Decision problems on geometric tilings

TL;DR

This paper investigates decision problems in geometric tilings, proving undecidability results for variants of the Domino problem with arbitrary shapes and for the finite local complexity property, extending existing theoretical boundaries.

Contribution

It extends undecidability results to geometric tiles of arbitrary shape and to the property of finite local complexity in simple tiling settings.

Findings

Undecidability of the geometric Domino problem with arbitrary shapes

Undecidability of finite local complexity in simple geometric tilings

Results hold even under fixed tile sets and minimal modifications

Abstract

We study decision problems on geometric tilings. First, we study a variant of the Domino problem where square tiles are replaced by geometric tiles of arbitrary shape. We show that this variant is undecidable regardless of the shapes, extending previous results on rhombus tiles. This result holds even when the geometric tiling is forced to belong to a fixed set. Second, we consider the problem of deciding whether a geometric subshift has finite local complexity, which is a common assumption when studying geometric tilings. We show that this problem is undecidable even in a simple setting (square shapes with small modifications).