6,831 research outputs found
Complexity of Verification in Self-Assembly with Prebuilt Assemblies
We analyze the complexity of two fundamental verification problems within a generalization of the two-handed tile self-assembly model (2HAM) where initial system assemblies are not restricted to be singleton tiles, but may be larger pre-built assemblies. Within this model we consider the producibility problem, which asks if a given tile system builds, or produces, a given assembly, and the unique assembly verification (UAV) problem, which asks if a given system uniquely produces a given assembly. We show that producibility is NP-complete and UAV is coNP^{NP}-complete even when the initial assembly size and temperature threshold are both bounded by a constant. This is in stark contrast to results in the standard model with singleton input tiles where producibility is in P and UAV is in coNP for ?(1) bounded temperature and coNP-complete when temperature is part of the input. We further provide preliminary results for producibility and UAV in the case of 1-dimensional linear assemblies with pre-built assemblies, and provide polynomial time solutions
Optimal staged self-assembly of linear assemblies
We analyze the complexity of building linear assemblies, sets of linear assemblies, and O(1)-scale general shapes in the staged tile assembly model. For systems with at most b bins and t tile types, we prove that the minimum number of stages to uniquely assemble a 1 n line is (logt n + logb n t + 1). Generalizing to O(1) n lines, we prove the minimum number of stages is O( log n tb t log t b2 + log log b log t ) and
( log n tb t log t b2 ). Next, we consider assembling sets of lines and general shapes using t = O(1) tile types. We prove that the minimum number of stages needed to assemble a set of k lines of size at most O(1) n is O( k log n b2 + k p log n b + log log n) and ( k log n b2 ). In the case that b = O( p k), the minimum number of stages is (log n). The upper bound in this special case is then used to assemble \hefty shapes of at least logarithmic edge-length-to- edge-count ratio at O(1)-scale using O( p k) bins and optimal O(log n) stages
Exponential replication of patterns in the Signal Tile Assembly Model and experimental non-deterministic assembly of lines in the Probabilistic Tile Assembly Model
We introduce the problem of self-replication of rectangular two-dimensional patterns in the practically motivated Signal Tile Assembly Model (STAM), which is an extension of the aTAM. In the first part of this thesis, we construct an exponential pattern replicator that replicates a two-dimensional input pattern over some fixed alphabet of size Ï• with O(Ï•) tile types, O(Ï•) unique glues, and a signal complexity of O(1). In the second part of this thesis, we use a non-deterministic model of tile assembly to significantly reduce the tile complexity of specified-length linear assemblies, which are a particularly important substructure for building more complicated nanostructures
Two computational primitives for algorithmic self-assembly: Copying and counting
Copying and counting are useful primitive operations for computation and construction. We have made DNA crystals that copy and crystals that count as they grow. For counting, 16 oligonucleotides assemble into four DNA Wang tiles that subsequently crystallize on a polymeric nucleating scaffold strand, arranging themselves in a binary counting pattern that could serve as a template for a molecular electronic demultiplexing circuit. Although the yield of counting crystals is low, and per-tile error rates in such crystals is roughly 10%, this work demonstrates the potential of algorithmic self-assembly to create complex nanoscale patterns of technological interest. A subset of the tiles for counting form information-bearing DNA tubes that copy bit strings from layer to layer along their length
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
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
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