5,818 research outputs found
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 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
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
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
Optimal Staged Self-Assembly of General Shapes
We analyze the number of tile types , bins , and stages necessary to
assemble squares and scaled shapes in the staged tile assembly
model. For squares, we prove stages suffice and
are necessary for almost all .
For shapes with Kolmogorov complexity , we prove
stages suffice and are necessary to
assemble a scaled version of , for almost all . We obtain similarly tight
bounds when the more powerful flexible glues are permitted.Comment: Abstract version appeared in ESA 201
New Geometric Algorithms for Fully Connected Staged Self-Assembly
We consider staged self-assembly systems, in which square-shaped tiles can be
added to bins in several stages. Within these bins, the tiles may connect to
each other, depending on the glue types of their edges. Previous work by
Demaine et al. showed that a relatively small number of tile types suffices to
produce arbitrary shapes in this model. However, these constructions were only
based on a spanning tree of the geometric shape, so they did not produce full
connectivity of the underlying grid graph in the case of shapes with holes;
designing fully connected assemblies with a polylogarithmic number of stages
was left as a major open problem. We resolve this challenge by presenting new
systems for staged assembly that produce fully connected polyominoes in O(log^2
n) stages, for various scale factors and temperature {\tau} = 2 as well as
{\tau} = 1. Our constructions work even for shapes with holes and uses only a
constant number of glues and tiles. Moreover, the underlying approach is more
geometric in nature, implying that it promised to be more feasible for shapes
with compact geometric description.Comment: 21 pages, 14 figures; full version of conference paper in DNA2
Active Self-Assembly of Algorithmic Shapes and Patterns in Polylogarithmic Time
We describe a computational model for studying the complexity of
self-assembled structures with active molecular components. Our model captures
notions of growth and movement ubiquitous in biological systems. The model is
inspired by biology's fantastic ability to assemble biomolecules that form
systems with complicated structure and dynamics, from molecular motors that
walk on rigid tracks and proteins that dynamically alter the structure of the
cell during mitosis, to embryonic development where large-scale complicated
organisms efficiently grow from a single cell. Using this active self-assembly
model, we show how to efficiently self-assemble shapes and patterns from simple
monomers. For example, we show how to grow a line of monomers in time and
number of monomer states that is merely logarithmic in the length of the line.
Our main results show how to grow arbitrary connected two-dimensional
geometric shapes and patterns in expected time that is polylogarithmic in the
size of the shape, plus roughly the time required to run a Turing machine
deciding whether or not a given pixel is in the shape. We do this while keeping
the number of monomer types logarithmic in shape size, plus those monomers
required by the Kolmogorov complexity of the shape or pattern. This work thus
highlights the efficiency advantages of active self-assembly over passive
self-assembly and motivates experimental effort to construct general-purpose
active molecular self-assembly systems
Self-Assembly of 4-sided Fractals in the Two-handed Tile Assembly Model
We consider the self-assembly of fractals in one of the most well-studied
models of tile based self-assembling systems known as the Two-handed Tile
Assembly Model (2HAM). In particular, we focus our attention on a class of
fractals called discrete self-similar fractals (a class of fractals that
includes the discrete Sierpi\'nski carpet). We present a 2HAM system that
finitely self-assembles the discrete Sierpi\'nski carpet with scale factor 1.
Moreover, the 2HAM system that we give lends itself to being generalized and we
describe how this system can be modified to obtain a 2HAM system that finitely
self-assembles one of any fractal from an infinite set of fractals which we
call 4-sided fractals. The 2HAM systems we give in this paper are the first
examples of systems that finitely self-assemble discrete self-similar fractals
at scale factor 1 in a purely growth model of self-assembly. Finally, we show
that there exists a 3-sided fractal (which is not a tree fractal) that cannot
be finitely self-assembled by any 2HAM system
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
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