Abstract
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.
A. Luchsinger, R. Schweller and T. Wylie—This author’s research is supported in part by National Science Foundation Grant CCF-1817602.
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Notes
- 1.
We borrow the notion of freezing from the cellular automata literature [1, 6, 7]. There are two informal perspectives towards freezing that are equivalent in CA but not equivalent in TA. One is that a cell (tile) must never revisit the same state twice. The other is that a position in \(\mathbb {Z}^2\) must never revisit the same state twice. Intuitively, in TA, a position may see several tiles due to tiles attaching and detaching. Thus, the perspectives are different. We choose the first perspective, matching the notion that tiles themselves are stateful, and positions in space are not stateful.
- 2.
We note that \(\varSigma \) does not include an “empty” state. In tile self-assembly, unlike cellular automata, positions in \(\mathbb {Z}^2\) may have no tile (and thus no state).
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Chalk, C., Luchsinger, A., Martinez, E., Schweller, R., Winslow, A., Wylie, T. (2018). Freezing Simulates Non-freezing Tile Automata. In: Doty, D., Dietz, H. (eds) DNA Computing and Molecular Programming. DNA 2018. Lecture Notes in Computer Science(), vol 11145. Springer, Cham. https://doi.org/10.1007/978-3-030-00030-1_10
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