Self-Assembly of Infinite Structures
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. 215–225, doi:10.4204/EPTCS.1.21 c (cid:13)
M.J. Patitz and S.M. Summers
Self-Assembly of Infinite Structures
Matthew J. Patitz Scott M. Summers ∗ [email protected] [email protected] Department of ComputerScience, Iowa State University, Ames, IA 50011, USA. † We review some recent results related to the self-assembly of infinite structures in the Tile Assem-bly Model. These results include impossibility results, as well as novel tile assembly systems inwhich shapes and patterns that represent various notions of computation self-assemble. Several openquestions are also presented and motivated.
The simplest mathematical model of nanoscale self-assembly is the Tile Assembly Model (TAM), aneffectivization of Wang tiling [24, 25] that was introduced by Winfree [27] and refined by Rothemundand Winfree [19, 18]. (See also [1, 17, 22].) As a basic model for the self-assembly of matter, the TAMhas allowed researchers to explore an assortment of avenues into both laboratory-based and theoreticalapproaches to designing systems that self-assemble into desired shapes or autonomously coalesce intopatterns that, in doing so, perform computations.Actual physical experimentation has driven lines of research involving kinetic variations of the TAMto deal with molecular concentrations, reaction rates, etc. as in [26], as well as work focused on errorprevention and error correction [6, 28, 21]. For examples of the impressive progress in the physicalrealization of self-assembling systems, see [20, 23].Divergent from, but supplementary to, the laboratory work, much theoretical research involving theTAM has also been carried out. Interesting questions concerning the minimum number of tile typesrequired to self-assemble shapes have been addressed in [22, 19, 2, 4]. Different notions of runningtime and bounds thereof were explored in [14, 5, 7]. Variations of the model where temperature valuesare intentionally fluctuated and the ensuing benefits and tradeoffs can be found in [10, 4]. Systems forgenerating randomized shapes or approximations of target shapes were investigated in [11, 5]. This isjust a small sampling of the theoretical work in the field of algorithmic self-assembly.However, as different as they may be, the above mentioned lines of research share a common thread.They all tend to focus on the self-assembly of finite structures. Clearly, for experimental research, thisis a necessary limitation. Further, if the eventual goal of most of the theoretical research is to enable thedevelopment of fully functional, real world self-assembly systems, a valid question is: “Why care aboutanything other than finite structures?” This is the question that we address in this paper.This paper surveys a collection of recent findings related to the self-assembly of infinite structures inthe TAM. As a theoretical exploration of the TAM, this collection of results seeks to define absolute lim-itations on the classes of shapes that self-assemble. These results also help to explore how fundamentalaspects of the TAM, such as the inability of spatial locations to be reused and their immutability, affectand limit the constructions and computations that are achievable. ∗ This author’s research was supported in part by NSF-IGERT Training Project in Computational Molecular Biology Grantnumber DGE-0504304. † This research supported in part by National Science Foundation grants 0652569, and 0728806.
16 Self-Assembly of Infinite StructuresIn addition to providing concise statements and intuitive descriptions of results, we also define andmotivate a set of open questions in the hope of furthering this line of research. First, we begin with somepreliminary definitions and constructions that will be referenced throughout this paper.
