On the behavior of tile assembly system at high temperatures

TL;DR

This paper explores the complex behavior of tile assembly systems at high temperatures, establishing their computational hardness and the necessity of high temperatures for certain shape assemblies.

Contribution

It introduces a novel connection between SAT, threshold programming, and TAS behavior, proving NP-hardness results for temperature optimization and shape assembly.

Findings

High temperatures are necessary for assembling some shapes.

NP-hardness of optimizing TAS temperature settings.

Complexity results for minimal TAS size at given temperatures.

Abstract

Behaviors of Winfree's tile assembly systems (TASs) at high temperatures are investigated in combination with integer programming of a specific form called threshold programming. First, we propose a way to build bridges from the Boolean satisfiability problem (SAT) to threshold programming, and further to TAS's behavior, in order to prove the NP-hardness of optimizing temperatures of TASs that behave in a way given as input. These bridges will take us further to two important results on the behavior of TASs at high temperatures. The first says that arbitrarily high temperatures are required to assemble some shape by a TAS of "reasonable" size. The second is that for any temperature at least 4 given as a parameter, it is NP-hard to find the minimum size TAS that self-assembles a given shape and works at the given temperature or below.