158 research outputs found

### Fuel Efficient Computation in Passive Self-Assembly

In this paper we show that passive self-assembly in the context of the tile
self-assembly model is capable of performing fuel efficient, universal
computation. The tile self-assembly model is a premiere model of self-assembly
in which particles are modeled by four-sided squares with glue types assigned
to each tile edge. The assembly process is driven by positive and negative
force interactions between glue types, allowing for tile assemblies floating in
the plane to combine and break apart over time. We refer to this type of
assembly model as passive in that the constituent parts remain unchanged
throughout the assembly process regardless of their interactions. A
computationally universal system is said to be fuel efficient if the number of
tiles used up per computation step is bounded by a constant. Work within this
model has shown how fuel guzzling tile systems can perform universal
computation with only positive strength glue interactions. Recent work has
introduced space-efficient, fuel-guzzling universal computation with the
addition of negative glue interactions and the use of a powerful non-diagonal
class of glue interactions. Other recent work has shown how to achieve fuel
efficient computation within active tile self-assembly. In this paper we
utilize negative interactions in the tile self-assembly model to achieve the
first computationally universal passive tile self-assembly system that is both
space and fuel-efficient. In addition, we achieve this result using a limited
diagonal class of glue interactions

### Temperature 1 Self-Assembly: Deterministic Assembly in 3D and Probabilistic Assembly in 2D

We investigate the power of the Wang tile self-assembly model at temperature
1, a threshold value that permits attachment between any two tiles that share
even a single bond. When restricted to deterministic assembly in the plane, no
temperature 1 assembly system has been shown to build a shape with a tile
complexity smaller than the diameter of the shape. In contrast, we show that
temperature 1 self-assembly in 3 dimensions, even when growth is restricted to
at most 1 step into the third dimension, is capable of simulating a large class
of temperature 2 systems, in turn permitting the simulation of arbitrary Turing
machines and the assembly of $n\times n$ squares in near optimal $O(\log n)$
tile complexity. Further, we consider temperature 1 probabilistic assembly in
2D, and show that with a logarithmic scale up of tile complexity and shape
scale, the same general class of temperature $\tau=2$ systems can be simulated
with high probability, yielding Turing machine simulation and $O(\log^2 n)$
assembly of $n\times n$ squares with high probability. Our results show a sharp
contrast in achievable tile complexity at temperature 1 if either growth into
the third dimension or a small probability of error are permitted. Motivated by
applications in nanotechnology and molecular computing, and the plausibility of
implementing 3 dimensional self-assembly systems, our techniques may provide
the needed power of temperature 2 systems, while at the same time avoiding the
experimental challenges faced by those systems

### Size-Dependent Tile Self-Assembly: Constant-Height Rectangles and Stability

We introduce a new model of algorithmic tile self-assembly called
size-dependent assembly. In previous models, supertiles are stable when the
total strength of the bonds between any two halves exceeds some constant
temperature. In this model, this constant temperature requirement is replaced
by an nondecreasing temperature function $\tau : \mathbb{N} \rightarrow
\mathbb{N}$ that depends on the size of the smaller of the two halves. This
generalization allows supertiles to become unstable and break apart, and
captures the increased forces that large structures may place on the bonds
holding them together.
We demonstrate the power of this model in two ways. First, we give fixed tile
sets that assemble constant-height rectangles and squares of arbitrary input
size given an appropriate temperature function. Second, we prove that deciding
whether a supertile is stable is coNP-complete. Both results contrast with
known results for fixed temperature.Comment: In proceedings of ISAAC 201

### 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

### Optimal Staged Self-Assembly of General Shapes

We analyze the number of tile types $t$, bins $b$, and stages necessary to
assemble $n \times n$ squares and scaled shapes in the staged tile assembly
model. For $n \times n$ squares, we prove $\mathcal{O}(\frac{\log{n} - tb -
t\log t}{b^2} + \frac{\log \log b}{\log t})$ stages suffice and
$\Omega(\frac{\log{n} - tb - t\log t}{b^2})$ are necessary for almost all $n$.
For shapes $S$ with Kolmogorov complexity $K(S)$, we prove
$\mathcal{O}(\frac{K(S) - tb - t\log t}{b^2} + \frac{\log \log b}{\log t})$
stages suffice and $\Omega(\frac{K(S) - tb - t\log t}{b^2})$ are necessary to
assemble a scaled version of $S$, for almost all $S$. We obtain similarly tight
bounds when the more powerful flexible glues are permitted.Comment: Abstract version appeared in ESA 201

- …