Intractability and Undecidability in Small Sets of Wang Tiles

Imagine a never-ending checkerboard, red and black squares alternating forever in every direction. Now close your eyes, wait for a second, and open them again. There is still the checkerboard, but is it different? Has somebody moved the checkerboard over two squares? Four squares? One million squares? It still looks the same. This is the nature of periodic tilings. Wang tiles are squares, much like the red and black ones used on a checkerboard, except Wang tiles have colors on their edges instead of on the whole square. Also, Wang tiles can only be put edge-to-edge with each other where these colors are the same. So what\u27s so special about Wang tiles? If you cover the infinite plane with certain sets of Wang tiles, close your eyes, and open them again, you will always be able to tell if it has changed. In these sorts of tilings, there is always something that does not quite overlap when moved any amount in any direction. This is the nature of aperiodic tilings. The smallest known such set of Wang tiles has thirteen tiles. This paper computationally explores sets of six, seven, and eight Wang tiles, looking for the same aperiodic structure

OPTIMIZATION-BASED APPROACH TO TILING OF FINITE AREAS WITH ARBITRARY SETS OF WANG TILES

Wang tiles proved to be a convenient tool for the design of aperiodic tilings in computer graphics and in materials engineering. While there are several algorithms for generation of finite-sized tilings, they exploit the specific structure of individual tile sets, which prevents their general usage. In this contribution, we reformulate the NP-complete tiling generation problem as a binary linear program, together with its linear and semidefinite relaxations suitable for the branch and bound method. Finally, we assess the performance of the established formulations on generations of several aperiodic tilings reported in the literature, and conclude that the linear relaxation is better suited for the problem

On bounded Wang tilings

Wang tiles enable efficient pattern compression while avoiding the periodicity in tile distribution via programmable matching rules. However, most research in Wang tilings has considered tiling the infinite plane. Motivated by emerging applications in materials engineering, we consider the bounded version of the tiling problem and offer four integer programming formulations to construct valid or nearly-valid Wang tilings: a decision, maximum-rectangular tiling, maximum cover, and maximum adjacency constraint satisfaction formulations. To facilitate a finer control over the resulting tilings, we extend these programs with tile-based, color-based, packing, and variable-sized periodic constraints. Furthermore, we introduce an efficient heuristic algorithm for the maximum-cover variant based on the shortest path search in directed acyclic graphs and derive simple modifications to provide a $1/2$ approximation guarantee for arbitrary tile sets, and a $2/3$ guarantee for tile sets with cyclic transducers. Finally, we benchmark the performance of the integer programming formulations and of the heuristic algorithms showing that the heuristics provides very competitive outputs in a fraction of time. As a by-product, we reveal errors in two well-known aperiodic tile sets: the Knuth tile set contains a tile unusable in two-way infinite tilings, and the Lagae corner tile set is not aperiodic

Aperiodic Subshifts of Finite Type on Groups

In this note we prove the following results: $\bullet$ If a finitely presented group $G$ admits a strongly aperiodic SFT, then $G$ has decidable word problem. More generally, for f.g. groups that are not recursively presented, there exists a computable obstruction for them to admit strongly aperiodic SFTs. $\bullet$ On the positive side, we build strongly aperiodic SFTs on some new classes of groups. We show in particular that some particular monster groups admits strongly aperiodic SFTs for trivial reasons. Then, for a large class of group $G$, we show how to build strongly aperiodic SFTs over $\mathbb{Z}\times G$. In particular, this is true for the free group with 2 generators, Thompson's groups $T$ and $V$, $PSL_2(\mathbb{Z})$ and any f.g. group of rational matrices which is bounded.Comment: New version. Adding results about monster group

Tiling Problems on Baumslag-Solitar groups

We exhibit a weakly aperiodic tile set for Baumslag-Solitar groups, and prove that the domino problem is undecidable on these groups. A consequence of our construction is the existence of an arecursive tile set on Baumslag-Solitar groups.Comment: In Proceedings MCU 2013, arXiv:1309.104

OPTIMIZATION-BASED APPROACH TO TILING OF FINITE AREAS WITH ARBITRARY SETS OF WANG TILES

Wang tiles proved to be a convenient tool for the design of aperiodic tilings in computer graphics and in materials engineering. While there are several algorithms for generation of finite-sized tilings, they exploit the specific structure of individual tile sets, which prevents their general usage. In this contribution, we reformulate the NP-complete tiling generation problem as a binary linear program, together with its linear and semidefinite relaxations suitable for the branch and bound method. Finally, we assess the performance of the established formulations on generations of several aperiodic tilings reported in the literature, and conclude that the linear relaxation is better suited for the problem