A linear bound for the size of the finite terminal assembly of a directed non-cooperative tile assembly system

TL;DR

This paper proves that non-determinism is essential for assembling efficient paths in directed non-cooperative tile assembly systems, establishing an optimal bound for constructing squares with minimal tile types.

Contribution

It demonstrates the necessity of non-determinism in assembling efficient paths and establishes an asymptotic optimal bound for square assembly in non-cooperative aTAM.

Findings

Non-determinism is strictly necessary for efficient path assembly.

Constructing a square of width n with 2n-1 tile types is asymptotically optimal.

The techniques may improve understanding of non-directed tile assembly systems.

Abstract

The abstract tile assembly model (aTam) is a model of DNA self-assembly. Most of the studies focus on cooperative aTAM where a form of synchronization between the tiles is possible. Simulating Turing machines is achievable in this context. Few results and constructions are known for the non-cooperative case (a variant of Wang tilings where assemblies do not need to cover the whole plane and some mismatches may occur). Introduced by P.E. Meunier, efficient paths are a non-trivial construction for non-cooperative aTAM designed with $n$ different tile types and reaching a distance linearly greater than n. Later, efficient paths were improved to be able to reach a distance of n log(n). Assembling them relies heavily on a form of ``non-determinism''. Indeed, the set of tiles may produce different finite terminal assemblies but they all contain the same efficient path, a model called directed non-cooperative aTAM. This variant of aTAM is the only one who was shown to be decidable. In this paper, we prove that this non-determinism is strictly necessary for assembling the efficient paths. This result also implies that the construction of a square of width n using 2n-1 tiles types is asymptotically optimal. Moreover, we hope that the techniques introduced here will lead to a better comprehension of the non-directed case.