Fixed Parameter Undecidability for Wang Tilesets

TL;DR

This paper investigates the computational complexity of the Wang tiles tiling problem, proving undecidability persists under certain parameter bounds and introducing Wang bars as a new analytical tool.

Contribution

It establishes fixed parameter undecidability for Wang tilesets with a bounded difference between tiles and colors, and introduces Wang bars as a novel concept for analysis.

Findings

Decidability holds when tiles or colors are bounded.

Undecidability persists when the difference between tiles and colors is at most 43.

Wang bars are introduced as a new analytical tool.

Abstract

Deciding if a given set of Wang tiles admits a tiling of the plane is decidable if the number of Wang tiles (or the number of colors) is bounded, for a trivial reason, as there are only finitely many such tilesets. We prove however that the tiling problem remains undecidable if the difference between the number of tiles and the number of colors is bounded by 43. One of the main new tool is the concept of Wang bars, which are equivalently inflated Wang tiles or thin polyominoes.