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

Abstract

Motivated by applications in DNA-nanotechnology, theoretical investigations in algorithmic tile-assembly have blossomed into a mature theory. In addition to computational universality, the abstract Tile Assembly Model (aTAM) was shown to be intrinsically universal (FOCS 2012), a strong notion of completeness where a single tile set is capable of simulating the full dynamics of all systems within the model; however, this construction fundamentally required non-deterministic tile attachments. This was confirmed necessary when it was shown that the class of directed aTAM systems, those where all possible sequences of tile attachments result in the same terminal assembly, is not intrinsically universal (FOCS 2016). Furthermore, it was shown that the non-cooperative aTAM, where tiles only need to match on 1 side to bind rather than 2 or more, is not intrinsically universal (SODA 2014) nor computationally universal (STOC 2017). Building on these results to further investigate the other dynamics, Hader et al. examined several tile-assembly models which varied across (1) the numbers of dimensions used, (2) how tiles diffused through space, and (3) whether each system is directed, and determined which models exhibited intrinsic universality (SODA 2020). In this paper we extend those results to provide direct comparisons of the various models against each other by considering intrinsic simulations between models. Our results show that in some cases, one model is strictly more powerful than another, and in others, pairs of models have mutually exclusive capabilities. This paper is a greatly expanded version of that which appeared in ICALP 2023.