536 research outputs found
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
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
Combinatorial optimization problems in self-assembly
Self-assembly is the ubiquitous process by which simple objects autonomously assemble into intricate complexes. It has been suggested that intricate self-assembly processes will ultimately be used in circuit fabrication, nano-robotics, DNA computation, and amorphous computing. In this paper, we study two combinatorial optimization problems related to efficient self-assembly of shapes in the Tile Assembly Model of self-assembly proposed by Rothemund and Winfree [18]. The first is the Minimum Tile Set Problem, where the goal is to find the smallest tile system that uniquely produces a given shape. The second is the Tile Concentrations Problem, where the goal is to decide on the relative concentrations of different types of tiles so that a tile system assembles as quickly as possible. The first problem is akin to finding optimum program size, and the second to finding optimum running time for a "program" to assemble the shape.Self-assembly is the ubiquitous process by which simple objects autonomously assemble into intricate complexes. It has been suggested that intricate self-assembly processes will ultimately be used in circuit fabrication, nano-robotics, DNA computation, and amorphous computing. In this paper, we study two combinatorial optimization problems related to efficient self-assembly of shapes in the Tile Assembly Model of self-assembly proposed by Rothemund and Winfree [18]. The first is the Minimum Tile Set Problem, where the goal is to find the smallest tile system that uniquely produces a given shape. The second is the Tile Concentrations Problem, where the goal is to decide on the relative concentrations of different types of tiles so that a tile system assembles as quickly as possible. The first problem is akin to finding optimum program size, and the second to finding optimum running time for a "program" to assemble the shape. We prove that the first problem is NP-complete in general, and polynomial time solvable on trees and squares. In order to prove that the problem is in NP, we present a polynomial time algorithm to verify whether a given tile system uniquely produces a given shape. This algorithm is analogous to a program verifier for traditional computational systems, and may well be of independent interest. For the second problem, we present a polynomial time -approximation algorithm that works for a large class of tile systems that we call partial order systems
Program Size and Temperature in Self-Assembly
Winfree’s abstract Tile Assembly Model is a model of molecular self-assembly of DNA complexes known as tiles, which float freely in solution and attach one at a time to a growing “seed” assembly based on specific binding sites on their four sides. We show that there is a polynomial-time algorithm that, given an n×n square, finds the minimal tile system (i.e., the system with the smallest number of distinct tile types) that uniquely self-assembles the square, answering an open question of Adleman et al. (Combinatorial optimization problems in self-assembly, STOC 2002). Our investigation leading to this algorithm reveals other positive and negative results about the relationship between the size of a tile system and its “temperature” (the binding strength threshold required for a tile to attach)
Directed Percolation and the Abstract Tile Assembly Model
Self-assembly is a process by which simple components build complex structures through local interactions. Directed percolation is a statistical physical model for describing competitive spreading processes on lattices. The author describes an algorithm which can transform a tile assembly system in the abstract Tile Assembly Model into a directed percolation problem, and then shows simulations of the aTAM which support this algorithm. The author also investigates two new constructs designed for Erik Winfree\u27s abstract Tile Assembly Model called the NULL tile and temperature 1.5. These constructs aid the translation between self-assembly and directed percolation and may assist self-assembly researchers in designing tilesets in the aTAM with non-deterministic local properties, but guaranteed global properties. Temperature 1.5 results indicate the brittleness of the standard temperature 2 tile assembly system, and the NULL tile is shown to assist simulations of large assembly processes while also reinforcing the need for variable temperature models to more closely simulate laboratory self-assembly
Polyominoes Simulating Arbitrary-Neighborhood Zippers and Tilings
This paper provides a bridge between the classical tiling theory and the
complex neighborhood self-assembling situations that exist in practice. The
neighborhood of a position in the plane is the set of coordinates which are
considered adjacent to it. This includes classical neighborhoods of size four,
as well as arbitrarily complex neighborhoods. A generalized tile system
consists of a set of tiles, a neighborhood, and a relation which dictates which
are the "admissible" neighboring tiles of a given tile. Thus, in correctly
formed assemblies, tiles are assigned positions of the plane in accordance to
this relation. We prove that any validly tiled path defined in a given but
arbitrary neighborhood (a zipper) can be simulated by a simple "ribbon" of
microtiles. A ribbon is a special kind of polyomino, consisting of a
non-self-crossing sequence of tiles on the plane, in which successive tiles
stick along their adjacent edge. Finally, we extend this construction to the
case of traditional tilings, proving that we can simulate
arbitrary-neighborhood tilings by simple-neighborhood tilings, while preserving
some of their essential properties.Comment: Submitted to Theoretical Computer Scienc
COMPUTER SIMULATION AND COMPUTABILITY OF BIOLOGICAL SYSTEMS
The ability to simulate a biological organism by employing a computer is related to the
ability of the computer to calculate the behavior of such a dynamical system, or the "computability" of the system.* However, the two questions of computability and simulation are not equivalent. Since the question of computability can be given a precise answer in terms of recursive functions, automata theory and dynamical systems, it will be appropriate to consider it first. The more elusive question of adequate simulation of biological systems by a computer will be then addressed and a possible connection between the two answers given will be considered. A conjecture is formulated that suggests the possibility of employing an algebraic-topological, "quantum" computer (Baianu, 1971b)
for analogous and symbolic simulations of biological systems that may include chaotic processes that are not, in genral, either recursively or digitally computable. Depending on the biological network being modelled, such as the Human Genome/Cell Interactome or a trillion-cell Cognitive Neural Network system, the appropriate logical structure for such simulations might be either the Quantum MV-Logic (QMV) discussed in recent publications (Chiara, 2004, and references cited therein)or Lukasiewicz Logic Algebras that were shown to be isomorphic to MV-logic algebras (Georgescu et al, 2001)
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