Deterministic 2-Dimensional Temperature-1 Tile Assembly Systems Cannot Compute
Abstract
We consider non cooperative binding in so called `temperature 1', in deterministic (here called {\it confluent}) tile self-assembly systems (1-TAS) and prove the standing conjecture that such systems do not have universal computational power. We call a TAS whose maximal assemblies contain at least one ultimately periodic assembly path {\it para-periodic}. We observe that a confluent 1-TAS has at most one maximal producible assembly, $\alpha_{max}$, that can be considered a union of path assemblies, and we show that such a system is always para-periodic. This result is obtained through a superposition and a combination of two paths that produce a new path with desired properties, a technique that we call \emph{co-grow} of two paths. Moreover we provide a characterization of an $\alpha_{max}$ of a confluent 1-TAS as one of two possible cases, so called, a grid or a disjoint union of combs. To a given $\alpha_{max}$ we can associate a finite labeled graph, called \emph{quipu}, such that the union of all labels of paths in the quipu equals $\alpha_{max}$, therefore giving a finite description for $\alpha_{max}$. This finite description implies that $\alpha_{max}$ is a union of semi-affine subsets of $\mathbb{Z}^2$ and since such a finite description can be algorithmicly generated from any 1-TAS, 1-TAS cannot have universal computational power.
- Publication:
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arXiv e-prints
- Pub Date:
- January 2019
- DOI:
- 10.48550/arXiv.1901.08575
- arXiv:
- arXiv:1901.08575
- Bibcode:
- 2019arXiv190108575D
- Keywords:
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- Computer Science - Computational Complexity;
- Computer Science - Computational Geometry;
- Computer Science - Data Structures and Algorithms