Pattern overlap implies runaway growth in hierarchical tile systems

TL;DR

This paper proves that in hierarchical tile assembly, pattern overlaps lead to infinite growth, highlighting the importance of avoiding such overlaps for finite structure control.

Contribution

It establishes a fundamental link between pattern overlaps and unbounded growth in hierarchical tile systems, simplifying previous conditions for finite assembly.

Findings

Pattern overlaps imply arbitrarily large assemblies.

Finite, unique assemblies must avoid pattern overlaps.

Simplifies conditions for controlled assembly growth.

Abstract

We show that in the hierarchical tile assembly model, if there is a producible assembly that overlaps a nontrivial translation of itself consistently (i.e., the pattern of tile types in the overlap region is identical in both translations), then arbitrarily large assemblies are producible. The significance of this result is that tile systems intended to controllably produce finite structures must avoid pattern repetition in their producible assemblies that would lead to such overlap. This answers an open question of Chen and Doty (SODA 2012), who showed that so-called "partial-order" systems producing a unique finite assembly *and" avoiding such overlaps must require time linear in the assembly diameter. An application of our main result is that any system producing a unique finite assembly is automatically guaranteed to avoid such overlaps, simplifying the hypothesis of Chen and Doty's main theorem.