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：算法自组装发生在无序组件自主结合形成结构的过程中，且凭借其设计及系统动力学特性，迫使这些结构遵循算法执行。受DNA纳米技术应用驱动，基于算法瓦片自组装的研究已发展成成熟理论，研究者利用可计算性理论、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。除计算通用性外，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，即单一瓦片集能模拟该模型内的所有系统；然而，该结果需要非确定性瓦片附着。此后，当有向aTAM系统类被证明不具有IU时，这一条件被确认为必要。基于上述结果，为深入探究其他动力学因素的影响，Hader等人研究了多种瓦片组装模型，这些模型在以下三方面存在差异：（1）所用维度数量；（2）基于瓦片在空间中扩散的限制；（3）各系统是否具有有向性。他们证明了哪些模型具有IU。这些结果揭示了瓦片组装动力学的多方面因素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，直接对比了各类模型。我们证明在某些情况下，一种模型比另一种更强大；而在其他情况下，成对模型具有互斥能力。这种比较有助于揭示这三个重要因素的影响，并进一步定义瓦片组装模型的层次结构。