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

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 known results from Euclidean to hyperbolic geometries.

Contribution

It establishes the undecidability of injectivity for cellular automata in the hyperbolic plane, generalizing previous Euclidean plane results.

Findings

Injectivity problem is undecidable in the hyperbolic plane

Extends Euclidean plane results to hyperbolic geometry

Supports the universality of undecidability in cellular automata

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.