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Decision problems on geometric tilings

Benjamin Hellouin de Menibus, Victor Lutfalla, Pascal Vanier

TL;DR

Problem: determine decidability of domino tilings and finite local complexity in geometric tilings. Approach: reduce from classical undecidable tiling problems using quasi-isometries that link the integer lattice adjacency to geometric tiling graphs, enabling transfer from symbolic to geometric tilings via symbolic-geometric tilings. Contributions: prove undecidability of the domino problem for arbitrary compact, simply connected shapes and for fixed shape sets, and show that FLC is undecidable in simple geometric settings. Significance: establishes fundamental limits on algorithmic analysis of geometric tilings and clarifies how symbolic methods extend to geometric contexts.

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).

Decision problems on geometric tilings

TL;DR

Problem: determine decidability of domino tilings and finite local complexity in geometric tilings. Approach: reduce from classical undecidable tiling problems using quasi-isometries that link the integer lattice adjacency to geometric tiling graphs, enabling transfer from symbolic to geometric tilings via symbolic-geometric tilings. Contributions: prove undecidability of the domino problem for arbitrary compact, simply connected shapes and for fixed shape sets, and show that FLC is undecidable in simple geometric settings. Significance: establishes fundamental limits on algorithmic analysis of geometric tilings and clarifies how symbolic methods extend to geometric contexts.

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).
Paper Structure (6 sections, 2 theorems, 2 equations, 3 figures)

This paper contains 6 sections, 2 theorems, 2 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Definitions and settings
  3. Computability
  4. Symbolic tilings
  5. Geometric tilings
  6. Symbolic-geometric tiling and the domino problem

Key Result

Theorem 2

The classical domino problem is undecidable and $\Pi_1^0$-complete.

Figures (3)

  • Figure 1: The square-triangle shapeset $\mathcal{T}\xspace_\triangle$.
  • Figure 2: The two forbbiden patterns of $X_\triangle$ up to rotation and reflexion. There are in total $12$ forbidden patterns up to translation.
  • Figure 3: A typical fragment of a tiling in $X_\triangle$.

Theorems & Definitions (12)

  • Definition 1: Classical domino problem kahr1962
  • Theorem 2: berger1966
  • Definition 3: Geometric tilings
  • Remark 4: Shapes and tiles
  • Remark 5: Shapes and polygons
  • Proposition 6
  • Definition 7: Pattern and support
  • Definition 8: Geometric subshifts
  • Definition 9: Valid tiling
  • Definition 10: FLC
  • ...and 2 more