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Assembly Model

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Trent A. Rogers – 1st expert on this subject based on the ideXlab platform

  • FOCS – Universal Simulation of Directed Systems in the Abstract Tile Assembly Model Requires Undirectedness
    2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS), 2016
    Co-Authors: Jacob Hendricks, Matthew J. Patitz, Trent A. Rogers

    Abstract:

    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.

  • Universal Simulation of Directed Systems in the abstract Tile Assembly Model Requires Undirectedness
    arXiv: Computational Geometry, 2016
    Co-Authors: Jacob Hendricks, Matthew J. Patitz, Trent A. Rogers

    Abstract:

    As a mathematical Model of self-assembling systems, Winfree’s abstract Tile Assembly Model (aTAM) is a remarkable platform for studying the behaviors and powers of self-assembling systems. 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. Adding to its completeness, the aTAM was shown to also be intrinsically universal, which means that there exists a 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 previously given makes use of nondeterminism in terms of tile types placed in several key locations when different Assembly sequences are followed, even when simulating a directed system, meaning one that has exactly one unique terminal Assembly. The question then became whether or not that nondeterminism is fundamentally required. Here, we answer that 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 provides insight into the role of nondeterminism in self-Assembly, which is itself a fundamentally nondeterministic process. 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 design systems with precisely controllable space resources available to embedded computations. We also develop novel techniques for designing systems containing subsystems with disjoint, mutually exclusive computational powers.

  • the two handed tile Assembly Model is not intrinsically universal
    Algorithmica, 2016
    Co-Authors: Erik D Demaine, Scott M. Summers, Trent A. Rogers, Matthew J. Patitz, Robert T Schweller, Damien Woods

    Abstract:

    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 $$\tau $$?, where $$\tau $$? 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 $$U$$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 $$\tau ‘ < \tau $$??Assembly Model is intrinsically universal. On the positive side, we prove that, for every fixed temperature $$\tau \ge 2$$??2, temperature-$$\tau $$? 2HAM tile systems are indeed intrinsically universal. In other words, for each $$\tau $$? there is a single intrinsically universal 2HAM tile set $$U_{\tau }$$U? that, when appropriately initialized, is capable of simulating the behavior of any temperature-$$\tau $$? 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 $$\tau $$?, there is a temperature-$$\tau $$? 2HAM system that simultaneously simulates all temperature-$$\tau $$? 2HAM systems.

Matthew J. Patitz – 2nd expert on this subject based on the ideXlab platform

  • Hierarchical growth is necessary and (sometimes) sufficient to self-assemble discrete self-similar fractals
    Natural Computing, 2019
    Co-Authors: Jacob Hendricks, Matthew J. Patitz, Joseph Opseth, Scott M. Summers

    Abstract:

    In this paper, we prove that in the abstract Tile Assembly Model (aTAM), an accretion-based Model which only allows for a single tile to attach to a growing Assembly at each step, there are no tile Assembly systems capable of self-assembling the discrete self-similar fractals known as the “H” and “U” fractals. We then show that in a related Model which allows for hierarchical self-Assembly, the 2-Handed Assembly Model (2HAM), there does exist a tile Assembly system which self-assembles the “U” fractal and conjecture that the same holds for the “H” fractal. This is the first example of discrete self similar fractals which self-assemble in the 2HAM but not in the aTAM, providing a direct comparison of the Models and greater understanding of the power of hierarchical Assembly.

  • FOCS – Universal Simulation of Directed Systems in the Abstract Tile Assembly Model Requires Undirectedness
    2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS), 2016
    Co-Authors: Jacob Hendricks, Matthew J. Patitz, Trent A. Rogers

    Abstract:

    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.

  • Universal Simulation of Directed Systems in the abstract Tile Assembly Model Requires Undirectedness
    arXiv: Computational Geometry, 2016
    Co-Authors: Jacob Hendricks, Matthew J. Patitz, Trent A. Rogers

    Abstract:

    As a mathematical Model of self-assembling systems, Winfree’s abstract Tile Assembly Model (aTAM) is a remarkable platform for studying the behaviors and powers of self-assembling systems. 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. Adding to its completeness, the aTAM was shown to also be intrinsically universal, which means that there exists a 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 previously given makes use of nondeterminism in terms of tile types placed in several key locations when different Assembly sequences are followed, even when simulating a directed system, meaning one that has exactly one unique terminal Assembly. The question then became whether or not that nondeterminism is fundamentally required. Here, we answer that 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 provides insight into the role of nondeterminism in self-Assembly, which is itself a fundamentally nondeterministic process. 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 design systems with precisely controllable space resources available to embedded computations. We also develop novel techniques for designing systems containing subsystems with disjoint, mutually exclusive computational powers.

Jacob Hendricks – 3rd expert on this subject based on the ideXlab platform

  • Hierarchical growth is necessary and (sometimes) sufficient to self-assemble discrete self-similar fractals
    Natural Computing, 2019
    Co-Authors: Jacob Hendricks, Matthew J. Patitz, Joseph Opseth, Scott M. Summers

    Abstract:

    In this paper, we prove that in the abstract Tile Assembly Model (aTAM), an accretion-based Model which only allows for a single tile to attach to a growing Assembly at each step, there are no tile Assembly systems capable of self-assembling the discrete self-similar fractals known as the “H” and “U” fractals. We then show that in a related Model which allows for hierarchical self-Assembly, the 2-Handed Assembly Model (2HAM), there does exist a tile Assembly system which self-assembles the “U” fractal and conjecture that the same holds for the “H” fractal. This is the first example of discrete self similar fractals which self-assemble in the 2HAM but not in the aTAM, providing a direct comparison of the Models and greater understanding of the power of hierarchical Assembly.

  • FOCS – Universal Simulation of Directed Systems in the Abstract Tile Assembly Model Requires Undirectedness
    2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS), 2016
    Co-Authors: Jacob Hendricks, Matthew J. Patitz, Trent A. Rogers

    Abstract:

    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.

  • Universal Simulation of Directed Systems in the abstract Tile Assembly Model Requires Undirectedness
    arXiv: Computational Geometry, 2016
    Co-Authors: Jacob Hendricks, Matthew J. Patitz, Trent A. Rogers

    Abstract:

    As a mathematical Model of self-assembling systems, Winfree’s abstract Tile Assembly Model (aTAM) is a remarkable platform for studying the behaviors and powers of self-assembling systems. 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. Adding to its completeness, the aTAM was shown to also be intrinsically universal, which means that there exists a 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 previously given makes use of nondeterminism in terms of tile types placed in several key locations when different Assembly sequences are followed, even when simulating a directed system, meaning one that has exactly one unique terminal Assembly. The question then became whether or not that nondeterminism is fundamentally required. Here, we answer that 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 provides insight into the role of nondeterminism in self-Assembly, which is itself a fundamentally nondeterministic process. 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 design systems with precisely controllable space resources available to embedded computations. We also develop novel techniques for designing systems containing subsystems with disjoint, mutually exclusive computational powers.