1,452 research outputs found
Intrinsic universality in tile self-assembly requires cooperation
We prove a negative result on the power of a model of algorithmic
self-assembly for which it has been notoriously difficult to find general
techniques and results. Specifically, we prove that Winfree's abstract Tile
Assembly Model, when restricted to use noncooperative tile binding, is not
intrinsically universal. This stands in stark contrast to the recent result
that, via cooperative binding, the abstract Tile Assembly Model is indeed
intrinsically universal. Noncooperative self-assembly, also known as
"temperature 1", is where tiles bind to each other if they match on one or more
sides, whereas cooperative binding requires binding on multiple sides. Our
result shows that the change from single- to multi-sided binding qualitatively
improves the kinds of dynamics and behavior that these models of nanoscale
self-assembly are capable of. Our lower bound on simulation power holds in both
two and three dimensions; the latter being quite surprising given that
three-dimensional noncooperative tile assembly systems simulate Turing
machines. On the positive side, we exhibit a three-dimensional noncooperative
self-assembly tile set capable of simulating any two-dimensional noncooperative
self-assembly system.
Our negative result can be interpreted to mean that Turing universal
algorithmic behavior in self-assembly does not imply the ability to simulate
arbitrary algorithmic self-assembly processes.Comment: Added references. Improved presentation of definitions and proofs.
This article uses definitions from arXiv:1212.4756. arXiv admin note: text
overlap with arXiv:1006.2897 by other author
Intrinsic universality and the computational power of self-assembly
This short survey of recent work in tile self-assembly discusses the use of
simulation to classify and separate the computational and expressive power of
self-assembly models. The journey begins with the result that there is a single
universal tile set that, with proper initialization and scaling, simulates any
tile assembly system. This universal tile set exhibits something stronger than
Turing universality: it captures the geometry and dynamics of any simulated
system. From there we find that there is no such tile set in the
noncooperative, or temperature 1, model, proving it weaker than the full tile
assembly model. In the two-handed or hierarchal model, where large assemblies
can bind together on one step, we encounter an infinite set, of infinite
hierarchies, each with strictly increasing simulation power. Towards the end of
our trip, we find one tile to rule them all: a single rotatable flipable
polygonal tile that can simulate any tile assembly system. It seems this could
be the beginning of a much longer journey, so directions for future work are
suggested.Comment: In Proceedings MCU 2013, arXiv:1309.104
On the Equivalence of Cellular Automata and the Tile Assembly Model
In this paper, we explore relationships between two models of systems which
are governed by only the local interactions of large collections of simple
components: cellular automata (CA) and the abstract Tile Assembly Model (aTAM).
While sharing several similarities, the models have fundamental differences,
most notably the dynamic nature of CA (in which every cell location is allowed
to change state an infinite number of times) versus the static nature of the
aTAM (in which tiles are static components that can never change or be removed
once they attach to a growing assembly). We work with 2-dimensional systems in
both models, and for our results we first define what it means for CA systems
to simulate aTAM systems, and then for aTAM systems to simulate CA systems. We
use notions of simulate which are similar to those used in the study of
intrinsic universality since they are in some sense strict, but also
intuitively natural notions of simulation. We then demonstrate a particular
nondeterministic CA which can be configured so that it can simulate any
arbitrary aTAM system, and finally an aTAM tile set which can be configured so
that it can be used to simulate any arbitrary nondeterministic CA system which
begins with a finite initial configuration.Comment: In Proceedings MCU 2013, arXiv:1309.104
Noncooperative algorithms in self-assembly
We show the first non-trivial positive algorithmic results (i.e. programs
whose output is larger than their size), in a model of self-assembly that has
so far resisted many attempts of formal analysis or programming: the planar
non-cooperative variant of Winfree's abstract Tile Assembly Model.
This model has been the center of several open problems and conjectures in
the last fifteen years, and the first fully general results on its
computational power were only proven recently (SODA 2014). These results, as
well as ours, exemplify the intricate connections between computation and
geometry that can occur in self-assembly.
In this model, tiles can stick to an existing assembly as soon as one of
their sides matches the existing assembly. This feature contrasts with the
general cooperative model, where it can be required that tiles match on
\emph{several} of their sides in order to bind.
In order to describe our algorithms, we also introduce a generalization of
regular expressions called Baggins expressions. Finally, we compare this model
to other automata-theoretic models.Comment: A few bug fixes and typo correction
The 3D abstract Tile Assembly Model is Intrinsically Universal
In this paper, we prove that the three-dimensional abstract Tile Assembly Model (3DaTAM) is intrinsically universal. This means that there is a universal tile set in the 3DaTAM which can be used to simulate any 3DaTAM system. This result adds to a body of work on the intrinsic universality of models of self-assembly, and is specifically motivated by a result in FOCS 2016 showing that any intrinsically universal tile set for the 2DaTAM requires nondeterminism (i.e. undirectedness) even when simulating directed systems. To prove our result we have not only designed, but also fully implemented what we believe to be the first intrinsically universal tile set which has been implemented and simulated in any tile assembly model, and have made it and a simulator which can display it freely available
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
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