Abstract
Tile-based self-assembly systems are capable of universal computation and algorithmically-directed growth. Systems capable of such behavior typically make use of “glue cooperation” in which the glues on at least 2 sides of a tile must match and bind to those exposed on the perimeter of an assembly for that tile to attach. However, several models have been developed which utilize “weak cooperation”, where only a single glue needs to bind but other preventative forces (such as geometric, or steric, hindrance) provide additional selection for which tiles may attach, and where this allows for algorithmic behavior. In this paper we first work in a model where tiles are allowed to have geometric bumps and dents on their edges. We show how such tiles can simulate systems of square tiles with complex glue functions (using asymptotically optimal sizes of bumps and dents). We also show that with only weak cooperation via geometric hindrance, no system in any model can simulate even a class of tightly constrained, deterministic cooperative systems, further defining the boundary of what is possible using this tool.
This author’s research was supported in part by National Science Foundation Grants CCF-1422152 and CAREER-1553166.
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Change history
27 May 2019
The original version of this paper contained a mistake. Theorem 3 in “Section 4,” which claimed that for every temperature-1 system in the Dupled abstract Tile Assembly Model (DaTAM) there exists a temperature-1 system in the Geometric Tile Assembly Model (GTAM) which simulates it at scale factor 1 and using only 2 glues, was incorrect. That theorem and the section that contained it have been removed. Please note that that result was independent of all other results, and the fact that it was incorrect does not impact them or any of the other remaining portions of the paper.
Notes
- 1.
Note that rectilinear systems can also be simulated, and they have similar properties except that they may have multiple frontier locations available.
- 2.
Note that with negative glues, more complex dynamics, which include the breaking apart of assemblies, are possible [13].
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Hader, D., Patitz, M.J. (2019). Geometric Tiles and Powers and Limitations of Geometric Hindrance in Self-assembly. In: McQuillan, I., Seki, S. (eds) Unconventional Computation and Natural Computation. UCNC 2019. Lecture Notes in Computer Science(), vol 11493. Springer, Cham. https://doi.org/10.1007/978-3-030-19311-9_16
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