Exact Shapes and Turing Universality at Temperature 1 with a Single Negative Glue

Published on: 2011/05/06

Abstract

Is Winfree's abstract Tile Assembly Model (aTAM) "powerful?" Well, if certain tiles are required to "cooperate" in order to be able to bind to a growing tile assembly (a.k.a., temperature 2 self-assembly), then Turing universal computation and the efficient self-assembly of \(N \times N\) squares is achievable in the aTAM (Rotemund and Winfree, STOC 2000). So yes, in a computational sense, the aTAM is quite powerful. However, if one completely removes this cooperativity condition (a.k.a., temperature 1 self-assembly), then the computational "power" of the aTAM (i.e., its ability to support Turing universal computation and the efficient self-assembly of \(N \times N\) squares) becomes unknown. On the plus side, the aTAM, at temperature 1, is not only Turing universal but also supports the efficient self-assembly \(N \times N\) squares if self-assembly is allowed to utilize three spatial dimensions (Fu, Schweller and Cook, SODA 2011). In this paper, we investigate the theoretical "power" of a seemingly simple, restrictive variant of Winfree's aTAM in which (1) the absolute value of every glue strength is 1, (2) there is a single negative strength glue type and (3) unequal glues cannot interact (i.e., glue functions must be "diagonal"). We call this abstract model of self-assembly the restricted glue Tile Assembly Model (rgTAM). We achieve two positive results. First, we first show that the tile complexity of uniquely producing an \(N \times N\) square in the rgTAM is \(O(\log N)\). In our second result, we prove that the rgTAM is Turing universal.

Authors

Matthew J. Patitz, Robert T. Schweller, and Scott M. Summers

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Exact Shapes and Turing Universality at Temperature 1 with a Single Negative Glue