On the Behavior of Tile Assembly System at High Temperatures

Abstract

Behavior of Winfree's tile assembly system (TAS) at high temperatures is 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 τ ≥ 4 given as a parameter, it is NP-hard to find the minimum size TAS that self-assembles a given shape at a temperature at most τ.