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 Two-Handed Tile Assembly Model Is Not Intrinsically Universal

In this paper, we study the intrinsic universality of the well-studied Two-Handed Tile Assembly Model (2HAM), in which two “supertile” assemblies, each consisting of one or more unit-square tiles, can fuse together (self-assemble) whenever their total attachment strength is at least the global temperature τ. Our main result is that for all τ′ < τ, each temperature-τ′ 2HAM tile system cannot simulate at least one temperature-τ 2HAM tile system. This impossibility result proves that the 2HAM is not intrinsically universal, in stark contrast to the simpler abstract Tile Assembly Model which was shown to be intrinsically universal (The tile assembly model is intrinsically universal, FOCS 2012). On the positive side, we prove that, for every fixed temperature τ ≥ 2, temperature-τ 2HAM tile systems are intrinsically universal: for each τ there is a single universal 2HAM tile set U that, when appropriately initialized, is capable of simulating the behavior of any temperature τ 2HAM tile system. As a corollary of these results we find an infinite set of infinite hierarchies of 2HAM systems with strictly increasing power within each hierarchy. Finally, we show how to construct, for each τ, a temperature-τ 2HAM system that simultaneously simulates all temperature-τ 2HAM systems

The Two-Handed Tile Assembly Model is not Intrinsically Universal

The Two-Handed Tile Assembly Model (2HAM) is a model of algorithmic self-assembly in which large structures, or assemblies of tiles, are grown by the binding of smaller assemblies. In order to bind, two assemblies must have matching glues that can simultaneously touch each other, and stick together with strength that is at least the temperature τ, where τ is some fixed positive integer. We ask whether the 2HAM is intrinsically universal. In other words, we ask: is there a single 2HAM tile set U which can be used to simulate any instance of the model? Our main result is a negative answer to this question. We show that for all τ′ < τ, each temperature-τ′ 2HAM tile system does not simulate at least one temperature-τ 2HAM tile system. This impossibility result proves that the 2HAM is not intrinsically universal and stands in contrast to the fact that the (single-tile addition) abstract Tile Assembly Model is intrinsically universal. On the positive side, we prove that, for every fixed temperature τ ≥ 2, temperature-τ 2HAM tile systems are indeed intrinsically universal. In other words, for each τ there is a single intrinsically universal 2HAM tile set U_τ that, when appropriately initialized, is capable of simulating the behavior of any temperature-τ 2HAM tile system. As a corollary, we find an infinite set of infinite hierarchies of 2HAM systems with strictly increasing simulation power within each hierarchy. Finally, we show that for each τ, there is a temperature-τ 2HAM system that simultaneously simulates all temperature-τ 2HAM systems

Signal Transmission Across Tile Assemblies: 3D Static Tiles Simulate Active Self-Assembly by 2D Signal-Passing Tiles

The 2-Handed Assembly Model (2HAM) is a tile-based self-assembly model in which, typically beginning from single tiles, arbitrarily large aggregations of static tiles combine in pairs to form structures. The Signal-passing Tile Assembly Model (STAM) is an extension of the 2HAM in which the tiles are dynamically changing components which are able to alter their binding domains as they bind together. For our first result, we demonstrate useful techniques and transformations for converting an arbitrarily complex STAM$^+$ tile set into an STAM$^+$ tile set where every tile has a constant, low amount of complexity, in terms of the number and types of ``signals'' they can send, with a trade off in scale factor. Using these simplifications, we prove that for each temperature $\tau>1$ there exists a 3D tile set in the 2HAM which is intrinsically universal for the class of all 2D STAM$^+$ systems at temperature $\tau$ (where the STAM$^+$ does not make use of the STAM's power of glue deactivation and assembly breaking, as the tile components of the 2HAM are static and unable to change or break bonds). This means that there is a single tile set $U$ in the 3D 2HAM which can, for an arbitrarily complex STAM$^+$ system $S$, be configured with a single input configuration which causes $U$ to exactly simulate $S$ at a scale factor dependent upon $S$. Furthermore, this simulation uses only two planes of the third dimension. This implies that there exists a 3D tile set at temperature $2$ in the 2HAM which is intrinsically universal for the class of all 2D STAM$^+$ systems at temperature $1$. Moreover, we show that for each temperature $\tau>1$ there exists an STAM$^+$ tile set which is intrinsically universal for the class of all 2D STAM$^+$ systems at temperature $\tau$, including the case where $\tau = 1$.Comment: A condensed version of this paper will appear in a special issue of Natural Computing for papers from DNA 19. This full version contains proofs not seen in the published versio

