Limitations of Self-Assembly at Temperature One (extended abstract)
aa r X i v : . [ c s . CC ] J un T. Neary, D. Woods, A.K. Seda and N. Murphy (Eds.):The Complexity of Simple Programs 2008.EPTCS 1, 2009, pp. 67–69, doi:10.4204/EPTCS.1.6 c (cid:13)
D. Doty, M. Patitz and S. SummersThis work is licensed under theCreative Commons Attribution License.
Limitations of Self-Assembly at Temperature One (extendedabstract)
David Doty
Department of Computer ScienceIowa State UniversityAmes, IA 50011, USA [email protected]
Matthew J. Patitz
Department of Computer ScienceIowa State UniversityAmes, IA 50011, USA [email protected]
Scott M. Summers
Department of Computer ScienceIowa State UniversityAmes, IA 50011, USA [email protected]
We prove that if a set X ⊆ Z weakly self-assembles at temperature 1 in a deterministic (Winfree)tile assembly system satisfying a natural condition known as pumpability , then X is a finite union ofsemi-doubly periodic sets. This shows that only the most simple of infinite shapes and patterns canbe constructed using pumpable temperature 1 tile assembly systems, and gives evidence for the thesisthat temperature 2 or higher is required to carry out general-purpose computation in a tile assemblysystem. Finally, we show that general-purpose computation is possible at temperature 1 if negativeglue strengths are allowed in the tile assembly model. Self-assembly is a bottom-up process by which a small number of fundamental components automati-cally coalesce to form a target structure. In 1998, Winfree [7] introduced the (abstract) Tile AssemblyModel (TAM) – an “effectivization” of Wang tiling [5, 6] – as an over-simplified mathematical modelof the DNA self-assembly pioneered by Seeman [4]. In the TAM, the fundamental components are un-rotatable, but translatable square “tile types” whose sides are labeled with glue “colors” and “strengths.”Two tiles that are placed next to each other interact if the glue colors on their abutting sides match, andthey bind if the strength on their abutting sides matches with total strength at least a certain ambient“temperature,” usually taken to be 1 or 2.Despite its deliberate over-simplification, the TAM is a computationally and geometrically expressivemodel at temperature 2. The reason is that, at temperature 2, certain tiles are not permitted to bond until two tiles are already present to match the glues on the bonding sides, which enables cooperation betweendifferent tile types to control the placement of new tiles. Winfree [7] proved that at temperature 2 theTAM is computationally universal and thus can be directed algorithmically.In contrast, we aim to prove that the lack of cooperation at temperature 1 inhibits the sort of complexbehavior possible at temperature 2. Our main theorem concerns the weak self-assembly of subsets of Z using temperature 1 tile assembly systems. Informally, a set X ⊆ Z weakly self-assembles in a tileassembly system T if some of the tile types of T are painted black, and T is guaranteed to self-assembleinto an assembly a of tiles such that X is precisely the set of integer lattice points at which a containsblack tile types. As an example, Winfree [7] exhibited a temperature 2 tile assembly system, directed by aclever XOR-like algorithm, that weakly self-assembles a well-known set, the discrete Sierpinski triangle,onto the first quadrant. Note that the underlying shape (set of all points containing a tile, whether blackor not) of Winfree’s construction is an infinite canvas that covers the entire first quadrant, onto which amore sophisticated set, the discrete Sierpinski triangle, is painted.We show that under a plausible assumption, temperature 1 tile systems weakly self-assemble only alimited class of sets. To prove our main result, we require the hypothesis that the tile system is pumpable .8 Limitations ofSelf-Assembly atTemperature OneInformally, this means that every sufficiently long path of tiles in an assembly of this system contains asegment in which the same tile type repeats (a condition clearly implied by the pigeonhole principle), andthat furthermore, the subpath between these two occurrences can be repeated indefinitely (“pumped”)along the same direction as the first occurrence of the segment, without “colliding” with a previousportion of the path. We give a counterexample of a path in which the same tile type appears twice, yetthe segment