Abstract
In the classical model of tile self-assembly, unit square tiles translate in the plane and attach edgewise to form large crystalline structures. This model of self-assembly has been shown to be capable of asymptotically optimal assembly of arbitrary shapes and, via information-theoretic arguments, increasingly complex shapes necessarily require increasing numbers of distinct types of tiles.
We explore the possibility of complex and efficient assembly using systems consisting of a single tile. Our main result shows that any system of square tiles can be simulated using a system with a single tile that is permitted to flip and rotate. We also show that systems of single tiles restricted to translation only can simulate cellular automata for a limited number of steps given an appropriate seed assembly, and that any longer-running simulation must induce infinite assembly.
A full version of this paper can be found at http://arxiv.org/abs/1212.4756
Research of Matthew J. Patitz supported in part by NSF grant CCF-1117672. Research of Robert Schweller supported in part by NSF grant CCF-1117672. Research of Andrew Winslow supported in part by NSF grant CDI-0941538. Research of Damien Woods Supported by NSF grants 0832824 & 1317694 (the Molecular Programming Project), CCF-1219274, and CCF-1162589.
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Demaine, E.D. et al. (2014). One Tile to Rule Them All: Simulating Any Tile Assembly System with a Single Universal Tile. In: Esparza, J., Fraigniaud, P., Husfeldt, T., Koutsoupias, E. (eds) Automata, Languages, and Programming. ICALP 2014. Lecture Notes in Computer Science, vol 8572. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-43948-7_31
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