The two-handed tile assembly model is not intrinsically universal

TL;DR

This paper investigates the universality of the Two-Handed Tile Assembly Model (2HAM), proving it is not intrinsically universal across different temperatures but is universal within fixed temperature classes, revealing hierarchical simulation capabilities.

Contribution

The paper establishes that 2HAM is not intrinsically universal across all temperatures but is universal within each fixed temperature, and introduces hierarchies of simulation power.

Findings

2HAM systems at different temperatures cannot simulate each other across temperature thresholds

For each fixed temperature, a universal 2HAM tile set exists that can simulate all systems at that temperature

Hierarchies of increasing simulation power are demonstrated within the 2HAM model

Abstract

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 $\tau$, where $\tau$ 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 $\tau' < \tau$, each temperature-$\tau'$ 2HAM tile system does not 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 (single-tile addition only) abstract Tile Assembly Model which is intrinsically universal ("The tile assembly model is intrinsically universal", FOCS 2012). However, on the positive side, we prove that, for every fixed temperature $\tau \geq 2$, temperature-$\tau$ 2HAM tile systems are indeed intrinsically universal: in other words, 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 simulation power within each hierarchy. Finally, we show that for each $\tau$, there is a temperature-$\tau$ 2HAM system that simultaneously simulates all temperature-$\tau$ 2HAM systems.