Parallel computation using active self-assembly

Abstract

We study the computational complexity of the recently proposed nubots model of molecular-scale selfassembly. The model generalises asynchronous cellular automata 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 macroscale 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, we show that 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. Asynchronous cellular automata are not capable of such parallelism, and our result shows that adding a movement primitive to such a model, to get the nubots model, drastically increases parallel processing abilities.

Terminating distributed construction of shapes and patterns in a fair solution of automata

“…The main result of [43] was that any computable shape of size ≤ n × n can be built in time polylogarithmic in n, plus roughly the time needed to simulate a TM that computes whether or not a given pixel is in the final shape. One of the main differences between the Nubot model and our model is that in the former the nodes are equipped with an active actuation mechanism (see also [14] for another study of active self-assembly). This means that nodes (representing monomers there) are capable of firing transition rules that apart from changing their state can also change their relative position to neighboring nodes.…”

“…Still there are some important differences that set our model apart from the signal-passing tiles model. The most crucial one, is that in signal-passing tiles (and in the vast majority of algorithmic self-assembly models) there is an unlimited supply of tiles and any global parameter of the target configuration, such as its size n, must be somehow explicitly encoded in advance (as input), e.g., by assigning to each tile a number of glues that depends on n or, as in [14], by starting from an initial line of length log n. In contrast, in our model n is always the number of nodes in the system, their number remaining unmodified throughout the execution, and, additionally, the nodes do not know n in advance and have to coordinate in order to compute it and become capable of constructing a sufficiently large shape (i.e., one that depends on the size of the system). Other important differences are the existence of various types of glues in signal-passing tile assembly and also temperature and strength parameters that determine stability of a configuration, whereas in our model stability only depends on the local states of nodes and their position in the configuration.…”

A Stochastic Approach to Shortcut Bridging in Programmable Matter

“…In the context of molecular programming, our model most closely relates to the nubot model by Woods et al [29,7], which seeks to provide a framework for rigorous algorithmic research on selfassembly systems composed of active molecular components, emphasizing the interactions between molecular structure and active dynamics. This model shares many characteristics of our amoebot model (e.g., space is modeled as a triangular grid, nubot monomers have limited computational abilities, and there is no global orientation) but differs in that nubot monomers can replicate or die and can perform coordinated rigid body movements.…”

On the Runtime of Universal Coating for Programmable Matter

“…In the context of molecular programming, our model most closely relates to the nubot model by Woods et al [18,19], which seeks to provide a framework for rigorous algorithmic research on self-assembly systems composed of active molecular components, emphasizing the interactions between molecular structure and active dynamics. This model shares many characteristics of our amoebot model (e.g., space is modeled as a triangular grid, nubot monomers have limited computational abilities, and there is no global orientation) but differs in that nubot monomers can replicate or die and can perform coordinated rigid body movements.…”