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arXiv · 2405.18630

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

Sergiu Ivanov·Damien Regnault
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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.

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Sergiu Ivanov, Damien Regnault. 2026-05-07. A linear bound for the size of the finite terminal assembly of a directed non-cooperative tile assembly system. https://arxiv.org/abs/2405.18630

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