This section provides a very brief overview of the TAM. See [27, 19, 18, 13] for other developments ofthe model. Our notation is that of [13]. We work in the 2-dimensional discrete space Z . We write U for the set of all unit vectors , i.e., vectors of length 1 in Z . We write [ X ] for the set of all 2-elementsubsets of a set X . All graphs here are undirected graphs, i.e., ordered pairs G = ( V , E ) , where V is theset of vertices and E ⊆ [ V ] is the set of edges .A grid graph is a graph G = ( V , E ) in which V ⊆ Z and every edge { ~ a ,~ b } ∈ E has the property that ~ a − ~ b ∈ U . The full grid graph on a set V ⊆ Z is the graph G V = ( V , E ) in which E contains every { ~ a ,~ b } ∈ [ V ] such that ~ a − ~ b ∈ U .Intuitively, a tile type t is a unit square that can be translated, but not rotated, having a well-defined“side ~ u ” for each ~ u ∈ U . Each side ~ u of t has a “glue” of “color” col t ( ~ u ) - a string over some fixedalphabet S - and “strength” str t ( ~ u ) - a natural number - specified by its type t . Two tiles t and t ′ that areplaced at the points ~ a and ~ a + ~ u respectively, bind with strength str t ( ~ u ) if and only if ( col t ( ~ u ) , str t ( ~ u )) =( col t ′ ( − ~ u ) , str t ′ ( − ~ u )) .Given a set T of tile types, an assembly is a partial function a : Z T . An assembly is t - stable ,where t ∈ N , if it cannot be broken up into smaller assemblies without breaking bonds whose strengthssum to at least t .Self-assembly begins with a seed assembly s and proceeds asynchronously and nondeterministically,with tiles adsorbing one at a time to the existing assembly in any manner that preserves stability at alltimes. A tile assembly system ( TAS ) is an ordered triple T = ( T , s , t ) , where T is a finite set of tiletypes, s is a seed assembly with finite domain, and t is the temperature. An assembly sequence in aTAS T = ( T , s , ) is a (possibly infinite) sequence ~ a = ( a i | ≤ i < k ) of assemblies in which a = s and each a i + is obtained from a i by the “ t -stable” addition of a single tile. We write A [ T ] for the set of all producible assemblies of T . An assembly a is terminal , and we write a ∈ A (cid:3) [ T ] , if no tilecan be stably added to it. We write A (cid:3) [ T ] for the set of all terminal assemblies of T . A TAS T is directed , or produces a unique assembly , if it has exactly one terminal assembly i.e., | A (cid:3) [ T ] | =
1. Thereader is cautioned that the term “directed” has also been used for a different, more specialized notion inself-assembly [3].A set X ⊆ Z weakly self-assembles if there exists a TAS T = ( T , s , ) and a set B ⊆ T such that a − ( B ) = X holds for every assembly a ∈ A (cid:3) [ T ] . A set X strictly self-assembles if there is a TAS T for which every assembly a ∈ A (cid:3) [ T ] satisfies dom a = X . The reader is encouraged to consult [22] fora detailed discussion of local determinism - a general and powerful method for proving the correctnessof tile assembly systems. In this subsection we introduce discrete self-similar fractals, and zeta-dimension..J.Patitzand S.M.Summers 217
Definition
Let 1 < c ∈ N , and X ( N . We say that X is a c - discrete self-similar fractal , if thereis a (non-trivial) set V ⊆ { , . . . , c − } × { , . . . , c − } such that X = ¥ [ i = X i , where X i is the i th stage satisfying X = { ( , ) } , and X i + = X i ∪ (cid:0) X i + c i V (cid:1) . In this case, we say that V generates X . (a) X (b) V = X (c) X (d) X (scaled down) Figure 1:
Example of a c -discrete self-similar fractal ( c = The most commonly used dimension for discrete fractals is zeta-dimension, which we refer to in thispaper.
Definition [8] For each set A ⊆ Z , the zeta-dimension of A isDim z ( A ) = lim sup n → ¥ log | A ≤ n | log n , where A ≤ n = { ( k , l ) ∈ A | | k | + | l | ≤ n } . It is clear that 0 ≤ Dim z ( A ) ≤ A ⊆ Z . q 0
111 1q 1
0q 0 _* ___* __* _*_q _ Figure 2: Example of the first four rows of a sam-ple wedge construction which is simulating a Turingmachine M on the input string ‘01’In order to perform universal computation in theTAM, we make use of a particular TAS called the“wedge construction” [15]. The wedge construc-tion, based on Winfree’s proof of the universalityof the TAM [27], is used to simulate an arbitraryTuring machine M = ( Q , S , G , d , q , q A , q R ) on agiven input string w ∈ S ∗ in a temperature 2 TAS.The wedge construction works as follows. Ev-ery row of the assembly specifies the completeconfiguration of M at some time step. M startsin its initial state with the tape head on the left-most tape cell and we assume that the tape headnever moves left off the left end of the tape. Theseed row (bottommost) encodes the initial config-uration of M . There is a special tile representing