## Abstract

We study the computational complexity of the recently proposed nubots model of molecular-scale self-assembly. 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.

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## Notes

- 1.
\({\mathrm{NC}}\), or Nick’s class, is named after Nicholas Pippenger.

- 2.
By confluent we mean a kind of determinism where the system (rules with the input) is assumed to always make a unique single terminal assembly.

- 3.
In Woods et al. (2013) the nubots model includes “agitation”: each monomer is repeatedly subjected to random movements intended to model a nano-scale environment where there is Brownian motion, uncontrolled movements and turbulent fluid flows in all directions. Our constructions in this paper work with or without agitation, hence they are robust to random uncontrolled movements, but we choose to ignore this issue and not formally define agitation for ease of presentation.

- 4.
For simplicity, when counting the number of applicable rules for a configuration, a movement rule is counted twice, to account for the two choices of arm and base.

- 5.
Our choice of a 1D, rather than 2D, encoding simplifies our constructions. It would also be possible use a more direct 2D square encoding, which, it turns out, can be unfolded to and from our line encoding in expected time \(O(\log n)\). We omit the details.

- 6.
Our configurations include an output tape write symbol and an output tape head position which is not standard practice (Papadimitriou 1994), but will be useful in our construction.

- 7.
Each configuration is of length polynomial in \(|x| = O(|\) \([\widetilde{x}]\) \(|)\), hence including \([\widetilde{x}]\) here does not change the asymptotics.

- 8.
The line segments in an encoded matrix usually encode the matrix element’s \((i, j)\) coordinates, here we do things slightly differently: we are using encoded configurations, rather than natural numbers, as the matrix indices. This simplifies our constructions a little.

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## Acknowledgments

We thank Erik Winfree for valuable discussion and suggestions on our results, Paul Rothemund for stimulating conversations on molecular muscle, Niall Murphy for informative discussions on circuit complexity theory, and Dhiraj Holden and Dave Doty for useful discussions. Damien thanks Beverley Henley for introducing him to developmental biology many moons ago. Supported by National Science Foundation grants CCF-1219274, 0832824 (The Molecular Programming Project), and CCF-1162589.

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Preliminary version appeared at The 19th International Conference on DNA Computing and Molecular Programming (DNA 19).

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Chen, M., Xin, D. & Woods, D. Parallel computation using active self-assembly.
*Nat Comput* **14, **225–250 (2015). https://doi.org/10.1007/s11047-014-9432-y

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### Keywords

- Molecular robotics
- Self-assembly
- Computational complexity