354,055 research outputs found
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 squares in near optimal
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 systems can be simulated
with high probability, yielding Turing machine simulation and
assembly of 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
DNA Staged Self-Assembly at Temperature 1
We introduce alternate temperature 1 self-assembly constructions of an n x n square by efficiently utilizing bins and stages to achieve desirable results. These bins are able to contain a variety of tiles or supertiles, which are then mixed together in a pre-determined sequence of distinct stages. The basic 2D tile assembly model at temperature 1 uses 2n-1 tile types to construct a square. The model only utilizes one bin and occurs all in one stage. We will demonstrate how the use of bins and stages will allow for the construction of these squares more efficiently
Optimal self-assembly of finite shapes at temperature 1 in 3D
Working in a three-dimensional variant of Winfree's abstract Tile Assembly
Model, we show that, for an arbitrary finite, connected shape , there is a tile set that uniquely self-assembles into a 3D
representation of a scaled-up version of 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 and 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)
The Power of Duples (in Self-Assembly): It's Not So Hip To Be Square
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 self-assemble 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
Doubles and Negatives are Positive (in Self-Assembly)
In the abstract Tile Assembly Model (aTAM), the phenomenon of cooperation
occurs when the attachment of a new tile to a growing assembly requires it to
bind to more than one tile already in the assembly. Often referred to as
``temperature-2'' systems, those which employ cooperation are known to be quite
powerful (i.e. they are computationally universal and can build an enormous
variety of shapes and structures). Conversely, aTAM systems which do not
enforce cooperative behavior, a.k.a. ``temperature-1'' systems, are conjectured
to be relatively very weak, likely to be unable to perform complex computations
or algorithmically direct the process of self-assembly. Nonetheless, a variety
of models based on slight modifications to the aTAM have been developed in
which temperature-1 systems are in fact capable of Turing universal computation
through a restricted notion of cooperation. Despite that power, though, several
of those models have previously been proven to be unable to perform or simulate
the stronger form of cooperation exhibited by temperature-2 aTAM systems.
In this paper, we first prove that another model in which temperature-1
systems are computationally universal, namely the restricted glue TAM (rgTAM)
in which tiles are allowed to have edges which exhibit repulsive forces, is
also unable to simulate the strongly cooperative behavior of the temperature-2
aTAM. We then show that by combining the properties of two such models, the
Dupled Tile Assembly Model (DTAM) and the rgTAM into the DrgTAM, we derive a
model which is actually more powerful at temperature-1 than the aTAM at
temperature-2. Specifically, the DrgTAM, at temperature-1, can simulate any
aTAM system of any temperature, and it also contains systems which cannot be
simulated by any system in the aTAM
Algorithmic Temperature 1 Self-Assembly
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. Our work shows 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
Signal Transmission Across Tile Assemblies: 3D Static Tiles Simulate Active Self-Assembly by 2D Signal-Passing Tiles
The 2-Handed Assembly Model (2HAM) is a tile-based self-assembly model in
which, typically beginning from single tiles, arbitrarily large aggregations of
static tiles combine in pairs to form structures. The Signal-passing Tile
Assembly Model (STAM) is an extension of the 2HAM in which the tiles are
dynamically changing components which are able to alter their binding domains
as they bind together. For our first result, we demonstrate useful techniques
and transformations for converting an arbitrarily complex STAM tile set
into an STAM tile set where every tile has a constant, low amount of
complexity, in terms of the number and types of ``signals'' they can send, with
a trade off in scale factor.
Using these simplifications, we prove that for each temperature
there exists a 3D tile set in the 2HAM which is intrinsically universal for the
class of all 2D STAM systems at temperature (where the STAM does
not make use of the STAM's power of glue deactivation and assembly breaking, as
the tile components of the 2HAM are static and unable to change or break
bonds). This means that there is a single tile set in the 3D 2HAM which
can, for an arbitrarily complex STAM system , be configured with a
single input configuration which causes to exactly simulate at a scale
factor dependent upon . Furthermore, this simulation uses only two planes of
the third dimension. This implies that there exists a 3D tile set at
temperature in the 2HAM which is intrinsically universal for the class of
all 2D STAM systems at temperature . Moreover, we show that for each
temperature there exists an STAM tile set which is intrinsically
universal for the class of all 2D STAM systems at temperature ,
including the case where .Comment: A condensed version of this paper will appear in a special issue of
Natural Computing for papers from DNA 19. This full version contains proofs
not seen in the published versio
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