21 research outputs found
Self-Healing Tile Sets
Biology provides the synthetic chemist with a tantalizing and frustrating challenge:
to create complex objects, defined from the molecular scale up to meters,
that construct themselves from elementary components, and perhaps
even reproduce themselves. This is the challenge of bottom-up fabrication.
The most compelling answer to this challenge was formulated in the early
1980s by Ned Seeman, who realized that the information carried by DNA
strands provides a means to program molecular self-assembly, with potential
applications including DNA scaffolds for crystallography [19] or for molecular
electronic circuits [15]. This insight opened the doors to engineering with the
rich set of phenomena available in nucleic acid chemistry [20]
The Two-Handed Tile Assembly Model Is Not Intrinsically Universal
In this paper, we study the intrinsic universality of the well-studied Two-Handed Tile Assembly Model (2HAM), in which two “supertile” assemblies, each consisting of one or more unit-square tiles, can fuse together (self-assemble) whenever their total attachment strength is at least the global temperature τ. Our main result is that for all τ′ < τ, each temperature-τ′ 2HAM tile system cannot simulate at least one temperature-τ 2HAM tile system. This impossibility result proves that the 2HAM is not intrinsically universal, in stark contrast to the simpler abstract Tile Assembly Model which was shown to be intrinsically universal (The tile assembly model is intrinsically universal, FOCS 2012). On the positive side, we prove that, for every fixed temperature τ ≥ 2, temperature-τ 2HAM tile systems are intrinsically universal: for each τ there is a single universal 2HAM tile set U that, when appropriately initialized, is capable of simulating the behavior of any temperature τ 2HAM tile system. As a corollary of these results we find an infinite set of infinite hierarchies of 2HAM systems with strictly increasing power within each hierarchy. Finally, we show how to construct, for each τ, a temperature-τ 2HAM system that simultaneously simulates all temperature-τ 2HAM systems
Pattern overlap implies runaway growth in hierarchical tile systems
We show that in the hierarchical tile assembly model, if there is a
producible assembly that overlaps a nontrivial translation of itself
consistently (i.e., the pattern of tile types in the overlap region is
identical in both translations), then arbitrarily large assemblies are
producible. The significance of this result is that tile systems intended to
controllably produce finite structures must avoid pattern repetition in their
producible assemblies that would lead to such overlap. This answers an open
question of Chen and Doty (SODA 2012), who showed that so-called
"partial-order" systems producing a unique finite assembly *and" avoiding such
overlaps must require time linear in the assembly diameter. An application of
our main result is that any system producing a unique finite assembly is
automatically guaranteed to avoid such overlaps, simplifying the hypothesis of
Chen and Doty's main theorem
The Two-Handed Tile Assembly Model is not Intrinsically Universal
The Two-Handed Tile Assembly Model (2HAM) is a model of algorithmic self-assembly in which large structures, or assemblies of tiles, are grown by the binding of smaller assemblies. In order to bind, two assemblies must have matching glues that can simultaneously touch each other, and stick together with strength that is at least the temperature τ, where τ is some fixed positive integer. We ask whether the 2HAM is intrinsically universal. In other words, we ask: is there a single 2HAM tile set U which can be used to simulate any instance of the model? Our main result is a negative answer to this question. We show that for all τ′ < τ, each temperature-τ′ 2HAM tile system does not simulate at least one temperature-τ 2HAM tile system. This impossibility result proves that the 2HAM is not intrinsically universal and stands in contrast to the fact that the (single-tile addition) abstract Tile Assembly Model is intrinsically universal. On the positive side, we prove that, for every fixed temperature τ ≥ 2, temperature-τ 2HAM tile systems are indeed intrinsically universal. In other words, for each τ there is a single intrinsically universal 2HAM tile set U_τ that, when appropriately initialized, is capable of simulating the behavior of any temperature-τ 2HAM tile system. As a corollary, we find an infinite set of infinite hierarchies of 2HAM systems with strictly increasing simulation power within each hierarchy. Finally, we show that for each τ, there is a temperature-τ 2HAM system that simultaneously simulates all temperature-τ 2HAM systems
Producibility in hierarchical self-assembly
Three results are shown on producibility in the hierarchical model of tile self-assembly. It is shown that a simple greedy polynomial-time strategy decides whether an assembly α is producible. The algorithm can be optimized to use O(|α|log^2 |α|) time. Cannon et al. (STACS 2013: proceedings of the thirtieth international symposium on theoretical aspects of computer science. pp 172–184, 2013) showed that the problem of deciding if an assembly α is the unique producible terminal assembly of a tile system T can be solved in O(|α|^2 |T|+|α||T|^2) time for the special case of noncooperative “temperature 1” systems. It is shown that this can be improved to O(|α||T|log|T|) time. Finally, it is shown that if two assemblies are producible, and if they can be overlapped consistently—i.e., if the positions that they share have the same tile type in each assembly—then their union is also producible
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