ablank tape symbol as the rightmost tile in the seed row. Every subsequent row grows by one additionalcell to the right. This gives the assembly the wedge shape responsible for its name. Figure 2 shows thefirst four rows of a wedge construction for a particular TM, with arrows depicting a possible assemblysequence.18 Self-Assembly of Infinite Structures The self-assembly of shapes (i.e., subsets of Z ) in the TAM is most naturally characterized by strictself-assembly. In searching for absolute limitations of strict self-assembly in the TAM, it is necessary toconsider infinite shapes because any finite, connected shape strictly self-assembles via a spanning treeconstruction in which there is a unique tile type created for each point. In this section we discuss (bothpositive and negative) results pertaining to the strict self-assembly of infinite shapes in the TAM. In [16], Patitz and Summers defined a class C of (non-tree) “pinch-point” discrete self-similar fractals,and proved that if X ∈ C , then X does not strictly self-assemble. Definition A pinch-point discrete self-similar fractal is a discrete self-similar fractal satisfying (1) { ( , ) , ( , c − ) , ( c − , ) } ⊆ V , (2) V ∩ ( { , . . . c − } × { c − } ) = /0, (3), V ∩ ( { c − } × { , . . . , c − } ) = /0, and G V is connectedA famous example of a pinch-point fractal is the standard discrete Sierpinski triangle S . The impos-sibility of the strict self-assembly of S was first shown in [13]. Figure 3 shows another example of apinch-point discrete self-similar fractal. Note that any fractal X such that G X is a tree is necessarily apinch-point discrete self-similar fractal.The following (slight) generalization to [13] was shown in [16]. Theorem 3.1
If X ( N is a pinch-point discrete self-similar fractal, then X does not strictly self-assemble in the TAM. The idea behind the proof of Theorem 3.1 can be seen in Figure 3. Note that the black points arepinch-points in the sense that arbitrarily large aperidic sub-structures appear on the far-side of the blacktile from the origin. Figure 3:
An example of the first four stages ofpinch-point fractal with the first three pinch-pointshighlighted in black.
Theorem 3.1 motivates the following question.
Open Problem 3.2
Does any non-trivial discrete self-similar fractal strictly self-assemble in the TAM? Weconjecture that the answer is ‘no’, for any tempera-ture t ∈ N . However, proving that there exists a (non-trivial) discrete self-similar fractal that does strictlyself-assemble would likely involve a novel and usefulalgorithm for directing the behavior self-assembly. As shown above, there is a class of discrete self-similarfractals that do not strictly self-assemble (at any tem-perature) in the TAM. However, in [16], Patitz andSummers introduced a particular set of “nice” dis-crete self-similar fractals that contains some but notall pinch-point discrete self-similar fractals. Further,they proved that any element of the former class has a“fibered” version that strictly self-assembles..J.Patitzand S.M.Summers 219 A nice discrete self-similar fractal is a discrete self-similar fractal such that ( { , . . . , c − } ×{ } ) ∪ ( { } × { , . . . , c − } ) ⊆ V , and G V is connected.See Figure 4 for examples of both nice, and non-nice discrete self-similar fractals. (a) Nice (b) Non-nice Figure 4:
Stage 2 of some discrete self-similar fractals.
The inability of pinch-point fractals (and the conjectured inability of any discrete self-similar fractal)to strictly self-assemble in the TAM is based on the intuition that the necessary amount of informationcannot be transmitted through available connecting tiles during self-assembly.Thus, for any nice discrete self-similar fractal X , Patitz and Summers [16] defined a fibered operator F ( X ) (a routine extension of [13]) which adds, in a zeta-dimension-preserving manner, additional band-width to X . Strict self-assembly of F ( X ) is achieved via a “modified binary counter” algorithm that isembedded into the additional bandwidth of F ( X ) .For any nice discrete self-similar fractal X , F ( X ) is defined recursively. Figure 5 shows an exampleof the construction of F ( X ) , where X is the discrete Sierpinski carpet. Note that F ( X ) is only definedif X is a nice discrete self-similar fractal. Moreover, it appears non-trivial to extend F to other discreteself-similar fractals such as the ‘H’ fractal (the second-to-the-right most image in Figure 4). Open Problem 3.3
Does there exist a zeta-dimension-preserving fibered operator F for a class of dis-crete self-similar fractals which is a superset of the nice discrete self-similar fractals (e.g. it also in-cludes the ‘H’ fractal)? The above open question is intentionally vague. Not only should F preservezeta-dimension, but F ( X ) should also “look” like X in some reasonable visual sense. Figure 5:
