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

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+

Intrinsic universality and the computational power of self-assembly

This short survey of recent work in tile self-assembly discusses the use of simulation to classify and separate the computational and expressive power of self-assembly models. The journey begins with the result that there is a single universal tile set that, with proper initialization and scaling, simulates any tile assembly system. This universal tile set exhibits something stronger than Turing universality: it captures the geometry and dynamics of any simulated system. From there we find that there is no such tile set in the noncooperative, or temperature 1, model, proving it weaker than the full tile assembly model. In the two-handed or hierarchal model, where large assemblies can bind together on one step, we encounter an infinite set, of infinite hierarchies, each with strictly increasing simulation power. Towards the end of our trip, we find one tile to rule them all: a single rotatable flipable polygonal tile that can simulate any tile assembly system. It seems this could be the beginning of a much longer journey, so directions for future work are suggested.Comment: In Proceedings MCU 2013, arXiv:1309.104

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

Doubles and Negatives are Positive (in Self-Assembly)

In the abstract Tile Assembly Model (aTAM), the phenomenon of cooperation occurs when the attachment of a new tile to a growing assembly requires it to bind to more than one tile already in the assembly. Often referred to as ``temperature-2'' systems, those which employ cooperation are known to be quite powerful (i.e. they are computationally universal and can build an enormous variety of shapes and structures). Conversely, aTAM systems which do not enforce cooperative behavior, a.k.a. ``temperature-1'' systems, are conjectured to be relatively very weak, likely to be unable to perform complex computations or algorithmically direct the process of self-assembly. Nonetheless, a variety of models based on slight modifications to the aTAM have been developed in which temperature-1 systems are in fact capable of Turing universal computation through a restricted notion of cooperation. Despite that power, though, several of those models have previously been proven to be unable to perform or simulate the stronger form of cooperation exhibited by temperature-2 aTAM systems. In this paper, we first prove that another model in which temperature-1 systems are computationally universal, namely the restricted glue TAM (rgTAM) in which tiles are allowed to have edges which exhibit repulsive forces, is also unable to simulate the strongly cooperative behavior of the temperature-2 aTAM. We then show that by combining the properties of two such models, the Dupled Tile Assembly Model (DTAM) and the rgTAM into the DrgTAM, we derive a model which is actually more powerful at temperature-1 than the aTAM at temperature-2. Specifically, the DrgTAM, at temperature-1, can simulate any aTAM system of any temperature, and it also contains systems which cannot be simulated by any system in the aTAM