Limitations of Self-Assembly at Temperature 1

Details Limitations of Self-Assembly at Temperature 1 Limitations of Self-Assembly at Temperature 1

2009-03-10

arxiv.org/abs/0903.1857v1

Computer Science

Discrete Mathematics

10 page conference submission with additional technical appendix containing proofs

Scientific paper

We prove that if a set $X \subseteq \Z^2$ weakly self-assembles at temperature 1 in a deterministic tile assembly system satisfying a natural condition known as \emph{pumpability}, then $X$ is a finite union of semi-doubly periodic sets. 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 tile assembly system. Finally, we show that general-purpose computation \emph{is} possible at temperature 1 if negative glue strengths are allowed in the tile assembly model.

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