Powers and Behaviors of Directed Self-assembly

In nature there are a variety of self-assembling systems occurring at varying scales which give rise to incredibly complex behaviors. Theoretical models of self-assembly allow us to gain insight into the fundamental nature of self-assembly independent of the specific physical implementation. In Winfree\u27s abstract tile assembly model (aTAM), the atomic components are unit square tiles which have glues on their four sides. Beginning from a seed assembly, these tiles attach one at a time during the assembly process in an asynchronous and nondeterministic manner. We can gain valuable insights into the nature of self-assembly by comparing different models of self-assembly which use fundamentally different mechanisms for local interactions. A powerful notion which allows us to compare models of self-assembly is simulation. The first result of this thesis examines the role of non-determinism in simulation. It shows that the universal simulation of directed aTAM systems requires undirectedness. A tile assembly model is said to be directed if it always assembles the same final assembly. We distinguish between two types of aTAM systems: cooperative systems and non-cooperative systems. In cooperative aTAM systems, we are able to enforce that in order for a tile to attach to an assembly, the glues of a tile must match two or more glues of neighboring tiles. On the other hand, in non-cooperative aTAM systems, tiles are able to attach to an assembly provided that one of the tile\u27s glues match an exposed glue on the assembly. It is well known that the cooperative aTAM is computationally universal, and it is conjectured that the non-cooperative aTAM is not computationally universal. For our second result, we show that if we allow tiles to be polygons with six or more sides, then the class of non-cooperative systems is capable of universal computation. On the other hand, we show that the class of systems consisting of polygons with six or less sides is not capable of computing using any of the currently known methods

The 3D abstract Tile Assembly Model is Intrinsically Universal

In this paper, we prove that the three-dimensional abstract Tile Assembly Model (3DaTAM) is intrinsically universal. This means that there is a universal tile set in the 3DaTAM which can be used to simulate any 3DaTAM system. This result adds to a body of work on the intrinsic universality of models of self-assembly, and is specifically motivated by a result in FOCS 2016 showing that any intrinsically universal tile set for the 2DaTAM requires nondeterminism (i.e. undirectedness) even when simulating directed systems. To prove our result we have not only designed, but also fully implemented what we believe to be the first intrinsically universal tile set which has been implemented and simulated in any tile assembly model, and have made it and a simulator which can display it freely available

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

Algorithmic self-assembly occurs when components in a disorganized collection autonomously combine to form structures and, by their design and the dynamics of the system, are forced to intrinsically follow the execution of algorithms. Motivated by applications in DNA-nanotechnology, theoretical investigations in algorithmic tile-based self-assembly have blossomed into a mature theory with research strongly leveraging tools from computability theory, complexity theory, information theory, and graph theory to develop a wide range of models and to show that many are computationally universal, while also exposing a wide variety of powers and limitations of each. 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 result fundamentally required non-deterministic tile attachments. This was later confirmed necessary when it was shown that the class of directed aTAM systems, those in which all possible sequences of tile attachments eventually 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 impacts of other dynamics, Hader et al. examined several tile-assembly models which varied across (1) the numbers of dimensions used, (2) restrictions imposed on the diffusion of tiles through space, and (3) whether each system is directed, and determined which models exhibited intrinsic universality (SODA 2020). Such results have shed much light on the roles of various aspects of the dynamics of tile-assembly and their effects on the universality of each model. In this paper we extend that previous work 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 direct comparison of models helps expose the impacts of these three important aspects of self-assembling systems, and further helps to define a hierarchy of tile-assembly models analogous to the hierarchies studied in traditional models of computation

Self-Assembly of 4-sided Fractals in the Two-handed Tile Assembly Model

We consider the self-assembly of fractals in one of the most well-studied models of tile based self-assembling systems known as the Two-handed Tile Assembly Model (2HAM). In particular, we focus our attention on a class of fractals called discrete self-similar fractals (a class of fractals that includes the discrete Sierpi\'nski carpet). We present a 2HAM system that finitely self-assembles the discrete Sierpi\'nski carpet with scale factor 1. Moreover, the 2HAM system that we give lends itself to being generalized and we describe how this system can be modified to obtain a 2HAM system that finitely self-assembles one of any fractal from an infinite set of fractals which we call 4-sided fractals. The 2HAM systems we give in this paper are the first examples of systems that finitely self-assemble discrete self-similar fractals at scale factor 1 in a purely growth model of self-assembly. Finally, we show that there exists a 3-sided fractal (which is not a tree fractal) that cannot be finitely self-assembled by any 2HAM system

The Need for Seed (in the abstract Tile Assembly Model)

In the abstract Tile Assembly Model (aTAM) square tiles self-assemble, autonomously binding via glues on their edges, to form structures. Algorithmic aTAM systems can be designed in which the patterns of tile attachments are forced to follow the execution of targeted algorithms. Such systems have been proven to be computationally universal as well as intrinsically universal (IU), a notion borrowed and adapted from cellular automata showing that a single tile set exists which is capable of simulating all aTAM systems (FOCS 2012). The input to an algorithmic aTAM system can be provided in a variety of ways, with a common method being via the "seed" assembly, which is a pre-formed assembly from which all growth propagates. In this paper we present a series of results which investigate the the trade-offs of using seeds consisting of a single tile, versus those containing multiple tiles. We show that arbitrary systems with multi-tile seeds cannot be converted to functionally equivalent systems with single-tile seeds without using a scale factor > 1. We prove tight bounds on the scale factor required, and also present a construction which uses a large scale factor but an optimal number of unique tile types. That construction is then used to develop a construction that performs simultaneous simulation of all aTAM systems in parallel, as well as to display a connection to other tile-based self-assembly models via the notion of intrinsic universality.Comment: To appear in the SODA 2023 proceeding

Contrasting Geometric Variations of Mathematical Models of Self-assembling Systems

Self-assembly is the process by which complex systems are formed and behave due to the interactions of relatively simple units. In this thesis, we explore multiple augmentations of well known models of self-assembly to gain a better understanding of the roles that geometry and space play in their dynamics. We begin in the abstract Tile Assembly Model (aTAM) with some examples and a brief survey of previous results to provide a foundation. We then introduce the Geometric Thermodynamic Binding Network model, a model that focuses on the thermodynamic stability of its systems while utilizing geometrically rigid components (dissimilar to other thermodynamic models). We show that this model is computationally universal, an ability conjectured to be impossible in similar models with non-rigid components. We continue by introducing the Flexible Tile Assembly Model, a generalization of the 2D aTAM that allows bonds between tiles to flex and assemblies to therefore reconfigure. We show how systems in this model can deterministically assemble shapes that adhere to a number of certain restrictions. Finally, we introduce the Spatial abstract Tile Assembly Model, a variation of the 3D aTAM that restricts tiles from attaching without a diffusion path. We show that this model is intrinsically universal, a property of computational models to simulate themselves which has been shown for the 3D aTAM and other similar models. We conclude this thesis with a summary of the presented results, a brief impact analysis, and potential directions for future research within this area

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum