The Need for Seed (in the abstract Tile Assembly Model)

TL;DR

This paper investigates the trade-offs between single-tile and multi-tile seed assemblies in the abstract Tile Assembly Model, establishing bounds on scale factors and demonstrating universal simulation capabilities.

Contribution

It proves that multi-tile seeds cannot be replaced by single-tile seeds without increased scale, and introduces constructions for universal simulation of all aTAM systems.

Findings

Multi-tile seeds cannot be replaced by single-tile seeds without scale factor > 1.

Tight bounds are established on the scale factor required for simulation.

A construction achieves universal simulation of all aTAM systems with optimal tile types.

Abstract

In the abstract Tile Assembly Model (aTAM) square tiles self-assemble, autonomously binding via glues on their edges, to form structures. Algorithmic aTAM systems can be designed in which the patterns of tile attachments are forced to follow the execution of targeted algorithms. Such systems have been proven to be computationally universal as well as intrinsically universal (IU), a notion borrowed and adapted from cellular automata showing that a single tile set exists which is capable of simulating all aTAM systems (FOCS 2012). The input to an algorithmic aTAM system can be provided in a variety of ways, with a common method being via the "seed" assembly, which is a pre-formed assembly from which all growth propagates. In this paper we present a series of results which investigate the the trade-offs of using seeds consisting of a single tile, versus those containing multiple tiles. We show that arbitrary systems with multi-tile seeds cannot be converted to functionally equivalent systems with single-tile seeds without using a scale factor > 1. We prove tight bounds on the scale factor required, and also present a construction which uses a large scale factor but an optimal number of unique tile types. That construction is then used to develop a construction that performs simultaneous simulation of all aTAM systems in parallel, as well as to display a connection to other tile-based self-assembly models via the notion of intrinsic universality.