Construction of the fibered Sierpinski carpet
20 Self-Assembly of Infinite Structures
Weak self-assembly is a natural way to define what it means for a tile assembly system to compute.There are examples of (decidable) sets that weakly self-assemble but do not strictly self-assemble (i.e.,the discrete Sierpinski triangle [13]). However, if a set X weakly self-assembles, then X is necessarilycomputably enumerable. In this section, we discuss results that pertain to the weak self-assembly of (1)discrete self-similar fractals [16], (2) decidable sets [15], and (3) computably enumerable sets [12]. Recall that pinch-point discrete self-similar fractals do not strictly self-assemble (at any temperature).Furthermore, Patitz and Summers [16] proved that no (non-trivial) discrete self-similar fractal weaklyself-assembles in a locally deterministic [22] temperature 1 tile assembly system. Theorem 4.1
If X ( N is a discrete self-similar fractal, and X weakly self-assembles in the locallydeterministic TAS T X = ( T , s , t ) , where s consists of a single tile placed at the origin, then t > . Intuitively, the proof relies on the aperiodic nature of discrete self-similar fractals and the fact thatthe binding (a.k.a. adjacency) graph of the terminal assembly of T X is an infinite tree, and every infinitebranch is composed of an infinite, periodically repeating sequence of tile types. Open Problem 4.2
Does Theorem 4.1 hold for any directed (not necessarily locally deterministic) TAS?We conjecture that it does, and that such a proof would provide useful new tools for impossibility proofsin the TAM.
We now shift gears and discuss the weak self-assembly of sets at temperature 2.
In [15], Patitz and Summers exhibited a novel characterization of decidable sets of positive integers interms of weak self-assembly in the TAM, where they proved the following theorem.
Theorem 4.3
Let A ⊆ N . Then A ⊆ N is decidable if and only if A × { } and A c × { } weakly self-assemble. Theorem 4.3 is the “self-assembly version” of the classical theorem, which says that a set A ⊆ N isdecidable if and only if A and A c are computably enumerable. The following lemma makes the proof ofthe reverse direction of Theorem 4.3 straight-forward. Lemma 4.4
Let X ⊆ Z . If X weakly self-assembles, then X is computably enumerable. The proof of Lemma 4.4 constructs a self-assembly simulator to enumerate X .To prove the forward direction of Theorem 4.3, it suffices to construct an infinite stack of wedgeconstructions and simply propagate the halting signals down to the negative y -axis. This is illustrated inFigure 6..J.Patitzand S.M.Summers 221PSfrag replacements M ( ) M ( ) M ( ) Figure 6:
The left-most (dark grey) vertical bars represent a binary counter that is embedded into the tile types ofthe TM; the darkest (black) rows represent the initial configuration of M on inputs 0, 1, and 2; and the (light grey)horizontal rows that contain a white/black tile represent halting configurations of M . Although this image seemsto imply that the embedded binary counter increases its width (to the left) each time it increments, this is not truein the construction. This image merely depicts the general shape of the counter as it increments. In addition to their positive result, Patitz and Summers [15] established that any tile assembly system T that “row-computes” a decidable language A ⊆ N having sufficient space complexity must place at leastone tile in each of two adjacent quadrants. A TAS T is said to row-compute a language A ⊆ N if T simulates a TM M with L ( M ) = A on every input n ∈ N , one row at a time, and uses single-tile-wide pathsof tiles to propagate the answer to the question, “does M accept input n ?” to the x -axis. Figure 6 depictsthe essence of what it means for a TAS to row-compute some language. This result, stated precisely, isas follows. Theorem 4.5
Let A ⊆ N . If A DSPACE ( n ) , and T is any TAS that “row-computes” A, then everyterminal assembly of T places at least one tile in each of two adjacent quadrants. Open Problem 4.6
Let A ⊆ N with A DSPACE ( n ) . Is it possible to construct a directed TAS T inwhich the sets A × { } and A c × { } weakly self-assemble, and every terminal assembly a ∈ A (cid:3) [ T ] iscontained in the first quadrant? We conjecture that the answer is ‘no’, and any proof would account forall, possibly exotic methods of computation in the TAM, not only by row-computing. In contrast to Theorem 4.3, Lathrop, Lutz, Patitz, and Summers [12] proved that there are decidable sets D ⊆ Z that do not weakly self-assemble. To see this, for each r ∈ N , define D r = { ( m , n ) ∈ Z (cid:12)(cid:12) | m | + | n | = r } .
