## 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.

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## Notes

- 1.
Note that this simulation result of Cannon et al. [5] does not imply that the 2HAM is intrinsically universal because (a) it is for 2HAM simulating aTAM, and (b) it is a “for all, there exists\(\ldots \)” statement, whereas intrinsic universality is a “there exists, for all\(\ldots \)” statement.

- 2.
We do not use this definition in this paper but have included it for the sake of completeness.

- 3.
with the convention that \(\infty = \infty + 1 = \infty - 1\)

- 4.
Note that a supertile \(\tilde{\alpha }\) could be non-terminal in the sense that there is a producible supertile \(\tilde{\beta }\) such that \(C^\tau _{\tilde{\alpha },\tilde{\beta }} \ne \emptyset \), yet it may not be possible to produce \(\tilde{\alpha }\) and \(\tilde{\beta }\) simultaneously if some tile types are given finite initial counts, implying that \(\tilde{\alpha }\) cannot be “grown” despite being non-terminal. If the count of each tile type in the initial state is \(\infty \), then all producible supertiles are producible from any state, and the concept of terminal becomes synonymous with “not able to grow”, since it would always be possible to use the abundant supply of tiles to assemble \(\tilde{\beta }\) alongside \(\tilde{\alpha }\) and then attach them.

- 5.
Note that in the glue-binding pad region there are no “single tile” bumps: this ensures that the simulator tile set \(U_{\tau }\) does not contain strength \(\tau \) glues, which in turn simplifies our construction.

- 6.
The crawlers, counters, computational primitives (guessing strings, computing simple numerical functions on bit strings, and even simulating Turing machines), and geometric primitives (copying bit sequences around in two-dimensional space) used in this and later constructions are relatively straightforward implementations similar to those used in the aTAM in [15], among others. These primitives are designed to assemble on the edges of existing supertiles (or assemblies in the aTAM), and can be made (and usually already are) “2HAM-safe” (essentially, “polyomino safe” as in [26]), meaning that in the 2HAM they function identically and correctly without danger of unwanted supertiles forming which are unattached to the desired supertiles. The general technique is to limit the number of \(\tau \)-strength glues on any particular tile type which assembles the primitive to \(1\), so that the largest unattached supertile which can form from them is a size \(2\) duple. All other attachments, and even the incorporation of the duples, requires cooperation provided by the surface of the supertile onto which the primitive is intended to form. Since the constructions for these primitives are standard and straightforward, we omit the details here.

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## 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.

## Author information

## Additional information

Matthew J. Patitz’s research was supported in part by National Science Foundation Grants CCF-1117672 and CCF-1422152.

Trent A. Rogers’s research was supported by the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE-1450079, and National Science Foundation Grants CCF-1117672 and CCF-1422152.

Robert T. Schweller’s research was supported in part by National Science Foundation Grant CCF-1117672.

Damien Woods’s research was supported by National Science Foundation Grants CCF-1219274, 0832824 (The Molecular Programming Project), CCF-1219274, and CCF-1162589.

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Demaine, E.D., Patitz, M.J., Rogers, T.A. *et al.* The Two-Handed Tile Assembly Model is not Intrinsically Universal.
*Algorithmica* **74, **812–850 (2016). https://doi.org/10.1007/s00453-015-9976-y

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### Keywords

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