Towards a neighborhood simplification of tile systems: From Moore to quasi-linear dependencies

Abstract

Self-assembly is the process by which objects aggregate independently and form complex structures. One of the theoretical frameworks in which the process of self-assembly can be embedded and formally studied is that of tile systems. A Wang tile is a square unit, with glues on its edges, attaching to other tiles which have matching glues, and forming larger and larger structures. In this paper we concentrate over two basic, but essential, self-assembling structures done by Wang tiles. The first one, called ribbon, is a non-self-crossing wire-like structure, in which successive tiles are adjacent along an edge, and where tiles are glued to their predecessor and successor by use of matching glues. The second one, called zipper, is a similar contiguous structure, only that here, all touching tiles must have matching glues on their abutting edges, independently of their position in the structure. In case of Wang tiles, it has been shown that these two structures are equivalent. Here we generalize this result for the case when the tiles have eight glues, four on their edges and four on their corners. Thus we show that an eight neighborhood dependency, namely the Moore neighborhood, can be simulated by a quasi-linear dependency.