellular automaton

Automaton is rotation invariant

Halting problem is undecidable for this automaton

Abstract

In this paper, following the way opened by a previous paper deposited on arXiv, we give an upper bound to the number of states for a hyperbolic cellular automaton in the pentagrid. Indeed, we prove that there is a hyperbolic cellular automaton which is rotation invariant and whose halting problem is undecidable and which has 9~states.