Two Hands Are Better Than One (up to constant factors)

TL;DR

This paper compares seeded and seedless two-handed tile self-assembly models, demonstrating the greater power of the latter in simulating systems, constructing shapes, and the complexity of verification.

Contribution

It shows how to simulate seeded systems with two-handed systems, constructs shapes with separation in tile complexity, and proves verification is co-NP-complete in the two-handed model.

Findings

Two-handed model can simulate seeded systems with only constant factor increase.

Finite shapes exhibit a busy-beaver separation in tile complexity.

Verification of unique assembly is co-NP-complete in the two-handed model.

Abstract

We study the difference between the standard seeded model of tile self-assembly, and the "seedless" two-handed model of tile self-assembly. Most of our results suggest that the two-handed model is more powerful. In particular, we show how to simulate any seeded system with a two-handed system that is essentially just a constant factor larger. We exhibit finite shapes with a busy-beaver separation in the number of distinct tiles required by seeded versus two-handed, and exhibit an infinite shape that can be constructed two-handed but not seeded. Finally, we show that verifying whether a given system uniquely assembles a desired supertile is co-NP-complete in the two-handed model, while it was known to be polynomially solvable in the seeded model.