On the Behavior of Tile Assembly System at High Temperatures

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 ($\ensuremath{\mbox{\rm S{\scriptsize AT}}}$) 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 and works at a temperature below τ.