Self-Assembly of Tiles: Theoretical Models, the Power of Signals, and Local Computing

DNA-based self-assembly is an autonomous process whereby a disordered system of DNA sequences forms an organized structure or pattern as a consequence of Watson-Crick complementarity of DNA sequences, without external direction. Here, we propose self-assembly (SA) hypergraph automata as an automata-theoretic model for patterned self-assembly. We investigate the computational power of SA-hypergraph automata and show that for every recognizable picture language, there exists an SA-hypergraph automaton that accepts this language. Conversely, we prove that for any restricted SA-hypergraph automaton, there exists a Wang Tile System, a model for recognizable picture languages, that accepts the same language. Moreover, we investigate the computational power of some variants of the Signal-passing Tile Assembly Model (STAM), as well as propose the concept of {\it Smart Tiles}, i.e., tiles with glues that can be activated or deactivated by signals, and which possess a limited amount of local computing capability. We demonstrate the potential of smart tiles to perform some robotic tasks such as replicating complex shapes

Tiling allowing rotations only

AbstractWe define a “rotating board” as a finite square with one tile fixed in each cell. These fixed tiles can only be rotated and, in addition, they belong to a particular set of four tiles constructed with two colors. In this paper we show that any set of tiles T may be coded in linear time into a “rotating board” B in the following sense: 1.i. There exists an injection from the colors of the tiles of T into the set of border conditions of the board B.2.ii. There is a one-to-one relation between the set T and the set of tilings of B (obtained by rotating its tiles) satisfying that each t ϵ T is associated to a tiling Bθ in such a way that the (north, south, east and west) colors of t are related to the (north, south, east and west) border conditions of Bθ by the injection of i.The existence of this coding means that we can efficiently transform an arbitrary degrees of freedom tiling problem (in which to each cell is assigned an arbitrary set of admissible tiles) into a restricted four degrees of freedom problem (in which the tiles, fixed in each cell, can only be rotated). Considering the classical tiling results, we conclude the NP-completeness (resp. undecidability) of the natural bounded (resp. unbounded) version of the rotation tiling problem