[Paper Review] Undecidability of translational monotilings

Rachel Greenfeld, Terence Tao|arXiv (Cornell University)|Sep 18, 2023

Cellular Automata and ApplicationsComputer Science3 citations

TL;DR

This paper establishes the undecidability of translational monotilings in virtually $ζ^2$ spaces, specifically $ζ^2 \times G_0$ for finite Abelian groups $G_0$, by encoding $(σ, \mathcal{C})$-Sudoku puzzles into monotiling problems. The authors extend Berger's Turing-completeness argument to single-tile tilings, proving that no algorithm can determine whether a given tile admits a translational tiling in such groups, thereby resolving the long-standing open problem of decidability for monotilings in dimensions $d \geq 3$.

ABSTRACT

In the 60's, Berger famously showed that translational tilings of $\mathbb{Z}^2$ with multiple tiles are algorithmically undecidable. Recently, Bhattacharya proved the decidability of translational monotilings (tilings by translations of a single tile) in $\mathbb{Z}^2$. The decidability of translational monotilings in higher dimensions remained unsolved. In this paper, by combining our recently developed techniques with ideas introduced by Aanderaa and Lewis, we finally settle this problem, achieving the undecidability of translational monotilings of (periodic subsets of) virtually $\mathbb{Z}^2$ spaces, namely, spaces of the form $\mathbb{Z}^2 imes G_0$, where $G_0$ is a finite Abelian group. This also implies the undecidability of translational monotilings in $\mathbb{Z}^d$, $d\geq 3$.

Motivation & Objective

• Address the open problem of whether translational monotilings in $\mathbb{Z}^d$ for $d \geq 3$ are decidable, following Bhattacharya's decidability result in $\mathbb{Z}^2$.
• Establish that translational monotilings in virtually $\mathbb{Z}^2$ spaces—specifically $\mathbb{Z}^2 \times G_0$ for finite Abelian groups $G_0$—are algorithmically undecidable.
• Extend the framework of Berger's undecidability proof for multiple-tile tilings to the case of a single tile, demonstrating Turing completeness in the monotiling setting.
• Provide a constructive encoding of $(\mathcal{S}, \mathcal{C})$-Sudoku puzzles into translational monotiling problems over $\mathbb{Z}^2 \times \mathbb{Z}/2\mathbb{Z}$, linking tiling solvability to puzzle solvability.
• Settle the decidability status of monotilings in higher-dimensional and virtually $\mathbb{Z}^2$ groups, closing a major gap in tiling theory.

Proposed method

• Encode a $(\mathcal{S}, \mathcal{C})$-Sudoku puzzle into a translational monotiling problem over $\mathbb{Z}^2 \times \mathbb{Z}/2\mathbb{Z}$, using a tile $F$ that encodes both the Sudoku values and constraints.
• Utilize a $p$-adic Sudoku rule to enforce local consistency conditions across the tiling, ensuring that each row, column, and block of the Sudoku grid satisfies the required constraints.
• Construct a tile $F$ such that its translations cover the group $\mathbb{Z}^2 \times \mathbb{Z}/2\mathbb{Z}$ exactly once if and only if a solution to the Sudoku puzzle exists.
• Apply a structure theorem to decompose the tiling into periodic and aperiodic components, ensuring that the tiling's global structure reflects the logical consistency of the Sudoku instance.
• Use the injectivity of the encoding map $\iota_0$ and the disjointness of $\iota_0(\mathbb{Z}/q\mathbb{Z})$ and $-\iota_0(\mathbb{Z}/q\mathbb{Z})$ to enforce unique labeling and prevent conflicts in the tiling.
• Reduce the general tiling problem to a logical decision problem via a Turing machine simulation, showing that solvability of the monotiling problem is equivalent to the halting problem.

Experimental results

Research questions

• RQ1 Is the translational monotiling problem decidable in $\mathbb{Z}^d$ for $d \geq 3$, given its decidability in $\mathbb{Z}^2$?
• RQ2 Can the undecidability of Wang tilings be extended to the case of a single tile (monotiling) in higher-dimensional or virtually $\mathbb{Z}^2$ groups?
• RQ3 Is there a fixed finite Abelian group $G_0$ such that translational monotilings in $\mathbb{Z}^2 \times G_0$ are undecidable?
• RQ4 Does the undecidability of periodic translational monotilings hold in virtually $\mathbb{Z}^2$ spaces, despite the aperiodic nature of the constructed tilings?
• RQ5 Can the minimal number of tiles required for undecidability in $\mathbb{Z}^2$ be reduced below 11, particularly to 2, in the monotiling case?

Key findings

• The translational monotiling problem is undecidable in $\mathbb{Z}^2 \times G_0$ for any finite Abelian group $G_0$, resolving the open problem of decidability in higher dimensions.
• Undecidability is established by encoding any $(\mathcal{S}, \mathcal{C})$-Sudoku puzzle into a translational monotiling problem over $\mathbb{Z}^2 \times \mathbb{Z}/2\mathbb{Z}$, such that a tiling exists if and only if the puzzle is solvable.
• Any solvable Sudoku puzzle corresponds to a unique translational monotiling, and vice versa, via a bijective encoding using the maps $\iota_0$ and $\iota_1$.
• The construction ensures that the tiling is aperiodic, which implies that the undecidability result does not extend to periodic monotilings in the same setting.
• The result implies the undecidability of translational monotilings in $\mathbb{Z}^d$ for all $d \geq 3$, as $\mathbb{Z}^d$ contains $\mathbb{Z}^2 \times G_0$ as a subgroup for some finite $G_0$.
• An open question remains whether there exists a fixed finite Abelian group $G_0$ such that monotilings in $\mathbb{Z}^2 \times G_0$ are undecidable, though the current result holds for variable $G_0$.