# The Two-Handed Tile Assembly Model is not Intrinsically Universal

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## Abstract

The Two-Handed Tile Assembly Model (2HAM) is a model of algorithmic self-assembly in which large structures, or assemblies of tiles, are grown by the binding of smaller assemblies. In order to bind, two assemblies must have matching glues that can simultaneously touch each other, and stick together with strength that is at least the temperature \(\tau \), where \(\tau \) is some fixed positive integer. We ask whether the 2HAM is intrinsically universal. In other words, we ask: is there a single 2HAM tile set \(U\) which can be used to simulate any instance of the model? Our main result is a negative answer to this question. We show that for all \(\tau ' < \tau \), each temperature-\(\tau '\) 2HAM tile system does not simulate at least one temperature-\(\tau \) 2HAM tile system. This impossibility result proves that the 2HAM is not intrinsically universal and stands in contrast to the fact that the (single-tile addition) abstract Tile Assembly Model is intrinsically universal. On the positive side, we prove that, for every fixed temperature \(\tau \ge 2\), temperature-\(\tau \) 2HAM tile systems are indeed intrinsically universal. In other words, for each \(\tau \) there is a single intrinsically universal 2HAM tile set \(U_{\tau }\) that, when appropriately initialized, is capable of simulating the behavior of any temperature-\(\tau \) 2HAM tile system. As a corollary, we find an infinite set of infinite hierarchies of 2HAM systems with strictly increasing simulation power within each hierarchy. Finally, we show that for each \(\tau \), there is a temperature-\(\tau \) 2HAM system that simultaneously simulates all temperature-\(\tau \) 2HAM systems.

## Keywords

Tile-assembly Intrinsic universality 2HAM Tile assembly model## Notes

### Acknowledgments

This work was initiated at the 27th Bellairs Winter Workshop on Computational Geometry held on February 11–17, 2012 in Holetown, Barbados. We thank the other participants of that workshop for a fruitful and collaborative environment. We would also like to thank an anonymous reviewer for very thorough and insightful comments, helping us to improve this version of the paper.

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