Parallel computation using active self-assembly

TL;DR

This paper demonstrates that the nubot model of molecular self-assembly can efficiently simulate complex parallel computations, significantly enhancing the computational power of asynchronous cellular automata by incorporating movement primitives.

Contribution

It introduces the nubot model's ability to perform polylogarithmic time parallel computation, surpassing traditional models like cellular automata, and provides algorithms for key computational problems within this framework.

Findings

Nubots can simulate Boolean circuits of polylogarithmic depth in polylogarithmic expected time.

Any problem in NC can be solved efficiently using nubots with polynomial workspace.

The model enables fast parallel algorithms for line growth, sorting, matrix multiplication, and Turing machine simulation.

Abstract

We study the computational complexity of the recently proposed nubot model of molecular-scale self-assembly. The model generalises asynchronous cellular automata to have non-local movement where large assemblies of molecules can be pushed and pulled around, analogous to millions of molecular motors in animal muscle effecting the rapid movement of macroscale arms and legs. We show that the nubot model 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 solvable in polylogarithmic expected time and polynomial workspace using nubots. Along the way, we give fast parallel nubot 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 the nubot model is a highly parallel model of computation in a formal sense. Asynchronous cellular automata are not capable of this parallelism, and our result shows that adding a rigid-body movement primitive to such a model, to get the nubot model, drastically increases parallel processing abilities.