The domino problem is decidable for robust tilesets

TL;DR

The paper tackles the domino problem for Wang tiles by introducing a transducer-based formalism that recasts tilings as computations. It defines two notions of tileset robustness—semantically robust and provably robust—and shows that robust tilesets admit a decidable domino problem via simple, provable invariants and recurrence relations. The authors apply the framework to Robinson’s, Jeandel–Rao’s, and Kari’s tilings, establishing provable robustness and linking tileability to inductive invariants expressed through transducer compositions. They also draw a deep analogy with Turing machines, showing that robustness aligns decidability for both tilings and computation, and extend the approach to general shapes and tiling equivalence. The results provide a unified explanation for observed patterns in known tilings and offer a sound, relatively complete method for proving plane tilings, with implications for understanding algorithmic structure in tiling theory and potential extensions to aperiodicity and broader dynamical systems.

Abstract

One of the most fundamental problems in tiling theory is the domino problem: given a set of tiles and tiling rules, decide if there exists a way to tile the plane using copies of tiles and following their rules. The problem is known to be undecidable in general and even for sets of Wang tiles, which are unit square tiles wearing colours on their edges which can be assembled provided they share the same colour on their common edge, as proven by Berger in the 1960s. In this paper, we focus on Wang tilesets. We prove that the domino problem is decidable for robust tilesets, i.e. tilesets that either cannot tile the plane or can but, if so, satisfy some particular invariant provably. We establish that several famous tilesets considered in the literature are robust. We give arguments that this is true for all tilesets unless they are produced from non-robust Turing machines: a Turing machine is said to be non-robust if it does not halt and furthermore does so non-provably. As a side effect of our work, we provide a sound and relatively complete method for proving that a tileset can tile the plane. Our analysis also provides explanations for the observed similarities between proofs in the literature for various tilesets, as well as of phenomena that have been observed experimentally in the systematic study of tilesets using computer methods.

Figures (8)

• Figure 1: Two graphical representations of a Wang tile as a transition in the associated transducer.
• Figure 2: An example a of meta-transducer derived from the tileset $\tau$ composed by the four tiles pictured on the top.
• Figure 3: Two graphical representations of a finite horizontal valid tiling as a transition in a meta-transducer derived from the tileset pictured on the left.
• Figure 4: To the left Robinson's tileset, where tiles can be rotated and reflected. To the right a pattern that appears in every tiling by Robinson's tileset.
• Figure 5: The transducer $\mathcal{T}_{Robi}$ corresponding to Robinson's tileset.

Theorems & Definitions (94)

• Definition 1: Transducer
• Definition 2: Meta-transducer
• Remark 3
• Remark 4
• Remark 5
• Remark 6
• Proposition 7: Folklore
• Example 8
• Definition 9: Notation $\mathcal{T} \circ \mathcal{T'}$
• Proposition 10