The Tiling Problem

A Wang tile [12] is a unit square with each edge colored from a finite set of colors Σ. A set S of Wang tiles is said to tile a planar grid Z if copies of tiles from S can be placed, one at each grid position, such that abutting edges of adjacent tiles have the same color. Multiple copies of any tile may be used, with no restriction on the number. If we allow the tiles to be rotated or reflected, any single Wang tile can tile the plane by itself. The question of whether such a tiling exists for a given set of tiles is interesting only in the case where we do not allow rotation or reflection, thus holding tile orientation fixed. This decision problem, called the tiling or domino problem, was first posed in 1961 by Wang in a seminal paper. He also discussed the relation of this problem to the decision problem for certain classes of formulae of predicate calculus arising in automated theorem proving. Wang incorrectly conjectured that every tile set that tiles the plane permits a periodic tiling, that is, has a translational symmetry. Based on this assumption, he gave a general procedure for deciding the tiling problem. His assumption was disproved in 1966 by Berger [3] who constructed a tile set that allowed only an aperiodic tiling. He used this tile set to show that the tiling problem is undecidable. Berger’s tile set was quite large, over twenty thousand tiles, and his proof quite involved. Robinson [10] reduced the number of tiles to just over fifty and gave a much simpler proof of undecidability. Previous to Berger’s result, Wang himself showed a restricted version of the tiling problem, where only a certain tile was allowed at the origin, to be undecidable by reducing the halting problem to it. This and later work in tilings gave a method for simulating Turing machines using tiles, paving the way for thinking of tiles as a model of Turing universal computation. Interest in tilings has renewed in recent years due to two unrelated developments. Firstly, rapid advancements in theoretical and experimental DNA self-assembly allow us to construct nanoscale physical approximations to Wang tiles that can be programmed to tile according to simple rules. This allows us to perform computation in the test-tube, with advantages like massive parallelism and energy efficiency over traditional circuit based silicon machines. Attempts to model and study self-assembly via tilings was introduced by Winfree [14, 13] who extended Wang tilings by adding a mechanism for modeling growth. Secondly, several authors [7, 8, 6] have recently been interested in providing simpler