An introduction to symbolic dynamics and coding
An introduction to symbolic dynamics and coding
Reconstructing convex polyominoes from horizontal and vertical projections
Theoretical Computer Science
Reconstructing hv-convex polyominoes from orthogonal projections
Information Processing Letters
The program-size complexity of self-assembled squares (extended abstract)
STOC '00 Proceedings of the thirty-second annual ACM symposium on Theory of computing
On the Decidability of Self-Assembly of Infinite Ribbons
FOCS '02 Proceedings of the 43rd Symposium on Foundations of Computer Science
Algorithmic self-assembly of dna
Algorithmic self-assembly of dna
Theory and experiments in algorithmic self-assembly
Theory and experiments in algorithmic self-assembly
Theory of cellular automata: a survey
Theoretical Computer Science
Reducing tile complexity for self-assembly through temperature programming
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
The Undecidability of the Infinite Ribbon Problem: Implications for Computing by Self-Assembly
SIAM Journal on Computing
Staged self-assembly: nanomanufacture of arbitrary shapes with O(1) glues
DNA13'07 Proceedings of the 13th international conference on DNA computing
Towards a neighborhood simplification of tile systems: From Moore to quasi-linear dependencies
Natural Computing: an international journal
Programmable control of nucleation for algorithmic self-assembly
DNA'04 Proceedings of the 10th international conference on DNA computing
Hi-index | 5.23 |
This paper provides a bridge between the classical tiling theory and the complex-neighborhood self-assembling situations that exist in practice. The neighborhood of a position in the plane is the set of coordinates which are considered adjacent to it. This includes classical neighborhoods of size four, as well as arbitrarily complex neighborhoods. A generalized tile system consists of a set of tiles, a neighborhood, and a relation which dictates which are the ''admissible'' neighboring tiles of a given tile. Thus, in correctly formed assemblies, tiles are assigned positions of the plane in accordance with this relation. We prove that any validly tiled path defined in a given but arbitrary neighborhood (a zipper) can be simulated by a simple ''ribbon'' of microtiles. A ribbon is a special kind of polyomino, consisting of a non-self-crossing sequence of tiles on the plane, in which successive tiles stick along their adjacent edge. Finally, we extend this construction to the case of traditional tilings, proving that we can simulate arbitrary-neighborhood tilings by simple-neighborhood tilings, while preserving some of their essential properties.