1. The Power of Nondeterminism in Self-Assembly
Nathaniel Bryans†, Ehsan Chiniforooshan†, David Doty‡, Lila Kari†, & Shinnosuke Seki†
†Department of Computer Science, University of Western Ontario, London, ON, N6A 5B7, Canada
‡Department of Computer Science, California Institute of Technology, Pasadena, CA 91125, USA
Will be ACM-SIAM Symposium on Discrete Algorithms (SODA11), San Francisco, California, USA, January 23-25, 2011.

2. Tile Self-Assembly
In a well-mixed chemical soup, randomly-floating molecules tend to interact with each other so as to assemble information-processing “biowares” (programmable crystal growth).
Tile self-assembly is an algorithmically rich model of this phenomena, in which tiles, which are one of the finite types, self-assemble
a unique target shape
due to only their local interactions (without any externally controlling device)

3. Practical Implementation
Self-assembling molecular tiles based on DNA complexes were experimentally implemented in 1982 by Seeman [Seeman 1982]
Experimental advances for reliable assembly with error rates:
10% per tile [Rothemund, Papadakis, Winfree 2004]
1.4% per tile [Fujibayashi et al. 2007]
0.13% per tile [Barish et al. 2009]

4. Abstract Tile-Assembly Model (aTAM)
Introduced by E. Winfree [Winfree 1998]
Wang tile + the notion of time (growth)
A simplified mathematical model of self-assembling DNA tiles
Computationally universal
Various resource bounds
Tile complexity (min # of tile types required to assemble a shape)
Discrete Sierpinski Triangle

5. Abstract Tile-Assembly System
A tile assembly system (TAS) is a triple (T, s, τ), where
T is a finite set of tile types:
A tile type is a tuple t ∈(Σ*×N)4, i.e., a square tile each of whose 4 sides has a glue consisting of
a finite string (label),
a non-negative integer (strength).
A tile cannot rotate.
An infinite # of copies of each tile type is available for computations, each copy referred to as a tile.
s is a seed-assembly
τ is the temperature
N, 1
E, 3
S, 2
W, 1

6. Growing process of TASs
Given T, an assembly is a partial function
The shape of α is dom α
Let’s consider the TAS shown right with 7 tile types τ= 2.
Starting from the seed placed on (0, 0).
A tile can attach to an assembly via its edge(s) if the sum of binding strengths associated with these abutting edges is at least the temperatureτ.
example: binary τ= 2
strength 1
stre
ngth 1
cooperation s

7. Determinism in TASs S
A TAS is directed (a.k.a. deterministic) if it has exactly one terminal assembly
Any assembly process by the TAS reaches the same assembly.
A TAS strictly self-assembles a shape S if all of its terminal assemblies are of the shape S. S

8. Main Problems
Is there any (infinite) shape which can be self-assembled by a non-deterministic TAS but cannot be by any deterministic one?
Does non-determinism decrease the tile complexity of some (finite) shape?
How difficult to exploit the non-determinism to design the “smallest” (and hence most economical) TAS for a given shape?

9. (Directed) Tile Complexity of a Shape
Tile complexity of a shape S is a measure of how complex the shape S is w.r.t. TAS: defined as
The directed tile complexity of S is
Ctc(S) ≤ Cdtc(S).
If S is finite, then Cdtc(S) ≤ dom S (hard-coding).

10. Assembly of Infinite Shapes
Theorem 1. There is a shape S such that some TAS strictly self-assembles S, but no directed one can.

11. Proof idea.
Let and M be a TM for L.
The simulation of M on the input n is carried out adjacent to the n-th ray.
For a roughly quadratic function f, a special tile is placed at (f(n), 0) for all n (circled points in the figure).
To the point (f(n), 0), a vertical pillar is extended from above iff n is in L.
This vertical pillar gets longer as n increases. Thus, for any directed TAS, there exists a sufficiently-long vertical pillar on which this TAS has to put two tiles of the same type.
Simulation of M(1)
Simulation of M(2)
Non-determinism is necessary
(f(n), 0)

12. Assembly of Finite Shapes
Theorem 2. There is a finite shape S s.t. Ctc(S) < Cdtc(S).
proof outline. We can construct a TAS (T, σ, 1) which strictly self-assembles the right shape with |T| = 2h+16 as green tiles for A and red tiles for B can be reused to assemble the loop L (non-deterministic choice is made at the top-middle point of L).
Since any directed TAS which assembles this shape has to avoid this non-deterministic choice, it has to have h+1 tiles which are neither green nor red tiles to assemble either the left or right pillars of L.
Thus, Ctc(S) ≤ 2h+16 and 3h+17 ≤ Cdtc(S) 

13. Minimum Tile Set Problem
It is of significant practical interest to find the “smallest” TAS which strictly self-assembles an objective shape.
The minimum tile set problem is
The minimum directed tile set problem is
MINDIRECTEDTILESET was proved to be NP-complete [Adleman et al. 2002].
We will prove that MINTILESET is -complete.

14. -completeness of MINTILESET
Theorem 3. MINTILESET is -complete.
MINTILESET being in : Observe that the following verification language is in P:
Then

15. -completeness of MINTILESET
2. Hardness of MINTILESET in :
We show that , where
This language is complete.
Our strategy is similar to the reduction of 3SAT to MINDIRECTEDTILESET shown in [Adleman et al. 2002].

16. Bibliography [Adleman et al. 2002]
L. M. Adleman, Q. Cheng, A. Goel, M-D. A. Huang, D. Kempe, P. M. de Espanes, and P. W. K. Rothemund. Combinatorial optimization problems in self-assembly. In STOC 2002: Proceedings of the 34th Annual ACM Symposium on Theory of Computing, pages 23-32, 2002.
[Barish et al. 2009]
R. D. Barish, R. Schulman, P. W. K. Rothemund, and E. Winfree. An information-bearing seed for nucleating algorithmic self-assembly. Proceedings of the National Academy of Sciences, 106(15): , 2009.
[Fujibayashi et al. 2007]
K. Fujibayashi, R. Hariadi, S. H. Park, E. Winfree, and S. Murata. Toward reliable algorithmic self-assembly of DNA tiles: A fixed-width cellular automaton pattern. Nano Letters, 8(7): , 2007.

17. Bibliography [Rothemund, Papadakis, Winfree 2004]
P. W. K. Rothemund, N. Papadakis, and E. Winfree. Algorithmic self-assembly of DNA Sierpinski triangles. PLoS Biology, 2(12): , 2004.
[Seeman 1982] N. Seeman. Nucleic-acid junctions and lattices. Journal of Theoretical Biology, 99: , 1982.
[Winfree 1998] E. Winfree. Algorithmic Self-Assembly of DNA. Ph.D. thesis, California Institute of Technology, June 1998.