between the appearances cannot be pumped without eventually resulting in a collision thatprevents additional pumping. The hypothesis of pumpability states (roughly) that in every sufficientlylong path, despite the presence of some repeating tiles that cannot be pumped, there exists a segment inwhich the same tile type repeats that can be pumped. In the above-mentioned counterexample, the pathsconstructed to create a blocked segment always contain some previous segment that is pumpable. Weconjecture that this phenomenon, pumpability, occurs in every temperature 1 tile assembly system thatproduces a unique infinite structure.A semi-doubly periodic set X ⊆ Z is a set of integer lattice points with the property that thereare three vectors ~ b (the “base point” of the set), ~ u , and ~ v (the two periods of the set), such that X = n ~ b + n · ~ u + m · ~ v (cid:12)(cid:12)(cid:12) n , m ∈ N o . That is, a semi-doubly periodic set is a set that repeats infinitely alongtwo vectors (linearly independent vectors in the non-degenerate case), starting at some base point ~ b .We show that any directed, pumpable, temperature 1 tile assembly system weakly self-assembles a set X ⊆ Z that is a finite union of semi-doubly periodic sets.It is our contention that weak self-assembly captures the intuitive notion of what it means to “com-pute” with a tile assembly system. For example, the use of tile assembly systems to build shapes iscaptured by requiring all tile types to be black, in order to ask what set of integer lattice points containany tile at all (so-called strict self-assembly ). However, weak self-assembly is a more general notion.For example, Winfree’s above mentioned result shows that the discrete Sierpinski triangle weakly self-assembles at temperature 2 [3], yet this shape does not strictly self-assemble at any temperature [1].Hence weak self-assembly allows for a more relaxed notion of set building, in which intermediate spacecan be used for computation, without requiring that the space filled to carry out the computation alsorepresent the final result of the computation.As another example, there is a standard construction [7] by which a single-tape Turing machinemay be simulated by a temperature 2 tile assembly system. Regardless of the semantics of the Turingmachine (whether it decides a language, enumerates a language, computes a function, etc.), it is rou-tine to represent the result of the computation by the weak self-assembly of some set. For example,Patitz and Summers [2] showed that for any decidable language A ⊆ N , A ’s projection along the X -axis (cid:0) the set (cid:8) ( x , ) ∈ N (cid:12)(cid:12) x ∈ A (cid:9)(cid:1) weakly self-assembles in a temperature 2 tile assembly system. As an-other example, if a Turing machine computes a function f : N → N , it is routine to design a tile assemblysystem based on Winfree’s construction such that, if the seed assembly is used to encode the binaryrepresentation of a number n ∈ N , then the tile assembly system weakly self-assembles the set ( ( k , ) ∈ N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) the k th least significant bit of thebinary representation of f ( n ) is 1 ) . Our result is motivated by the thesis that if a tile assembly system can reasonably be said to “com-pute”, then the result of this computation can be represented in a straightforward manner as a set X ⊆ Z that weak self-assembles in the tile assembly system, or a closely related tile assembly system. Our ex-amples above provide evidence for this thesis, although it is as informal and unprovable as the Church-Turing thesis. On the basis of this claim, and the triviality of semi-doubly periodic sets, we conclude.Doty, M.Patitzand S.Summers 69that our main result implies that directed, pumpable, temperature 1 tile assembly systems are incapableof general-purpose deterministic computation, without further relaxing the model. Acknowledgments
We wish to thank Maria Axenovich, Matt Cook, and Jack Lutz for useful discussions, and anonymousreferees for corrections. We would especially like to thank Niall Murphy, Turlough Neary, Anthony K.Seda and Damien Woods for inviting us to present a preliminary version of this research at the Interna-tional Workshop on The Complexity of Simple Programs, University College Cork, Ireland on December6th and 7th, 2008.
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