22 Self-Assembly of Infinite StructuresThis set is a “diamond” in Z with radius r and center at the origin. For each A ⊆ N , let D A = [ r ∈ A D r . This set is the “system of concentric diamonds” centered at the origin with radii in A . Using Lemma 4.4,one can establish the following result. Lemma 4.7
Let A ∈ N . If D A weakly self-assembles, then there exists an algorithm that, given r ∈ N ,halts and accepts in time O ( n ) , where n = ⌊ lg r ⌋ + , if and only if r ∈ A. The proof of Lemma 4.7 is based on the simple observation that each diamond is finite, and once atile is placed at some point, it cannot be removed. The time hierarchy theorem [9] can be employed toshow that there exists a set A ∈ N such that A ∈ DTIME (cid:0) n (cid:1) − DTIME (cid:0) n (cid:1) . Lemma 4.7 with D = D A is sufficient to prove the following theorem. Theorem 4.8
There is a decidable set D ⊆ Z that does not weakly self-assemble. It is easy to see that if A ⊆ N , then D A ∈ DTIME (cid:0) linear (cid:1) because you can simulate self-assemblywith a Turing machine. Is it possible to do better? Open Problem 4.9 [12] Is there a polynomial-time decidable set D ∈ Z such that D does not weaklyself-assemble? The characterization of decidable sets in terms of weak self-assembly [15] is closely related to the char-acterization of computably enumerable sets in terms of weak self-assembly due to Lathrop, Lutz, Patitzand Summers [12].Let f : Z + → Z + be a function such that for all n ∈ N , f ( n ) ≥ n and f ( n ) = O (cid:0) n (cid:1) . For each set A ⊆ Z + , the set X A = { ( f ( n ) , ) | n ∈ A } is thus a straightforward representation of A as a set of points on the positive x -axis. The first main resultof [12] is stated as follows. Theorem 4.10
If f : Z + → Z + is a function as defined above, then, for all A ⊆ Z + , A is computablyenumerable if and only if the set X A = { ( f ( n ) , ) | n ∈ A } self-assembles. The reverse direction of the proof follows easily from Lemma 4.4. To prove the forward direction, itis sufficient to exhibit, for any TM M , a directed TAS T M that correctly simulates M on all inputs x ∈ Z + in Z . A snapshot of the main construction of [12] is shown in Figure 7.Intuitively, T M self-assembles a “gradually thickening bar” immediately below the positive x -axiswith upward growths emanating from well-defined intervals of points. For each x ∈ Z + , there is anupward growth, in which a modifed wedge construction carries out a simulation of M on x . If M haltson x , then (a portion of) the upward growth associated with the simulation of M ( x ) eventually stops, andsends a signal down along the right side of the upward growth via a one-tile-wide-path of tiles to thepoint ( f ( x ) , ) , where a black tile is placed.Note that Theorem 4.3 is exactly Theorem 4.10 with “computably enumerable” replaced with “de-cidable,” and f ( n ) = n ..J.Patitzand S.M.Summers 223Figure 7: Simulation of M on every input x ∈ N . Notice that M ( ) halts - indicated by the black tile alongthe x -axis. Open Problem 4.11 [12] Does Theorem 4.10 hold for any f such that f ( n ) = O ( n ) ? We conjecture thatthe answer is “no”, and that the construction of [12] is effectively optimal. If the answer to this questionis “yes,” then the proof would require a novel construction which manages to provide an infinite amountof space for each of an infinite number of perhaps non-halting computations in a more compact way than[12]. This paper surveyed a subset of recent theoretical results in algorithmic self-assembly relating to the self-assembly of infinite structures in the TAM. Specifically, in this paper we reviewed impossibility resultswith respect to the strict/weak self-assembly of various classes of discrete self-similar fractals [16],impossibility results for the weak self-assembly of exponential-time decidable sets [12], characterizationsof particular classes of languages in terms of weak self-assembly [12, 15], and the strict self-assembly offractal-like structures. Finally, we believe that the benefit of continued research along these lines has thepotential to shed light on the elusive relationship between geometry and computation.
