Intrinsic Universality in Self-Assembly

Abstract : We show that the Tile Assembly Model exhibits a strong notion of universality where the goal is to give a single tile assembly system that simulates the behavior of any other tile assembly system. We give a tile assembly system that is capable of simulating a very wide class of tile systems, including itself. Specifically, we give a tile set that simulates the assembly of any tile assembly system in a class of systems that we call \emph{locally consistent}: each tile binds with exactly the strength needed to stay attached, and that there are no glue mismatches between tiles in any produced assembly. Our construction is reminiscent of the studies of \emph{intrinsic universality} of cellular automata by Ollinger and others, in the sense that our simulation of a tile system $T$ by a tile system $U$ represents each tile in an assembly produced by $T$ by a $c \times c$ block of tiles in $U$, where $c$ is a constant depending on $T$ but not on the size of the assembly $T$ produces (which may in fact be infinite). Also, our construction improves on earlier simulations of tile assembly systems by other tile assembly systems (in particular, those of Soloveichik and Winfree, and of Demaine et al.) in that we simulate the actual process of self-assembly, not just the end result, as in Soloveichik and Winfree's construction, and we do not discriminate against infinite structures. Both previous results simulate only temperature 1 systems, whereas our construction simulates tile assembly systems operating at temperature 2.
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Jean-Yves Marion and Thomas Schwentick. 27th International Symposium on Theoretical Aspects of Computer Science - STACS 2010, Mar 2010, Nancy, France. pp.275-286, 2010, Proceedings of the 27th Annual Symposium on the Theoretical Aspects of Computer Science
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David Doty, Jack H. Lutz, Matthew J. Patitz, Scott M. Summers, Damien Woods. Intrinsic Universality in Self-Assembly. Jean-Yves Marion and Thomas Schwentick. 27th International Symposium on Theoretical Aspects of Computer Science - STACS 2010, Mar 2010, Nancy, France. pp.275-286, 2010, Proceedings of the 27th Annual Symposium on the Theoretical Aspects of Computer Science. <inria-00455225>

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