Size-Dependent Tile Self-Assembly: Constant-Height Rectangles and Stability

TL;DR

This paper introduces a size-dependent tile self-assembly model where stability depends on a variable temperature function, enabling the assembly of arbitrary shapes and revealing computational complexity in stability decisions.

Contribution

It proposes a novel size-dependent assembly model and demonstrates its power in shape assembly and computational complexity analysis.

Findings

Able to assemble arbitrary size rectangles and squares with variable temperature functions

Deciding supertile stability is coNP-complete

Contrasts with fixed temperature models in stability and assembly capabilities

Abstract

We introduce a new model of algorithmic tile self-assembly called size-dependent assembly. In previous models, supertiles are stable when the total strength of the bonds between any two halves exceeds some constant temperature. In this model, this constant temperature requirement is replaced by an nondecreasing temperature function $\tau : \mathbb{N} \rightarrow \mathbb{N}$ that depends on the size of the smaller of the two halves. This generalization allows supertiles to become unstable and break apart, and captures the increased forces that large structures may place on the bonds holding them together. We demonstrate the power of this model in two ways. First, we give fixed tile sets that assemble constant-height rectangles and squares of arbitrary input size given an appropriate temperature function. Second, we prove that deciding whether a supertile is stable is coNP-complete. Both results contrast with known results for fixed temperature.