The Two-Handed Tile Assembly Model Is Not Intrinsically Universal

In this paper, we study the intrinsic universality of the well-studied Two-Handed Tile Assembly Model (2HAM), in which two "supertile" assemblies, each consisting of one or more unit-square tiles, can fuse together (self-assemble) whenever their total attachment strength is at least the global temperature tau. Our main result is that for all tau' < tau, each temperature-J-1 2HAM tile system cannot simulate at least one temperature-tau 2HAM tile system. This impossibility result proves that the 2HAM is not intrinsically universal, in stark contrast to the simpler abstract Tile Assembly Model which was shown to be intrinsically universal (The tile assembly model is intrinsically universal, FOCS 2012). On the positive side, we prove that, for every fixed temperature tau >= 2, temperature-tau 2HAM tile systems are intrinsically universal: for each tau there is a single universal 2HAM tile set U that, when appropriately initialized, is capable of simulating the behavior of any temperature tau 2HAM tile system. As a corollary of these results we find an infinite set of infinite hierarchies of 2HAM systems with strictly increasing power within each hierarchy. Finally, we show how to construct, for each tau, a temperature-tau 2HAM system that simultaneously simulates all temperature-tau 2HAM systems.