The periodic domino problem revisited

In this article we give a new proof of the undecidability of the periodic domino problem. The main difference with the previous proofs is that this one does not start from a proof of the undecidability of the (general) domino problem but only from the existence of an aperiodic tileset. The formalism of Wang tiles was introduced in [Wan61] to study decision procedures for the ∀∃∀ fragment of the first-order logic. The earliest and most fundamental question is the domino problem: decide, given a finite set of Wang tiles if it tiles the plane. It turns out that this is not possible, this so-called domino problem was proven undecidable [Ber64]. There are until now to the author’s knowledge 6 different proofs of the undecidability of the domino problem. Five of them encode the halting problem [Ber64, Rob71, AL74, Oll08, DRS12] while the last one [Kar07] encode the immortality problem for Turing machines [Hoo66]. This problem is intimately linked with the existence of aperiodic tilesets. An aperiodic tileset is a tileset that can tile the plane, but cannot tile it periodically. Wang conjectured that no such tileset exists. Were the conjecture true, the domino problem would be decidable [Wan61]. As a consequence, every proof of the undecidability of the domino problem gives as a byproduct the existence of an aperiodic tileset. In fact, almost any known proof first builds an aperiodic tileset then explains how to code computation in its tilings. This is indeed the case in [Ber64, Rob71, AL74, Oll08, DRS12]. This is not the case in [Kar07]. However, we can still build an aperiodic tileset from the proof: The immortality problem being undecidable [Hoo66], there must exist by compactness a Turing machine with no periodic points. [BCN02] gives such a machine. Encoding this machine with the construction in [Kar07] will give an aperiodic tileset. A proof of the undecidability of the domino problem gives a new aperiodic tileset. Is the converse true ? Can we use any aperiodic tileset as the first step in a proof of the undecidability of the domino problem ? In the constructions of [Ber64, Rob71, AL74, Oll08], each tileset is indeed handmade so that encoding of computation (by Turing machines) is easily done. However, can we do the same with any aperiodic tileset, not a specific one ? We do not know an answer to this question. As a specific example, we do not know how to encode a computation in the Ammann tileset [AGS92, GS87] or in the Kari-Culik tilesets [Kar96, II96]. A related problem is the periodic domino problem, where one asks whether a tileset can produce a periodic tiling. As aperiodic tilesets exist, this problem is not trivial. As a matter of fact, it is also undecidable. Interestingly, all known proofs are obtained by looking carefully to a proof of the undecidability of the domino problem and making some adjustements: [AL74, DRS12] already contain the two 1