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Self-assembly is the process by which a system of particles randomly agitate and combine, through local interactions, to form larger complex structures. In this work, we fuse a particular well-studied generalization of tile assembly (the 2-Handed or Hierarchical Tile Assembly Model) with concepts from cellular automata such as states and state transitions characterized by neighboring states. This allows for a simplification of the concepts from active self-assembly, and gives us machinery to relate the disparate existing models. We show that this model, coined Tile Automata, is invariant with respect to freezing and non-freezing transition rules via a simulation theorem showing that any non-freezing tile automata system can be simulated by a freezing one. Freezing tile automata systems restrict state transitions such that each tile may visit a state only once, i.e., a tile may undergo only a finite number of transitions. We conjecture that this result can be used to show that the Signal-passing Tile Assembly Model is also invariant to this constraint via a series of simulation results between that model and the Tile Automata model. Further, we conjecture that this model can be used to consolidate the several oft-studied models of self-assembly wherein assemblies may break apart, such as the Signal-passing Tile Assembly Model, the negative-glue 2-Handed Tile Assembly Model, and the Size-Dependent Tile Assembly Model. Lastly, the Tile Automata model may prove useful in combining results in cellular automata with self-assembly.
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Brief Announcement: Intrinsic Universality in Seeded Active Tile Self-Assembly
The Tile Automata (TA) model describes self-assembly systems in which monomers can build structures and transition with an adjacent monomer to change their states. This paper shows that seeded TA is a non-committal intrinsically universal model of self-assembly. We present a single universal Tile Automata system containing approximately 4600 states that can simulate (a) the output assemblies created by any other Tile Automata system Γ, (b) the dynamics involved in building Γ’s assemblies, and (c) Γ’s internal state transitions. It does so in a non-committal way: it preserves the full non-deterministic dynamics of a tile’s potential attachment or transition by selecting its state in a single step, considering all possible outcomes until the moment of selection. The system uses supertiles, each encoding the complete system being simulated. The universal system builds supertiles from its seed, each representing a single tile in Γ, transferring the information to simulate Γ to each new tile. Supertiles may also asynchronously transition states according to the rules of Γ. This result also implies IU for pairwise asynchronous Cellular Automata.
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- Award ID(s):
- 2329918
- PAR ID:
- 10657122
- Editor(s):
- Meeks, Kitty; Scheideler, Christian
- Publisher / Repository:
- Schloss Dagstuhl – Leibniz-Zentrum für Informatik
- Date Published:
- Volume:
- 330
- ISSN:
- 1868-8969
- Page Range / eLocation ID:
- 24:1-24:6
- Subject(s) / Keyword(s):
- Intrinsic Universality Tile Automata Cellular Automata Self-assembly Theory of computation → Models of computation
- Format(s):
- Medium: X Size: 6 pages; 691657 bytes Other: application/pdf
- Size(s):
- 6 pages 691657 bytes
- Sponsoring Org:
- National Science Foundation
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This paper answers a long-standing open question in tile-assembly theory, namely that it is possible to strictly assemble discrete self-similar fractals (DSSFs) in the abstract Tile-Assembly Model (aTAM). We prove this in 2 separate ways, each taking advantage of a novel set of tools. One of our constructions shows that specializing the notion of a quine, a program which prints its own output, to the language of tile-assembly naturally induces a fractal structure. The other construction introduces self-describing circuits as a means to abstractly represent the information flow through a tile-assembly construction and shows that such circuits may be constructed for a relative of the Sierpinski carpet, and indeed many other DSSFs, through a process of fixed-point iteration. This later result, or more specifically the machinery used in its construction, further enable us to provide a polynomial time procedure for deciding whether any given subset of ℤ2 will generate an aTAM producible DSSF. To this end, we also introduce the Tree Pump Theorem, a result analogous to the important Window Movie Lemma, but with requirements on the set of productions rather than on the self-assembling system itself.more » « less
- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.4230/lipics.sand.2025.24
