The injectivity of the global function of a cellular automaton in the hyperbolic plane is undecidable

TL;DR

This paper proves that determining whether the global function of a cellular automaton in the hyperbolic plane is injective is an undecidable problem, extending previous results from the Euclidean plane.

Contribution

It demonstrates that the undecidability of injectivity for cellular automata's global functions also holds in the hyperbolic plane, a significant extension of prior Euclidean results.

Findings

Injectivity problem is undecidable in the hyperbolic plane

Extends Euclidean plane results to hyperbolic geometry

Shows limitations of algorithmic analysis for hyperbolic cellular automata

Abstract

In this paper, we look at the following question. We consider cellular automata in the hyperbolic plane 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.