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

Intrinsic universality in tile self-assembly requires cooperation

We prove a negative result on the power of a model of algorithmic self-assembly for which it has been notoriously difficult to find general techniques and results. Specifically, we prove that Winfree's abstract Tile Assembly Model, when restricted to use noncooperative tile binding, is not intrinsically universal. This stands in stark contrast to the recent result that, via cooperative binding, the abstract Tile Assembly Model is indeed intrinsically universal. Noncooperative self-assembly, also known as "temperature 1", is where tiles bind to each other if they match on one or more sides, whereas cooperative binding requires binding on multiple sides. Our result shows that the change from single- to multi-sided binding qualitatively improves the kinds of dynamics and behavior that these models of nanoscale self-assembly are capable of. Our lower bound on simulation power holds in both two and three dimensions; the latter being quite surprising given that three-dimensional noncooperative tile assembly systems simulate Turing machines. On the positive side, we exhibit a three-dimensional noncooperative self-assembly tile set capable of simulating any two-dimensional noncooperative self-assembly system. Our negative result can be interpreted to mean that Turing universal algorithmic behavior in self-assembly does not imply the ability to simulate arbitrary algorithmic self-assembly processes.Comment: Added references. Improved presentation of definitions and proofs. This article uses definitions from arXiv:1212.4756. arXiv admin note: text overlap with arXiv:1006.2897 by other author

Universality in algorithmic self-assembly

Tile-based self-assembly is a model of algorithmic crystal growth in which square tiles represent molecules that bind to each other via specific and variable-strength bonds on their four sides, driven by random mixing in solution but constrained by the local binding rules of the tile bonds. In the late 1990s, Erik Winfree introduced a discrete mathematical model of DNA tile assembly called the abstract Tile Assembly Mode. Winfree proved that the Tile Assembly Model is computationally universal, i.e., that any Turing machine can be encoded into a finite set of tile types whose self-assembly simulates that Turing machine. In this thesis, we investigate tile-based self-assembly systems that exhibit Turing universality, geometric universality and intrinsic universality. We first establish a novel characterization of the computably enumerable languages in terms of self-assembly--the proof of which is a novel proof of the Turing-universality of the Tile Assembly Model in which a particular Turing machine is simulated on all inputs in parallel in the two-dimensional discrete Euclidean plane. Then we prove that the multiple temperature tile assembly model (introduced by Aggarwal, Cheng, Goldwasser, Kao, and Schweller) exhibits a kind of geometric universality in the sense that there is a small (constant-size) universal tile set that can be programmed via deliberate changes in the system temperature to uniquely produce any finite shape. Just as other models of computation such as the Turing machine and cellular automaton are known to be intrinsically universal, (e.g., Turing machines can simulate other Turing machines, and cellular automata other cellular automata), we show that tile assembly systems satisfying a natural condition known as local consistency are able to simulate other locally consistent tile assembly systems. In other words, we exhibit a particular locally consistent tile assembly system that can simulate the behavior--as opposed to only the final result--of any other locally consistent tile assembly system. Finally, we consider the notion of universal fault-tolerance in algorithmic self-assembly with respect to the two-handed Tile Assembly Model, in which large aggregations of tiles may attach to each other, in contrast to the seeded Tile Assembly Model, in which tiles aggregate one at a time to a single specially-designated seed assembly. We introduce a new model of fault-tolerance in self-assembly: the fuzzy temperature model of faults having the following informal characterization: the system temperature is normally 2, but may drift down to 1, allowing unintended temperature-1 growth for an arbitrary period of time. Our main construction, which is a tile set capable of uniquely producing an $n \times n$ square with log n unique tile types in the fuzzy temperature model, is not universal but presents novel technique that we hope will ultimately pave the way for a universal fuzzy-fault-tolerant tile assembly system in the future

Simulation in Algorithmic Self-assembly

Winfree introduced a model of self-assembling systems called the abstract Tile Assembly Model (aTAM) where square tiles with glues on their edges attach spontaneously via matching glues to form complex structures. A generalization of the aTAM called the 2HAM (two-handed aTAM) not only allows for single tiles to bind, but also for supertile assemblies consisting of any number of tiles to attach. We consider a variety of models based on either the aTAM or the 2HAM. The underlying commonality of the work presented here is simulation. We introduce the polyTAM, where a tile system consists of a collection of polyomino tiles, and show that for any polyomino P of size greater than or equal to 3 and any Turing machine M , there exists a temperature-1 polyTAM system containing only shape-P tiles that simulates M . We introduce the RTAM (Reflexive Tile Assembly Model) that works like the aTAM except that tiles can nondeterministically flip prior to binding. We show that the temperature-1 RTAM cannot simulate a Turing machine by showing the much stronger result that the RTAM can only self-assemble periodic patterns. We then define notions of simulation which serve as relations between two tile assembly systems (possibly belonging to different models). Using simulation as a basis of comparison, we first show that cellular automata and the class of all tile assembly systems in the aTAM are equivalent. Next, we introduce the Dupled aTAM (DaTAM) and show that the temperature-2 aTAM and the temperature-1 DaTAM are mutually exclusive by showing that there is an aTAM system that cannot be simulated by any DaTAM system, and vice versa. Third, we consider the restricted glues Tile Assembly Model (rgTAM) and show that there is an aTAM system that cannot be simulated by any rgTAM system. We introduce the Dupled restricted glues Tile Assembly Model (DrgTAM), and show that the DrgTAM is intrinsically universal for the aTAM. Finally, we consider a variation of the Signal-passing Tile Assembly Model (STAM) called the STAM+ and show that the STAM+ is intrinsically universal and that the 3-D 2HAM is intrinsically universal for the STAM+

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