19 research outputs found
Complexity of graph self-assembly in accretive systems and self-destructible systems
AbstractSelf-assembly is a process in which small objects autonomously associate with each other to form larger complexes. It is ubiquitous in biological constructions at the cellular and molecular scale and has also been identified by nanoscientists as a fundamental method for building nano-scale structures. Recent years have seen convergent interest and efforts in studying self-assembly from mathematicians, computer scientists, physicists, chemists, and biologists. However most complexity theoretical studies of self-assembly utilize mathematical models with two limitations: (1) only attraction, while no repulsion, is studied; (2) only assembled structures of two dimensional square grids are studied. In this paper, we study the complexity of the assemblies resulting from the cooperative effect of repulsion and attraction in a more general setting of graphs. This allows for the study of a more general class of self-assembled structures than the previous tiling model. We define two novel assembly models, namely the accretive graph assembly model and the self-destructible graph assembly model, and identify a fundamental problem in them: the sequential construction of a given graph. We refer to it as the Accretive Graph Assembly Problem (AGAP) and the Self-Destructible Graph Assembly Problem (DGAP), in the respective models. Our main results are: (i) AGAP is NP-complete even if the maximum degree of the graph is restricted to 4 or the graph is restricted to be planar with maximum degree 5; (ii) counting the number of sequential assembly orderings that result in a target graph (#AGAP) is #P-complete; and (iii) DGAP is PSPACE-complete even if the maximum degree of the graph is restricted to 6 (this is the first PSPACE-complete result in self-assembly). We also extend the accretive graph assembly model to a stochastic model, and prove that determining the probability of a given assembly in this model is #P-complete
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
Negative Interactions in Irreversible Self-Assembly
This paper explores the use of negative (i.e., repulsive) interaction the
abstract Tile Assembly Model defined by Winfree. Winfree postulated negative
interactions to be physically plausible in his Ph.D. thesis, and Reif, Sahu,
and Yin explored their power in the context of reversible attachment
operations. We explore the power of negative interactions with irreversible
attachments, and we achieve two main results. Our first result is an
impossibility theorem: after t steps of assembly, Omega(t) tiles will be
forever bound to an assembly, unable to detach. Thus negative glue strengths do
not afford unlimited power to reuse tiles. Our second result is a positive one:
we construct a set of tiles that can simulate a Turing machine with space bound
s and time bound t, while ensuring that no intermediate assembly grows larger
than O(s), rather than O(s * t) as required by the standard Turing machine
simulation with tiles
Optimization of supply diversity for the self-assembly of simple objects in two and three dimensions
The field of algorithmic self-assembly is concerned with the design and
analysis of self-assembly systems from a computational perspective, that is,
from the perspective of mathematical problems whose study may give insight into
the natural processes through which elementary objects self-assemble into more
complex ones. One of the main problems of algorithmic self-assembly is the
minimum tile set problem (MTSP), which asks for a collection of types of
elementary objects (called tiles) to be found for the self-assembly of an
object having a pre-established shape. Such a collection is to be as concise as
possible, thus minimizing supply diversity, while satisfying a set of stringent
constraints having to do with the termination and other properties of the
self-assembly process from its tile types. We present a study of what we think
is the first practical approach to MTSP. Our study starts with the introduction
of an evolutionary heuristic to tackle MTSP and includes results from extensive
experimentation with the heuristic on the self-assembly of simple objects in
two and three dimensions. The heuristic we introduce combines classic elements
from the field of evolutionary computation with a problem-specific variant of
Pareto dominance into a multi-objective approach to MTSP.Comment: Minor typos correcte
Exponential Replication of Patterns in the Signal Tile Assembly Model
Chemical self-replicators are of considerable interest in the field of nanomanufacturing and as a model for evolution. We introduce the problem of self-replication of rectangular two-dimensional patterns in the practically motivated Signal Tile Assembly Model (STAM) [9]. The STAM is based on the Tile Assembly Model (TAM) which is a mathematical model of self-assembly in which DNA tile monomers may attach to other DNA tile monomers in a programmable way. More abstractly, four-sided tiles are assigned glue types to each edge, and self-assembly occurs when singleton tiles bind to a growing assembly, if the glue types match and the glue binding strength exceeds some threshold. The signal tile extension of the TAM allows signals to be propagated across assemblies to activate glues or break apart assemblies. Here, we construct a pattern replicator that replicates a two-dimensional input pattern over some fixed alphabet of size φ with O(φ) tile types, O(φ) unique glues, and a signal complexity of O(1). Furthermore, we show that this replication system displays exponential growth in n, the number of replicates of the initial patterned assembly
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
Universal Shape Replicators via Self-Assembly with Attractive and Repulsive Forces
We show how to design a universal shape replicator in a self- assembly system with both attractive and repulsive forces. More precisely, we show that there is a universal set of constant-size objects that, when added to any unknown holefree polyomino shape, produces an unbounded number of copies of that shape (plus constant-size garbage objects). The constant-size objects can be easily constructed from a constant number of individual tile types using a constant number of preprocessing self-assembly steps. Our construction uses the well-studied 2-Handed Assembly Model (2HAM) of tile self-assembly, in the simple model where glues interact only with identical glues, allowing glue strengths that are either positive (attractive) or negative (repulsive), and constant temperature (required glue strength for parts to hold together). We also require that the given shape has specified glue types on its surface, and that the feature size (smallest distance between nonincident edges) is bounded below by a constant. Shape replication necessarily requires a self-assembly model where parts can both attach and detach, and this construction is the first to do so using the natural model of negative/repulsive glues (also studied before for other problems such as fuel-efficient computation); previous replication constructions require more powerful global operations such as an “enzyme” that destroys a subset of the tile types.National Science Foundation (U.S.) (Grant EFRI1240383)National Science Foundation (U.S.) (Grant CCF-1138967
Algorithmic Assembly of Nanoscale Structures
The development of nanotechnology has become one of the most significant endeavors of our time. A natural objective of this field is discovering how to engineer nanoscale structures. Limitations of current top-down techniques inspire investigation into bottom-up approaches to reach this objective. A fundamental precondition for a bottom-up approach is the ability to control the behavior of nanoscale particles. Many abstract representations have been developed to model systems of particles and to research methods for controlling their behavior. This thesis develops theories on two such approaches for building complex structures: the self-assembly of simple particles, and the use of simple robot swarms. The concepts for these two approaches are straightforward. Self-assembly is the process by which simple particles, following the rules of some behavior-governing system, naturally coalesce into a more complex form. The other method of bottom-up assembly involves controlling nanoscale particles through explicit directions and assembling them into a desired form. Regarding the self-assembly of nanoscale structures, we present two construction methods in a variant of a popular theoretical model known as the 2-Handed Tile Self-Assembly Model. The first technique achieves shape construction at only a constant scale factor, while the second result uses only a constant number of unique particle types. Regarding the use of robot swarms for construction, we first develop a novel technique for reconfiguring a swarm of globally-controlled robots into a desired shape even when the robots can only move maximally in a commanded direction. We then expand on this work by formally defining an entire hierarchy of shapes which can be built in this manner and we provide a technique for doing so
Universal Shape Replicators via Self-Assembly with Attractive and Repulsive Forces
We show how to design a universal shape replicator in a self- assembly system with both attractive and repulsive forces. More precisely, we show that there is a universal set of constant-size objects that, when added to any unknown holefree polyomino shape, produces an unbounded number of copies of that shape (plus constant-size garbage objects). The constant-size objects can be easily constructed from a constant number of individual tile types using a constant number of preprocessing self-assembly steps. Our construction uses the well-studied 2-Handed Assembly Model (2HAM) of tile self-assembly, in the simple model where glues interact only with identical glues, allowing glue strengths that are either positive (attractive) or negative (repulsive), and constant temperature (required glue strength for parts to hold together). We also require that the given shape has specified glue types on its surface, and that the feature size (smallest distance between nonincident edges) is bounded below by a constant. Shape replication necessarily requires a self-assembly model where parts can both attach and detach, and this construction is the first to do so using the natural model of negative/repulsive glues (also studied before for other problems such as fuel-efficient computation); previous replication constructions require more powerful global operations such as an “enzyme” that destroys a subset of the tile types