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.

翻译：抽象瓦片组装模型（aTam）是一种DNA自组装模型。多数研究集中于协同型aTam，其中瓦片间可实现某种同步，且在该背景下可模拟图灵机。而非协同情形（王浩瓦片的变体，其中组装体无需覆盖整个平面，且可能发生错配）的研究成果与构造较少。由P.E. Meunier引入的高效路径是针对非协同aTam的一种非平凡构造，该构造使用$n$种不同瓦片类型，可达距离线性大于$n$。随后，高效路径被改进至可达距离$n \log(n)$。其组装过程高度依赖某种形式的“非确定性”。事实上，瓦片集可能产生不同的有限终端组装体，但这些组装体均包含同一高效路径，该模型称为有向非协同aTam。aTam的这一变体是唯一被证明可判定的。本文证明了这种非确定性对于高效路径的组装严格必要。该结果还表明，使用$2n-1$种瓦片类型构造宽度为$n$的正方形在渐近意义下是最优的。此外，我们希望本文引入的技术能促进对非有向情形更深入的理解。