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Algorithmic self-assembly occurs when disorganized components autonomouslycombine to form structures and, by their design and the dynamics of the system,are forced to follow the execution of algorithms. Motivated by applications inDNA-nanotechnology, investigations in algorithmic tile-based self-assembly haveblossomed into a mature theory with research leveraging tools fromcomputability theory, complexity theory, information theory, and graph theoryto develop a wide range of models and show that many are computationallyuniversal, while also exposing powers and limitations of each. Beyondcomputational universality, the abstract Tile Assembly Model (aTAM) was shownto be intrinsically universal (IU), a strong notion of completeness where asingle tile set is capable of simulating all systems within the model; however,this result required non-deterministic tile attachments. This was laterconfirmed necessary when it was shown that the class of directed aTAM systemsis not IU. Building on these results to further investigate the impacts ofother dynamics, Hader et al. examined several tile-assembly models which variedacross (1) the numbers of dimensions used, (2) restrictions based on diffusionof tiles through space, and (3) whether each system is directed, and showedwhich models are IU. Such results have shed much light on the roles of variousaspects of the dynamics of tile-assembly and their effects on the intrinsicuniversality of each model. Here we provide direct comparisons of the variousmodels by considering intrinsic simulations between models. We show that insome cases one model is more powerful than another, and in others, pairs ofmodels have mutually exclusive capabilities. This comparison helps to exposethe impacts of these three important aspects and further helps define ahierarchy of tile-assembly models.

The Impacts of Dimensionality, Diffusion, and Directedness on Intrinsic Cross-Model Simulation in Tile-Based Self-Assembly