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
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We study the computational complexity of the recently proposed nubots model of molecular-scale self-assembly. The model generalizes asynchronous cellular automaton to have non-local movement where large assemblies of molecules can be moved around, analogous to millions of molecular motors in animal muscle effecting the rapid movement of large arms and legs. We show that nubots is capable of simulating Boolean circuits of polylogarithmic depth and polynomial size, in only polylogarithmic expected time. In computational complexity terms, any problem from the complexity class NC is solved in polylogarithmic expected time on nubots that use a polynomial amount of workspace. Along the way, we give fast parallel algorithms for a number of problems including line growth, sorting, Boolean matrix multiplication and space-bounded Turing machine simulation, all using a constant number of nubot states monomer types. Circuit depth is a well-studied notion of parallel time, and our result implies that nubots is a highly parallel model of computation in a formal sense. Thus, adding a movement primitive to an asynchronous non-deterministic cellular automation, as in nubots, drastically increases its parallel processing abilities. 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.