The Tile Assembly Model is Intrinsically Universal

Doty, David; Lutz, Jack H.; Patitz, Matthew J.; Schweller, Robert T.; Summers, Scott M.; Woods, Damien

doi:10.1109/focs.2012.76

Cited by 109 publications

(159 citation statements)

References 38 publications

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“…a tile assembly system) of C itself. For example, the abstract Tile Assembly Model has been shown to be intrinsically universal [46]. Specifically, this means that there is a single set of tiles U that, when appropriately initialized, is capable of simulating an arbitrary tile assembly system T .…”

Section: Simulation and A Result: The Abstract Tile Assembly Model Ismentioning

confidence: 99%

Intrinsic universality and the computational power of self-assembly

Woods

2015

Phil. Trans. R. Soc. A.

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Molecular self-assembly, the formation of large structures by small pieces of matter sticking together according to simple local interactions, is a ubiquitous phenomenon. A challenging engineering goal is to design a few molecules so that large numbers of them can self-assemble into desired complicated target objects. Indeed, we would like to understand the ultimate capabilities and limitations of this bottom-up fabrication process. We look to theoretical models of algorithmic self-assembly, where small square tiles stick together according to simple local rules in order to carry out a crystal growth process. In this survey, we focus on the use of simulation between such models to classify and separate their computational and expressive powers. Roughly speaking, one model simulates another if they grow the same structures, via the same dynamical growth processes. Our journey begins with the result that there is a single intrinsically universal tile set that, with appropriate initialization and spatial scaling, simulates any instance of Winfree's abstract Tile Assembly Model. This universal tile set exhibits something stronger than Turing universality: it captures the geometry and dynamics of any simulated system in a very direct way. From there we find that there is no such tile set in the more restrictive non-cooperative model, proving it weaker than the full Tile Assembly Model. In the two-handed model, where large structures can bind together in one step, we encounter an infinite set of infinite hierarchies of strictly increasing simulation power. Towards the end of our trip, we find one tile to rule them all: a single rotatable flipable polygonal tile that simulates any tile assembly system. We find another tile that aperiodically tiles the plane (but with small gaps). These and other recent results show that simulation is giving rise to a kind of computational complexity theory for self-assembly. It seems this could be the beginning of a much longer journey, so directions for future work are suggested.

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Section: Simulation and A Result: The Abstract Tile Assembly Model Ismentioning

confidence: 99%

Intrinsic universality and the computational power of self-assembly

Woods

2015

Phil. Trans. R. Soc. A.

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“…In standard simulation definitions (e.g. those in [10,12,14,19]), fuzz is restricted to being laterally or vertically adjacent to macrotile positions in S which map to non-empty tiles in T . We follow this convention for the definition of simulation of aTAM systems by DaTAM systems.…”

Section: Simulationmentioning

confidence: 99%

The power of duples (in self-assembly): It's not so hip to be square

Hendricks

Patitz

Rogers

et al. 2018

Theoretical Computer Science

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Abstract. In this paper we define the Dupled abstract Tile Assembly Model (DaTAM), which is a slight extension to the abstract Tile Assembly Model (aTAM) that allows for not only the standard square tiles, but also "duple" tiles which are rectangles pre-formed by the joining of two square tiles. We show that the addition of duples allows for powerful behaviors of self-assembling systems at temperature 1, meaning systems which exclude the requirement of cooperative binding by tiles (i.e., the requirement that a tile must be able to bind to at least 2 tiles in an existing assembly if it is to attach). Cooperative binding is conjectured to be required in the standard aTAM for Turing universal computation and the efficient self-assembly of shapes, but we show that in the DaTAM these behaviors can in fact be exhibited at temperature 1. We then show that the DaTAM doesn't provide asymptotic improvements over the aTAM in its ability to efficiently build thin rectangles. Finally, we present a series of results which prove that the temperature-2 aTAM and temperature-1 DaTAM have mutually exclusive powers. That is, each is able to selfassemble shapes that the other can't, and each has systems which cannot be simulated by the other. Beyond being of purely theoretical interest, these results have practical motivation as duples have already proven to be useful in laboratory implementations of DNA-based tiles.

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“…Manuch, Stacho and Stoll [13] showed that the previous tile complexity is also true when the final assembly cannot contain even any glue mismatches. Moreover, at the time of this writing, the only way in which non-cooperative self-assembly has been shown to be unconditionally weaker than cooperative self-assembly is in the sense of intrinsic universality [5,6]. First, Doty, Lutz, Patitz, Schweller, Summers and Woods [5] proved the existence of a universal cooperative tile set that can be programmed to simulate the behavior of any tile set (i.e., the aTAM is intrinsically universal for itself).…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, at the time of this writing, the only way in which non-cooperative self-assembly has been shown to be unconditionally weaker than cooperative self-assembly is in the sense of intrinsic universality [5,6]. First, Doty, Lutz, Patitz, Schweller, Summers and Woods [5] proved the existence of a universal cooperative tile set that can be programmed to simulate the behavior of any tile set (i.e., the aTAM is intrinsically universal for itself). Then, Meunier, Patitz, Summers, Theyssier, Winslow and Woods [15] showed, via a combinatorial argument, that there is no universal non-cooperative tile set that can be programmed to simulate the behavior of an arbitrary (cooperative) tile set.…”

Section: Introductionmentioning

confidence: 99%

Optimal Self-assembly of Finite Shapes at Temperature 1 in 3D

Furcy

Summers

2015

Combinatorial Optimization and Applications

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Working in a three-dimensional variant of Winfree's abstract Tile Assembly Model, we show that, for an arbitrary finite, connected shape X ⊂ Z 2 , there is a tile set that uniquely self-assembles into a 3D representation of X at temperature 1 in 3D with optimal program-size complexity (the program-size complexity, also known as tile complexity, of a shape is the minimum number of tile types required to uniquely self-assemble it). Moreover, our construction is "just barely" 3D in the sense that it only places tiles in the z = 0 and z = 1 planes. Our result is essentially a just-barely 3D temperature 1 simulation of a similar 2D temperature 2 result by Soloveichik and Winfree (SICOMP 2007).

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The Tile Assembly Model is Intrinsically Universal

Cited by 109 publications

References 38 publications

Intrinsic universality and the computational power of self-assembly

Intrinsic universality and the computational power of self-assembly

The power of duples (in self-assembly): It's not so hip to be square

Optimal Self-assembly of Finite Shapes at Temperature 1 in 3D

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