Turning Machines: a simple algorithmic model for molecular robotics

TL;DR

This paper introduces Turning Machines, a simple abstract model for molecular robotics that uses asynchronous folding instructions to fold linear chains into complex shapes, analyzing their capabilities and limitations.

Contribution

It presents a novel 1D to 2D folding model with tools to analyze shape transformations and demonstrates fundamental primitives for shape folding in molecular robotics.

Findings

Line rotations up to 5π/3 radians are possible and efficient.

Arbitrarily large zig-zag shapes can be folded.

Some shapes are proven impossible to fold despite traversable paths.

Abstract

Molecular robotics is challenging, so it seems best to keep it simple. We consider an abstract molecular robotics model based on simple folding instructions that execute asynchronously. Turning Machines are a simple 1D to 2D folding model, also easily generalisable to 2D to 3D folding. A Turning Machine starts out as a line of connected monomers in the discrete plane, each with an associated turning number. A monomer turns relative to its neighbours, executing a unit-distance translation that drags other monomers along with it, and through collective motion the initial set of monomers eventually folds into a programmed shape. We provide a suite of tools for reasoning about Turning Machines by fully characterising their ability to execute line rotations: executing an almost-full line rotation of $5\pi/3$ radians is possible, yet a full $2\pi$ rotation is impossible. Furthermore, line rotations up to $5\pi/3$ are executed efficiently, in $O(\log n)$ expected time in our continuous time Markov chain time model. We then show that such line-rotations represent a fundamental primitive in the model, by using them to efficiently and asynchronously fold shapes. In particular, arbitrarily large zig-zag-rastered squares and zig-zag paths are foldable, as are $y$-monotone shapes albeit with error (bounded by perimeter length). Finally, we give shapes that despite having paths that traverse all their points, are in fact impossible to fold, as well as techniques for folding certain classes of (scaled) shapes without error. Our approach relies on careful geometric-based analyses of the feats possible and impossible by a very simple robotic system, and pushes conceptional hardness towards mathematical analysis and away from molecular implementation.