A Time Lower Bound for Multiple Nucleation on a Surface

TL;DR

This paper proves a fundamental time lower bound for multiple nucleation in tile self-assembly and robotic systems, showing such systems cannot solve certain problems faster than a constant time when limited to local rules.

Contribution

It establishes a theoretical lower bound on the speed of multiple nucleation in local-rule tile and robotic self-assembly systems, extending to 3D robotic assembly.

Findings

Multiple nucleation cannot solve simple problems in constant time under local rules.

The lower bound applies to both 2D tile systems and 3D robotic assembly.

A new distributed computing model was introduced for the proof.

Abstract

Majumder, Reif and Sahu have presented a stochastic model of reversible, error-permitting, two-dimensional tile self-assembly, and showed that restricted classes of tile assembly systems achieved equilibrium in (expected) polynomial time. One open question they asked was how much computational power would be added if the model permitted multiple nucleation, i.e., independent groups of tiles growing before attaching to the original seed assembly. This paper provides a partial answer, by proving that if a tile assembly model uses only local binding rules, then it cannot use multiple nucleation on a surface to solve certain "simple" problems in constant time (time independent of the size of the surface). Moreover, this time bound applies to macroscale robotic systems that assemble in a three-dimensional grid, not just to tile assembly systems on a two-dimensional surface. The proof technique defines a new model of distributed computing that simulates tile (and robotic) self-assembly. Keywords: self-assembly, multiple nucleation, locally checkable labeling.