Simulation of Programmable Matter Systems Using Active Tile-Based Self-Assembly

Alumbaugh, John Calvin; Daymude, Joshua J.; Demaine, Erik D.; Patitz, Matthew J.; Richa, AndrÃ©a W.

doi:10.1007/978-3-030-26807-7_8

Cited by 10 publications

(10 citation statements)

References 26 publications

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“…While the table is still missing a few entries, our contributions have brought the number of known entries up to 52 from the 16 which previously existed in published literature (8 of which were technically not explicitly stated, but were trivial observations based on the tile sets and proofs presented in [7]). 1 The rest of our paper is laid out as follows. In Section 2, we provide definitions of the various models and concepts used.…”

Section: Our Resultsmentioning

confidence: 99%

“…Definition 3. We say that S models T (under R), and we write S |= R T , if for every α ∈ A[T ], there exists Π ⊂ A[S] where Π = ∅ and R * (α ) = α for all α ∈ Π, such that, for every β ∈ A[T ] where α → T β, (1) for every α ∈ Π there exists β ∈ A[S] where R * (β ) = β and α → S β , and (2) for every α ∈ A[S] where α → S β , β ∈ A[S], R * (α ) = α, and R * (β ) = β, there exists α ∈ Π such that α → S α . Definition 4.…”

Section: Formal Definition Of Simulationmentioning

confidence: 99%

See 1 more Smart Citation

The Impacts of Dimensionality, Diffusion, and Directedness on Intrinsic Universality in the abstract Tile Assembly Model

Hader¹,

Koch²,

Patitz³

et al. 2020

Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

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In this paper we present a series of results related to mathematical models of self-assembling systems of tiles and the impacts that three diverse properties have on their dynamics. In these self-assembling systems, initially unorganized collections of tiles undergo random motion and can bind together, if they collide and enough of their incident glues match, to form assemblies. Here we greatly expand upon a series of prior results which showed that (1) the abstract Tile Assembly Model (aTAM) is intrinsically universal (FOCS 2012), and (2) the class of directed aTAM systems is not intrinsically universal (FOCS 2016). Intrinsic universality (IU) for a model (or class of systems within a model) means that there is a universal tile set which can be used to simulate an arbitrary system within that model (or class). Furthermore, the simulation must not only produce the same resultant structures, it must also maintain the full dynamics of the systems being simulated and display the same behaviors modulo a scale factor. While the FOCS 2012 result showed that the standard, two-dimensional (2D) aTAM is IU, here we show that this is also the case for the three-dimensional (3D) version. Conversely, the FOCS 2016 result showed that the class of aTAM systems which are directed (a.k.a. deterministic, or confluent) is not IU, meaning that there is no universal simulator which can simulate directed aTAM systems while itself always remaining directed, implying that nondeterminism is fundamentally required for such simulations. Here, however, we show that in 3D the class of directed aTAM systems is actually IU, i.e. there is a universal directed simulator for them. This implies that the constraint of tiles binding only in the plane forced the necessity of nondeterminism for the simulation of 2D directed systems. This then leads us to continue to explore the impacts of dimensionality and directedness on simulation of tile-based self-assembling systems by considering the influence of more rigid notions of dimensionality. Namely, we introduce the Planar aTAM, where tiles are not only restricted to binding in the plane, but they are also restricted to traveling within the plane, and we prove that the Planar aTAM is not IU, and prove that the class of directed systems within the Planar aTAM also is not IU. Finally, analogous to the Planar aTAM, we introduce the Spatial aTAM, its 3D counterpart, and prove that the Spatial aTAM is IU.This paper adds to a broad set of results which have been used to classify and compare the relative powers of differing models and classes of self-assembling systems, and also helps to further the understanding of the roles of dimension and nondeterminism on the dynamics of self-assembling systems. Furthermore, to prove our positive results we have not only designed, but also implemented what we believe to be the first IU tile set ever implemented and simulated in any tile assembly model, and have made it, along with a simulator which can demonstrate it, freely available.1 Our implementation omits two relati...

