Abstract
We define the Reflexive Tile Assembly Model (RTAM), which is obtained from the abstract Tile Assembly Model (aTAM) by allowing tiles to reflect across their horizontal and/or vertical axes. We show that the class of directed temperature-1 RTAM systems is not computationally universal, which is conjectured but unproven for the aTAM, and like the aTAM, the RTAM is computationally universal at temperature 2. We then show that at temperature 1, when starting from a single tile seed, the RTAM is capable of assembling \(n \times n\) squares for n odd using only n tile types, but incapable of assembling \(n \times n\) squares for n even. Moreover, we show that n is a lower bound on the number of tile types needed to assemble \(n \times n\) squares for n odd in the temperature-1 RTAM. The conjectured lower bound for temperature-1 aTAM systems is \(2n-1\). Finally, we give preliminary results toward the classification of which finite connected shapes in \(\mathbb {Z}^2\) can be assembled (strictly or weakly) by a singly seeded (i.e. seed of size 1) RTAM system, including a complete classification of which finite connected shapes can be strictly assembled by mismatch-free singly seeded RTAM systems.
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Notes
It is interesting to note that there are interesting temperature-2 RTAM systems that assemble a single assembly even when individual tile orientations are taken into account. For example, rectilinear tile assembly systems in the RTAM at temperature-2 can produce a single terminal assembly (even when taking tile orientations into account) when the seed assembly of the system is L-shaped. Note that the terminal assembly cannot have width or height exceeding that of the seed. See Kari et al. (2015) for interesting examples of such rectilinear tile assembly systems with L-shaped seeds.
Essentially, a compact zig-zag system is one in which assembly proceeds along a single possible assembly sequence which grows one row completely from left-to-right, then immediately above that grows the next complete row right-to-left, and so on. Also, each row may grow to a length of one tile longer than the row below it.
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Acknowledgements
We thank our anonymous reviewers for their careful reading of this work and their insightful comments that have significantly improved this paper. Supported in part by National Science Foundation Grants CCF-1117672 and CCF-1422152. T. Rogers’s research was supported by the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE-1450079.
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Hendricks, J., Patitz, M.J. & Rogers, T.A. Reflections on tiles (in self-assembly). Nat Comput 16, 295–316 (2017). https://doi.org/10.1007/s11047-017-9617-2
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DOI: https://doi.org/10.1007/s11047-017-9617-2