Self-Assembly of Infinite Structures

We review some recent results related to the self-assembly of infinite structures in the Tile Assembly Model. These results include impossibility results, as well as novel tile assembly systems in which shapes and patterns that represent various notions of computation self-assemble. Several open questions are also presented and motivated

Universality in algorithmic self-assembly

Tile-based self-assembly is a model of algorithmic crystal growth in which square tiles represent molecules that bind to each other via specific and variable-strength bonds on their four sides, driven by random mixing in solution but constrained by the local binding rules of the tile bonds. In the late 1990s, Erik Winfree introduced a discrete mathematical model of DNA tile assembly called the abstract Tile Assembly Mode. Winfree proved that the Tile Assembly Model is computationally universal, i.e., that any Turing machine can be encoded into a finite set of tile types whose self-assembly simulates that Turing machine. In this thesis, we investigate tile-based self-assembly systems that exhibit Turing universality, geometric universality and intrinsic universality. We first establish a novel characterization of the computably enumerable languages in terms of self-assembly--the proof of which is a novel proof of the Turing-universality of the Tile Assembly Model in which a particular Turing machine is simulated on all inputs in parallel in the two-dimensional discrete Euclidean plane. Then we prove that the multiple temperature tile assembly model (introduced by Aggarwal, Cheng, Goldwasser, Kao, and Schweller) exhibits a kind of geometric universality in the sense that there is a small (constant-size) universal tile set that can be programmed via deliberate changes in the system temperature to uniquely produce any finite shape. Just as other models of computation such as the Turing machine and cellular automaton are known to be intrinsically universal, (e.g., Turing machines can simulate other Turing machines, and cellular automata other cellular automata), we show that tile assembly systems satisfying a natural condition known as local consistency are able to simulate other locally consistent tile assembly systems. In other words, we exhibit a particular locally consistent tile assembly system that can simulate the behavior--as opposed to only the final result--of any other locally consistent tile assembly system. Finally, we consider the notion of universal fault-tolerance in algorithmic self-assembly with respect to the two-handed Tile Assembly Model, in which large aggregations of tiles may attach to each other, in contrast to the seeded Tile Assembly Model, in which tiles aggregate one at a time to a single specially-designated seed assembly. We introduce a new model of fault-tolerance in self-assembly: the fuzzy temperature model of faults having the following informal characterization: the system temperature is normally 2, but may drift down to 1, allowing unintended temperature-1 growth for an arbitrary period of time. Our main construction, which is a tile set capable of uniquely producing an $n \times n$ square with log n unique tile types in the fuzzy temperature model, is not universal but presents novel technique that we hope will ultimately pave the way for a universal fuzzy-fault-tolerant tile assembly system in the future

Watching the Village

Page 15

Self-Assembly of Arbitrary Shapes Using RNAse Enzymes: Meeting the Kolmogorov Bound with Small Scale Factor (extended abstract)

We consider a model of algorithmic self-assembly of geometric shapes out of square Wang tiles studied in SODA 2010, in which there are two types of tiles (e.g., constructed out of DNA and RNA material) and one operation that destroys all tiles of a particular type (e.g., an RNAse enzyme destroys all RNA tiles). We show that a single use of this destruction operation enables much more efficient construction of arbitrary shapes. In particular, an arbitrary shape can be constructed using an asymptotically optimal number of distinct tile types (related to the shape's Kolmogorov complexity), after scaling the shape by only a logarithmic factor. By contrast, without the destruction operation, the best such result has a scale factor at least linear in the size of the shape, and is connected only by a spanning tree of the scaled tiles. We also characterize a large collection of shapes that can be constructed efficiently without any scaling