Limitations of self-assembly at temperature 1

Authors:
David Doty;Matthew J. Patitz;Scott M. Summers
Affiliations:
Department of Computing and Mathematical Sciences, California Institute of Technology, Pasadena, CA 91125, USA;Department of Computer Science, University of Texas-Pan American, Edinburg, TX 78539, USA;Department of Computer Science and Software Engineering, University of WisconsinPlatteville, Platteville, WI 53818, USA
Venue:
Theoretical Computer Science
Year:
2011

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Abstract

We prove that if a set X@?Z^2 weakly self-assembles at temperature 1 in a deterministic (Winfree) tile assembly system satisfying a natural condition known as pumpability, then X is a semilinear set. This shows that only the most simple of infinite shapes and patterns can be constructed using pumpable temperature 1 tile assembly systems, and gives evidence for the thesis that temperature 2 or higher is required to carry out general-purpose computation in a deterministic two-dimensional tile assembly system. We employ this result to show that, unlike the case of temperature 2 self-assembly, no discrete self-similar fractal weakly self-assembles at temperature 1 in a pumpable tile assembly system.