Arithmetic computation in the tile assembly model: Addition and multiplication
Theoretical Computer Science
An Architectural Style for Solving Computationally Intensive Problems on Large Networks
SEAMS '07 Proceedings of the 2007 International Workshop on Software Engineering for Adaptive and Self-Managing Systems
Fault and adversary tolerance as an emergent property of distributed systems' software architectures
Proceedings of the 2007 workshop on Engineering fault tolerant systems
Nondeterministic polynomial time factoring in the tile assembly model
Theoretical Computer Science
Solving NP-complete problems in the tile assembly model
Theoretical Computer Science
Solving satisfiability in the tile assembly model with a constant-size tileset
Journal of Algorithms
Complexity classes for self-assembling flexible tiles
Theoretical Computer Science
Strict self-assembly of discrete Sierpinski triangles
Theoretical Computer Science
Path finding in the tile assembly model
Theoretical Computer Science
Perfectly quilted rectangular snake tilings
Theoretical Computer Science
Self-assembly of discrete self-similar fractals
Natural Computing: an international journal
Infinite snake tiling problems
DLT'02 Proceedings of the 6th international conference on Developments in language theory
Snakes and cellular automata: reductions and inseparability results
CSR'11 Proceedings of the 6th international conference on Computer science: theory and applications
On stoichiometry for the assembly of flexible tile DNA complexes
Natural Computing: an international journal
Polyominoes simulating arbitrary-neighborhood zippers and tilings
Theoretical Computer Science
Expectation and variance of self-assembled graph structures
DNA'05 Proceedings of the 11th international conference on DNA Computing
A self-assembly model of time-dependent glue strength
DNA'05 Proceedings of the 11th international conference on DNA Computing
A computational model for self-assembling flexible tiles
UC'05 Proceedings of the 4th international conference on Unconventional Computation
Self-replication and evolution of DNA crystals
ECAL'05 Proceedings of the 8th European conference on Advances in Artificial Life
Spectrum of a pot for DNA complexes
DNA'06 Proceedings of the 12th international conference on DNA Computing
On the computation of colored domino tilings of simple and non-simple orthogonal polygons
ISAAC'05 Proceedings of the 16th international conference on Algorithms and Computation
Programmable control of nucleation for algorithmic self-assembly
DNA'04 Proceedings of the 10th international conference on DNA computing
Efficient 3-SAT algorithms in the tile assembly model
Natural Computing: an international journal
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Self-assembly, the process by which objects autonomously come together to form complex structures, is omnipresent in the physical world. A systematic study of self-assembly as a mathematical process has been initiated. The individual components are modelled as square tiles on the infinite two-dimensional plane. Each side of a tile is covered by a specific "glue", and two adjacent tiles will stick if they have matching glues on their abutting edges. Tiles that stick to each other may form various two-dimensional "structures" such as squares, rectangles, or may cover the entire plane. In this paper we focus on a special type of structure, called ribbon: A non-self-crossing sequence of tiles on the plane, in which successive tiles are adjacent along an edge, and abutting edges of consecutive tiles have matching glues. We prove that it is undecidable whether an arbitrary finite set of tiles with glues (infinite supply of each tile type available) can be used to assemble an infinite ribbon. The proof is based on a construction, due to Robinson, of a special set of tiles that allow onlyaperiodic tilings of the plane. This construction is used to create a special set of directed tiles (tiles with arrows painted on the top) with the "strong plane-filling property" - a variation of the "plane-filling property" previously defined by J. Kari. A construction of "sandwich" tiles is then used in conjunction with this special tile set, to reduce the well-known undecidable Tiling Problem to the problem of the existence of an infinite directed zipper (a special kind of ribbon). A "motif" construction is then introduced that allows one tile system to simulate another by using geometry to represent glues. Using motifs, the infinite directed zipper problem is reduced to the infinite ribbon problem, proving the latter undecidable.The result settles an open problemformerly known as the "unlimited infinite snake problem". Moreover, an immediate consequence is the undecidability of the existence of arbitrarily large structures self-assembled using tiles from a given tile set.