An aperiodic set of 11 Wang tiles

A new aperiodic tile set containing 11 Wang tiles on 4 colors is presented. This tile set is minimal in the sense that no Wang set with less than 11 tiles is aperiodic, and no Wang set with less than 4 colors is aperiodic. Wang tiles are square tiles with colored edges. A tiling of the plane by Wang tiles consists in putting a Wang tile in each cell of the grid Z so that contiguous edges share the same color. The formalism of Wang tiles was introduced by Wang [Wan61] to study decision procedures for a specific fragment of logic (see section 1.1 for details). Wang asked the question of the existence of an aperiodic tile set: A set of Wang tiles which tiles the plane but cannot do so periodically. His student Berger quickly gave an example of such a tile set, with a tremendous number of tiles. The number of tiles needed for an aperiodic tileset was reduced during the years, first by Berger himself, then by others, to obtain in 1996 the previous record of an aperiodic set of 13 Wang tiles. (see section 1.2 for an overview of previous aperiodic sets of Wang tiles). While reducing the number of tiles may seem like a tedious exercise in itself, the articles also introduced different techniques to build aperiodic tilesets, and different techniques to prove aperiodicity. A few lower bounds exist on the number of Wang tiles needed to obtain an aperiodic tile set, the only reference [GS87] citing the impossibility to have one with 4 tiles or less. On the other hand, recent results show that an aperiodic set of Wang tiles need to have at least 4 different colors [CHLL14]. In this article, we fill all the gaps: We prove that there are no aperiodic tile set with less than 11 Wang tiles, and that there is an aperiodic tile set with 11 Wang tiles and 4 colors. The discovery of this tile set, and the proof that there is no aperiodic tile set with a smaller number of tiles was done by a computer search: We generated in particular all possible candidates with 10 tiles or less, and prove they were not aperiodic. Surprisingly it was somewhat easy to do so for all of them except one. The situation is different for 11 tiles: While we have found an aperiodic tileset, we also have a short list of tile set for which we do not know anything. The description of this computer search is described in section 3 of the paper, and 1 ar X iv :1 50 6. 06 49 2v 1 [ cs .D M ] 2 2 Ju n 20 15 can possibly be skipped by a reader only interested in the tile set itself. This section also contains a result of independent interest: the tile set from Culik with one tile omitted does not tile the plane. The tile set itself is presented in section 4, and the remaining sections prove that it is indeed an aperiodic tileset. 1 Aperiodic sets of Wang tiles Here is a brief summary of the known aperiodic sets of Wang tiles. Explanations about some of them may be found in [GS87]. We stay clear in this history about aperiodic sets of geometric figures, and focus only on Wang tiles. 1.1 Wang tiles and the ∀∃∀ problem Wang tiles were introduced by Wang [Wan61] in 1961 to study the decidability of the ∀∃∀ fragment of first order logic. Wang showed in this article how to build, starting from a ∀∃∀ formula φ, a set of tiles τ and a subset τ ′ ⊆ τ so that there exists a tiling by τ of the upper quadrant with tiles in the first row in τ ′ iff φ is satisfiable. If this particular tiling problem was decidable, this would imply that the satisfiability of ∀∃∀ formulas was decidable. Wang asked more generally in this article whether the more general tiling problem (with no particular tiles in the first row) is decidable and gave the fundamental conjecture: Every tileset either admits a periodic tiling or does not tile. Regardless of the status of this particular conjecture, Kahr, Moore and Wang [KMW62] proved the next year that the ∀∃∀ problem is indeed undecidable by reducing to another tiling problem: now we fix a subset τ ′ of tiles so that every tile on the diagonal of the first quadrant is in τ ′. This proof was later simplified by Hermes [Her71, Her70]. From the point of view of first order logic, the problem is thus solved. Formally speaking, the tiling problem with a constraint diagonal is reduced to a formula of the form ∀x∃y∀zφ(x, y, z) where φ contains a binary predicate P and some occurences of the subformula P (x, x) (to code the diagonal constraint). If we look at ∀∃∀ formulas that do not contain the subformula P (x, x) and P (z, z), the decidability of this particular fragment remained open. A few years later, Berger proved however [Ber64] that the domino problem is undecidable, and that an aperiodic tileset existed. This implies in particular that the particular fragment of ∀x∃y∀z where the only occurences of the binary predicates P are of the form P (x, z), P (y, z), P (z, y), P (z, x) was undecidable. A few other subcases of ∀∃∀ were done over the years. In 1975, Aanderaa and Lewis [AL74] proved the undecidability of the fragment of ∀∃∀ where the binary predicates P can only appear in the form P (x, z) and P (z, y). It has in particular the following consequence: The domino problem for deterministic tilesets is undecidable