In this paper we study the relationship between mathematical models of tile-based self-assembly which differ in terms of the synchronicity of tile additions. In the standard abstract Tile Assembly Model (aTAM), each step of assembly consists of a single tile being added to an assembly. At any given time, each location on the perimeter of an assembly to which a tile can legally bind is called a frontier location, and for each step of assembly one frontier location is randomly selected and a tile is added. In the Synchronous Tile Assembly Model (syncTAM), at each step of assembly every frontier location simultaneously receives a tile. Our results show that while directed, non-cooperative syncTAM systems are capable of universal computation (while directed, non-cooperative aTAM systems are known not to be), and they are capable of building shapes that can't be built within the aTAM, the non-cooperative aTAM is also capable of building shapes that can't be built within the syncTAM even cooperatively. We show a variety of results that demonstrate the similarities and differences between these two models.

In this paper, we extend existing results about simulation and intrinsic universality in a model of tile-based self-assembly. Namely, we work within the 2-Handed Assembly Model (2HAM), which is a model of self-assembly in which assemblies are formed by square tiles that are allowed to combine, usin g glues along their edges, individually or as pairs of arbitrarily large assemblies in a hierarchical manner, and we explore the abilities of these systems to simulate each other when the simulating systems have a higher “temperature” parameter, which is a system wide threshold dictating how many glue bonds must be formed between two assemblies to allow them to combine. It has previously been shown that systems with lower temperatures cannot simulate arbitrary systems with higher temperatures, and also that systems at some higher temperatures can simulate those at particular lower temperatures, creating an infinite set of infinite hierarchies of 2HAM systems with strictly increasing simulation power within each hierarchy. These previous results relied on two different definitions of simulation, one (strong simulation) seemingly more restrictive than the other (standard simulation), but which have previously not been proven to be distinct. Here we prove distinctions between them by first fully characterizing the set of pairs of temperatures such that the high temperature systems are intrinsically universal for the lower temperature systems (i.e. one tile set at the higher temperature can simulate any at the lower) using strong simulation. This includes the first impossibility result for simulation downward in temperature. We then show that lower temperature systems which cannot be simulated by higher temperature systems using the strong definition, can in fact be simulated using the standard definition, proving the distinction between the types of simulation.

RNA sequences start folding immediately as they are synthesized by RNA polymerase (cotranscriptional folding). The oritatami system (OS) is a novel mathematical model to study computational aspects of cotranscriptional folding. In this paper, we first provide a survey throughout existing research t opics and results on oritatami systems and offer research directions of significance. Simulation of an oritatami system in a different ratio (delay) of transcription speed to the speed of folding is one of them. We will introduce a simple notion of simulation, and prove that there is an OS of delay δ that cannot be simulated by any oritatami system at larger delay.

As a mathematical model of tile-based self-assembling systems, Winfree's abstract Tile Assembly Model (aTAM) has proven to be a remarkable platform for studying and understanding the behaviors and powers of self-assembling systems. Furthermore, as it is capable of Turing universal computation, the aTAM allows algorithmic self-assembly, in which the components can be designed so that the rules governing their behaviors force them to inherently execute prescribed algorithms as they combine. This power has yielded a wide variety of theoretical results in the aTAM utilizing algorithmic self-assembly to design systems capable of performing complex computations and forming extremely intricate structures. Adding to the completeness of the model, in FOCS 2012 the aTAM was shown to also be intrinsically universal, which means that there exists one single tile set such that for any arbitrary input aTAM system, that tile set can be configured into a "seed" structure which will then cause self-assembly using that tile set to simulate the input system, capturing its full dynamics modulo only a scale factor. However, the "universal simulator" of that result makes use of nondeterminism in terms of the tiles placed in several key locations when different assembly sequences are followed. This nondeterminism remains even when the simulator is simulating a system which is directed, meaning that it has exactly one unique terminal assembly and for any given location, no matter which assembly sequence is followed, the same tile type is always placed there. The question which then arose was whether or not that nondeterminism is fundamentally required, and if any universal simulator must in fact utilize more nondeterminism than directed systems when simulating them. In this paper, we answer that question in the affirmative: the class of directed systems in the aTAM is not intrinsically universal, meaning there is no universal simulator for directed systems which itself is always directed. This result provides a powerful insight into the role of nondeterminism in self-assembly, which is itself a fundamentally nondeterministic process occurring via unguided local interactions. Furthermore, to achieve this result we leverage powerful results of computational complexity hierarchies, including tight bounds on both best and worst-case complexities of decidable languages, to tailor design systems with precisely controllable space resources available to computations embedded within them. We also develop novel techniques for designing systems containing subsystems with disjoint, mutually exclusive computational powers. The main result will be important in the development of future simulation systems, and the supporting design techniques and lemmas will provide powerful tools for the development of future aTAM systems as well as proofs of their computational abilities.

In this paper we investigate the computational power of the polygonal tile assembly model (polygonal TAM) at temperature 1, i.e. in non-cooperative systems. The polygonal TAM is an extension of Winfree's abstract tile assembly model (aTAM) which not only allows for square tiles (as in the aTAM) but also allows for tile shapes which are arbitrary polygons. Although a number of self-assembly results have shown computational universality at temperature 1, these are the first results to do so by fundamentally relying on tile placements in continuous, rather than discrete, space. With the square tiles of the aTAM, it is conjectured that the class of temperature 1 systems is not computationally universal. Here we show that for each n > 6, the class of systems whose tiles are the shape of the regular polygon P with n sides is computationally universal. On the other hand, we show that the class of systems whose tiles consist of a regular polygon P with n ≤ 6 sides cannot compute using any known techniques. In addition, we show a number of classes of systems whose tiles consist of a non-regular polygon with n ≥ 3 sides are computationally universal.

In this paper we explore the power of geometry to overcome the limitations of non-cooperative self-assembly. We define a generalization of the abstract Tile Assembly Model (aTAM), such that a tile system consists of a collection of polyomino tiles, the Polyomino Tile Assembly Model (polyTAM), and investigate the computational powers of polyTAM systems at temperature 1, where attachment among tiles occurs without glue cooperation (i.e., without the enforcement that more than one tile already existing in an assembly must contribute to the binding of a new tile). Systems composed of the unit-square tiles of the aTAM at temperature 1 are believed to be incapable of Turing universal computation (while cooperative systems, with temperature > 1, are able). As our main result, we prove that for any polyomino P of size 3 or greater, there exists a temperature-1 polyTAM system containing only shape-P tiles that is computationally universal. Our proof leverages the geometric properties of these larger (relative to the aTAM) tiles and their abilities to effectively utilize geometric blocking of particular growth paths of assemblies, while allowing others to complete. In order to prove the computational powers of polyTAM systems, we also prove a number of geometric properties held by all polyominoes of size ≥ 3. To round out our main result, we provide strong evidence that size-1 (i.e. aTAM tiles) and size-2 polyomino systems are unlikely to be computationally universal by showing that such systems are incapable of geometric bitreading, which is a technique common to all currently known temperature-1 computationally universal systems. We further show that larger polyominoes with a limited number of binding positions are unlikely to be computationally universal, as they are only as powerful as temperature-1 aTAM systems. Finally, we connect our work with other work on domino self-assembly to show that temperature-1 assembly with at least 2 distinct shapes, regardless of the shapes or their sizes, allows for universal computation.