Algorithmic self-assembly occurs when disorganized components autonomously combine to form structures and, by their design and the dynamics of the system, are forced to follow the execution of algorithms. Motivated by applications in DNA-nanotechnology, investigations in algorithmic tile-based self-assembly have blossomed into a mature theory with research leveraging tools from computability theory, complexity theory, information theory, and graph theory to develop a wide range of models and show that many are computationally universal, while also exposing powers and limitations of each. Beyond computational universality, the abstract Tile Assembly Model (aTAM) was shown to be intrinsically universal (IU), a strong notion of completeness where a single tile set is capable of simulating all systems within the model; however, this result required non-deterministic tile attachments. This was later confirmed necessary when it was shown that the class of directed aTAM systems is not IU. Building on these results to further investigate the impacts of other dynamics, Hader et al. examined several tile-assembly models which varied across (1) the numbers of dimensions used, (2) restrictions based on diffusion of tiles through space, and (3) whether each system is directed, and showed which models are IU. Such results have shed much light on the roles of various aspects of the dynamics of tile-assembly and their effects on the intrinsic universality of each model. Here we provide direct comparisons of the various models by considering intrinsic simulations between models. We show that in some cases one model is more powerful than another, and in others, pairs of models have mutually exclusive capabilities. This comparison helps to expose the impacts of these three important aspects and further helps define a hierarchy of tile-assembly models.

翻译：摘要：算法自组装是指无序组件通过自身设计与系统动力学自主组合形成结构，并强制遵循算法执行的过程。受DNA纳米技术应用的推动，基于算法瓦片自组装的研究已发展为一套成熟理论，其研究利用可计算性理论、复杂性理论、信息论和图论工具，开发了多种模型，并证明了其中许多模型具有计算通用性，同时揭示了各模型的能力与局限性。在计算通用性之外，抽象瓦片组装模型（aTAM）被证明具有内在通用性（IU），这是一种强完备性概念——单一瓦片集能够模拟该模型内的所有系统；然而，该结果依赖非确定性瓦片附着。后续研究证实此条件不可或缺：有向aTAM系统类被证明不具备IU特性。基于这些成果，Hader等人进一步探究其他动力学的影响，考察了多个在（1）维度数、（2）瓦片空间扩散限制、（3）系统是否有向性三方面存在差异的瓦片组装模型，并揭示了哪些模型具有IU特性。这些结果深刻揭示了瓦片组装动力学的多方面作用及其对模型内在通用性的影响。本文通过考虑模型间的内在模拟，直接比较了不同模型。我们证明某些模型的能力强于其他模型，而另一些模型对则具有互斥能力。这种比较不仅揭示了上述三个重要方面的影响，更有助于定义瓦片组装模型的层次结构。