Many proposals have already been made for realizing programmable matter, ranging from shape-changing molecules, DNA tiles, and synthetic cells to reconfigurable modular robotics. Envisioning systems of nano-sensors devices, we are particularly interested in programmable matter consisting of systems of simple computational elements, called particles, that can establish and release bonds and can actively move in a self-organized way, and in shape formation problems relevant for programmable matter in those self-organizing particle systems (SOPS). In this paper, we present a general algorithmic framework for shape formation problems in SOPS, and show direct applications of this framework to the problems of having the particle system self-organize to form a hexagonal or triangular shape. Our algorithms utilize only local control, require only constant-size memory particles, and are asymptotically optimal both in terms of the total number of movements needed to reach the desired shape configuration.
The model @cite_6 @cite_15 aims at providing the theoretical framework that would allow for a more rigorous algorithmic study of biomolecular-inspired systems, more specifically of self-assembly systems with active molecular components. While bio-molecular inspired systems share many similarities with our SOPS, there are many differences --- e.g., there is always an arbitrarily large supply of "extra" particles that can be added to the system as needed, and the system allows for an additional (non-local) notion of rigid-body movement.
@cite_15 @cite_6
2084011432 2949409400 2085992503 2963927118
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 its length. Our main results show how to grow arbitrary connected two-dimensional geometric shapes and patterns in expected time 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 monomers required by the Kolmogorov complexity of the shape or pattern. This work thus highlights the fundamental efficiency advantage of active self-assembly over passive self-assembly and motivates experimental effort to construct self-assembly systems with active molecular components. 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. Self-assembly of active, robotic agents, rather than of passive agents such as molecules, is an emerging research field that is attracting increasing attention. Active self-assembly techniques are especially attractive at very small spatial scales, where alternative construction methods are unavailable or have severe limitations. Building nanostructures by using swarms of very simple nanorobots is a promising approach for manufacturing nanoscale devices and systems. The method described in this paper allows a group of simple, physically identical, identically programmed and reactive (i.e., stateless) agents to construct and repair polygonal approximations to arbitrary structures in the plane. The distributed algorithms presented here are tolerant of robot failures and of externally-induced disturbances. The structures are self-healing, and self-replicating in a weak sense. Their components can be re-used once the structures are no longer needed. A specification of vertices at relative positions, and the edges between them, is translated by a compiler into reactive rules for assembly agents. These rules lead to the construction and repair of the specified shape. Simulation results are presented, which validate the proposed algorithms. Molecular self-assembly, the formation of large structures by small pieces of matter sticking together according to simple local interactions, is a ubiquitous phenomenon. A challenging engineering goal is to design a few molecules so that large numbers of them can self-assemble into desired complicated target objects. Indeed, we would like to understand the ultimate capabilities and limitations of this bottom-up fabrication process. We look to theoretical models of algorithmic self-assembly, where small square tiles stick together according to simple local rules in order to carry out a crystal growth process. In this survey, we focus on the use of simulation between such models to classify and separate their computational and expressive powers. Roughly speaking, one model simulates another if they grow the same structures, via the same dynamical growth processes. Our journey begins with the result that there is a single intrinsically universal tile set that, with appropriate initialization and spatial scaling, simulates any instance of Winfree's abstract Tile Assembly Model. This universal tile set exhibits something stronger than Turing universality: it captures the geometry and dynamics of any simulated system in a very direct way. From there we find that there is no such tile set in the more restrictive non-cooperative model, proving it weaker than the full Tile Assembly Model. In the two-handed model, where large structures can bind together in one step, we encounter an infinite set of infinite hierarchies of 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 simulates any tile assembly system. We find another tile that aperiodically tiles the plane (but with small gaps). These and other recent results show that simulation is giving rise to a kind of computational complexity theory for self-assembly. It seems this could be the beginning of a much longer journey, so directions for future work are suggested.