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Section: Our Resultsmentioning

confidence: 99%

Section: Formal Definition Of Simulationmentioning

confidence: 99%

The Impacts of Dimensionality, Diffusion, and Directedness on Intrinsic Universality in the abstract Tile Assembly Model

Hader¹,

Koch²,

Patitz³

et al. 2020

Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

View full text Add to dashboard Cite

show abstract

“…The authors showed that freezing tile automata (where a tile cannot repeat states) is capable of simulating non-freezing systems. This powerful model has also been shown to be capable of simulating models of programmable matter (Alumbaugh et al 2019). A model motivated by real-world implementations, the Signalpassing Tile Assembly Model, is able to simulate Tile Automata (Cantu et al 2020) meaning results shown in the TA model carry over to STAM at scale.…”

Section: Previous Workmentioning

confidence: 99%

“…By abstracting away implementation details, TA strives to serve as a proving ground for exploring the power of active algorithmic self-assembly, along with providing a central hub through which various disparate models of selfassembly can be related by way of comparison to TA. One example of this type of application includes (Alumbaugh et al 2019) in which TA is shown capable of simulating the Amoebots model (Daymude et al 2019) of programmable matter. More recently a connection between the STAM and Tile Automata was established in (Cantu et al 2020) where it was shown that the STAM is capable of simulating any Tile Automata system.…”

Section: Introductionmentioning

confidence: 99%

Verification and computation in restricted Tile Automata

et al. 2021

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Many models of self-assembly have been shown to be capable of performing computation. Tile Automata was recently introduced combining features of both Cellular Automata and the 2-Handed Model of self-assembly both capable of universal computation. In this work we study the complexity of Tile Automata utilizing features inherited from the two models mentioned above. We first present a construction for simulating Turing machines that performs both covert and fuel efficient computation. We then explore the capabilities of limited Tile Automata systems such as 1-dimensional systems (all assemblies are of height 1) and freezing systems (tiles may not repeat states). Using these results we provide a connection between the problem of finding the largest uniquely producible assembly using n states and the busy beaver problem for non-freezing systems and provide a freezing system capable of uniquely assembling an assembly whose length is exponential in the number of states of the system. We finish by exploring the complexity of the Unique Assembly Verification problem in Tile Automata with different limitations such as freezing and systems without the power of detachment.

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“…168 Similar modeling strategies have been used to design, program, and optimize the algorithmically directed assembly of DNA origami tiles. 169,170,187,188 A wide variety of DNA origami tile shapes and interactions are possible, 171 offering ways to build large, complex assemblies with nanoscale resolution. Experimentally implemented systems have been regulated by diverse physical processes, including hybridization, 172 strand displacement, 173 shape complementarity, 171,174 and base stacking.…”

Section: ■ Introductionmentioning

confidence: 99%

Prediction and Control in DNA Nanotechnology

DeLuca

Sensale

Lin

et al. 2023

ACS Appl. Bio Mater.

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DNA nanotechnology is a rapidly developing field that uses DNA as a building material for nanoscale structures. Key to the field's development has been the ability to accurately describe the behavior of DNA nanostructures using simulations and other modeling techniques. In this Review, we present various aspects of prediction and control in DNA nanotechnology, including the various scales of molecular simulation, statistical mechanics, kinetic modeling, continuum mechanics, and other prediction methods. We also address the current uses of artificial intelligence and machine learning in DNA nanotechnology. We discuss how experiments and modeling are synergistically combined to provide control over device behavior, allowing scientists to design molecular structures and dynamic devices with confidence that they will function as intended. Finally, we identify processes and scenarios where DNA nanotechnology lacks sufficient prediction ability and suggest possible solutions to these weak areas.

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Simulation of Programmable Matter Systems Using Active Tile-Based Self-Assembly

Cited by 10 publications

References 26 publications

The Impacts of Dimensionality, Diffusion, and Directedness on Intrinsic Universality in the abstract Tile Assembly Model

The Impacts of Dimensionality, Diffusion, and Directedness on Intrinsic Universality in the abstract Tile Assembly Model

Verification and computation in restricted Tile Automata

Prediction and Control in DNA Nanotechnology

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