Paper detail

The two-handed tile assembly model is not intrinsically universal

The well-studied Two-Handed Tile Assembly Model (2HAM) is a model of tile assembly in which pairs of large assemblies can bind, or self-assemble, together. In order to bind, two assemblies must have matching glues that can simultaneously touch each other, and stick together with strength that is at least the temperature $τ$, where $τ$ is some fixed positive integer. We ask whether the 2HAM is intrinsically universal, in other words we ask: is there a single universal 2HAM tile set $U$ which can be used to simulate any instance of the model? Our main result is a negative answer to this question. We show that for all $τ&#39; < τ$, each temperature-$τ&#39;$ 2HAM tile system does not simulate at least one temperature-$τ$ 2HAM tile system. This impossibility result proves that the 2HAM is not intrinsically universal, in stark contrast to the simpler (single-tile addition only) abstract Tile Assembly Model which is intrinsically universal (&#34;The tile assembly model is intrinsically universal&#34;, FOCS 2012). However, on the positive side, we prove that, for every fixed temperature $τ\geq 2$, temperature-$τ$ 2HAM tile systems are indeed intrinsically universal: in other words, for each $τ$ there is a single universal 2HAM tile set $U$ that, when appropriately initialized, is capable of simulating the behavior of any temperature-$τ$ 2HAM tile system. As a corollary of these results we find an infinite set of infinite hierarchies of 2HAM systems with strictly increasing simulation power within each hierarchy. Finally, we show that for each $τ$, there is a temperature-$τ$ 2HAM system that simultaneously simulates all temperature-$τ$ 2HAM systems.