A pumping lemma for non-cooperative self-assembly

TL;DR

This paper proves that non-cooperative tile assembly models are computationally weak by demonstrating that long paths are pumpable, limiting the complexity of shapes that can be programmed in this model.

Contribution

It introduces a pumping lemma for non-cooperative tile assembly, establishing fundamental limitations and developing the visible glues method for future geometric problems.

Findings

Long paths in non-cooperative assembly are pumpable.

Only simple shapes can be programmed in this model.

Non-cooperative assembly has limited computational power.

Abstract

We prove the computational weakness of a model of tile assembly that has so far resisted many attempts of formal analysis or positive constructions. Specifically, we prove that, in Winfree's abstract Tile Assembly Model, when restricted to use only noncooperative bindings, any long enough path that can grow in all terminal assemblies is pumpable, meaning that this path can be extended into an infinite, ultimately periodic path. This result can be seen as a geometric generalization of the pumping lemma of finite state automata, and closes the question of what can be computed deterministically in this model. Moreover, this question has motivated the development of a new method called visible glues. We believe that this method can also be used to tackle other long-standing problems in computational geometry, in relation for instance with self-avoiding paths. Tile assembly (including non-cooperative tile assembly) was originally introduced by Winfree and Rothemund in STOC 2000 to understand how to program shapes. The non-cooperative variant, also known as temperature 1 tile assembly, is the model where tiles are allowed to bind as soon as they match on one side, whereas in cooperative tile assembly, some tiles need to match on several sides in order to bind. In this work, we prove that only very simple shapes can indeed be programmed, whereas exactly one known result (SODA 2014) showed a restriction on the assemblies general non-cooperative self-assembly could achieve, without any implication on its computational expressiveness. With non-square tiles (like polyominos, SODA 2015), other recent works have shown that the model quickly becomes computationally powerful.