Intrinsic Universality in Self-Assembly

TL;DR

This paper demonstrates a universal tile assembly system capable of simulating a broad class of tile systems, including infinite and temperature-2 systems, advancing the understanding of intrinsic universality in self-assembly.

Contribution

It introduces a tile set that can simulate any locally consistent tile assembly system, including itself, at temperature 2, improving upon prior simulations by capturing the assembly process and infinite structures.

Findings

Simulates a wide class of tile systems, including infinite ones.

Represents each tile as a fixed-size block in the simulation.

Operates at temperature 2, unlike previous temperature 1 models.

Abstract

We show that the Tile Assembly Model exhibits a strong notion of universality where the goal is to give a single tile assembly system that simulates the behavior of any other tile assembly system. We give a tile assembly system that is capable of simulating a very wide class of tile systems, including itself. Specifically, we give a tile set that simulates the assembly of any tile assembly system in a class of systems that we call \emph{locally consistent}: each tile binds with exactly the strength needed to stay attached, and that there are no glue mismatches between tiles in any produced assembly. Our construction is reminiscent of the studies of \emph{intrinsic universality} of cellular automata by Ollinger and others, in the sense that our simulation of a tile system $T$ by a tile system $U$ represents each tile in an assembly produced by $T$ by a $c \times c$ block of tiles in $U$, where $c$ is a constant depending on $T$ but not on the size of the assembly $T$ produces (which may in fact be infinite). Also, our construction improves on earlier simulations of tile assembly systems by other tile assembly systems (in particular, those of Soloveichik and Winfree, and of Demaine et al.) in that we simulate the actual process of self-assembly, not just the end result, as in Soloveichik and Winfree's construction, and we do not discriminate against infinite structures. Both previous results simulate only temperature 1 systems, whereas our construction simulates tile assembly systems operating at temperature 2.