Limitations of Self-Assembly at Temperature One (extended abstract)

TL;DR

This paper proves that at temperature 1, tile assembly systems can only produce simple, periodic shapes unless negative glue strengths are used, indicating higher temperatures are needed for complex computation.

Contribution

It establishes a fundamental limitation of temperature 1 tile assembly systems under pumpability, showing they can only assemble simple patterns, and demonstrates that negative glues enable universal computation at temperature 1.

Findings

Temperature 1 systems produce only finite unions of doubly periodic sets.

Pumpability condition restricts the complexity of shapes at temperature 1.

Negative glue strengths enable general-purpose computation at temperature 1.

Abstract

We prove that if a subset X of the integer Cartesian plane weakly self-assembles at temperature 1 in a deterministic (Winfree) tile assembly system satisfying a natural condition known as *pumpability*, then X is a finite union of doubly periodic sets. This shows that only the most simple of infinite shapes and patterns can be constructed using pumpable temperature 1 tile assembly systems, and gives strong evidence for the thesis that temperature 2 or higher is required to carry out general-purpose computation in a tile assembly system. Finally, we show that general-purpose computation is possible at temperature 1 if negative glue strengths are allowed in the tile assembly model.