References [1] L. Adleman,
Towards a mathematical theory of self-assembly , Tech. report, University of Southern Califor-nia, 2000.[2] Leonard M. Adleman, Qi Cheng, Ashish Goel, Ming-Deh A. Huang, David Kempe, Pablo Moisset de Es-pan´es, and Paul W. K. Rothemund,
Combinatorial optimization problems in self-assembly , Proceedings ofthe Thirty-Fourth Annual ACM Symposium on Theory of Computing, 2002, pp. 23–32.
24 Self-Assembly of Infinite Structures [3] Leonard M. Adleman, Jarkko Kari, Lila Kari, and Dustin Reishus,
On the decidability of self-assembly ofinfinite ribbons , Proceedings of the 43rd Annual IEEE Symposium on Foundations of Computer Science,2002, pp. 530–537.[4] Gagan Aggarwal, Michael H. Goldwasser, Ming-Yang Kau, and Robert T. Schweller,
Complexities for gen-eralized models of self-assembly , Proceedings of ACM-SIAM Symposium on Discrete Algorithms, 2004.[5] Florent Becker, Ivan Rapaport, and Eric R´emila,
Self-assemblying classes of shapes with a minimum numberof tiles, and in optimal time , FSTTCS, 2006, pp. 45–56.[6] Ho-Lin Chen and Ashish Goel,
Error free self-assembly with error prone tiles , Proceedings of the 10thInternational Meeting on DNA Based Computers, 2004.[7] Qi Cheng, Ashish Goel, and Pablo Moisset de Espan´es,
Optimal self-assembly of counters at temperaturetwo , Proceedings of the First Conference on Foundations of Nanoscience: Self-assembled Architectures andDevices, 2004.[8] D. Doty, X. Gu, J.H. Lutz, E. Mayordomo, and P. Moser,
Zeta-Dimension , Proceedings of the Thirtieth In-ternational Symposium on Mathematical Foundations of Computer Science, Springer-Verlag, 2005, pp. 283–294.[9] J. Hartmanis and R. E. Stearns,
On the computational complexity of algorithms , Transactions of the AmericanMathematical Society (1965), 285–306.[10] Ming-Yang Kao and Robert Schweller,
Reducing tile complexity for self-assembly through temperature pro-gramming , Proceedings of the 17th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2006),Miami, Florida, Jan. 2006, pp. 571-580, 2007.[11] Ming-Yang Kao and Robert T. Schweller,
Randomized self-assembly for approximate shapes. , ICALP(1) (Luca Aceto, Ivan Damgrd, Leslie Ann Goldberg, Magns M. Halldrsson, Anna Inglfsdttir, and IgorWalukiewicz, eds.), Lecture Notes in Computer Science, vol. 5125, Springer, 2008, pp. 370–384.[12] James I. Lathrop, Jack H. Lutz, Matthew J. Patitz, and Scott M. Summers,
Computability and complexityin self-assembly , Proceedings of The Fourth Conference on Computability in Europe (Athens, Greece, June15-20, 2008), 2008.[13] James I. Lathrop, Jack H. Lutz, and Scott M. Summers,
Strict self-assembly of discrete Sierpinski triangles ,Theoretical Computer Science. To appear.[14] Ashish Goel Leonard Adleman, Qi Cheng and Ming-Deh Huang,
Running time and program size for self-assembled squares , STOC ’01: Proceedings of the thirty-third annual ACM symposium on Theory of com-puting (New York, NY, USA), ACM, 2001, pp. 740–748.[15] Matthew J. Patitz and Scott M. Summers,
Self-assembly of decidable sets , Proceedings of The Seventh Inter-national Conference on Unconventional Computation (Vienna, Austria, August 25-28, 2008), 2008.[16] ,
Self-assembly of discrete self-similar fractals (extended abstract) , Proceedings of The FourteenthInternational Meeting on DNA Computing (Prague, Czech Republic, June 2-6, 2008). To appear., 2008.[17] John H. Reif,
Molecular assembly and computation: From theory to experimental demonstrations , Pro-ceedings of the Twenty-Ninth International Colloquium on Automata, Languages and Programming, 2002,pp. 1–21.[18] Paul W. K. Rothemund,
Theory and experiments in algorithmic self-assembly , Ph.D. thesis, University ofSouthern California, December 2001.[19] Paul W. K. Rothemund and Erik Winfree,
The program-size complexity of self-assembled squares (extendedabstract). , STOC, 2000, pp. 459–468.[20] Paul W.K. Rothemund, Nick Papadakis, and Erik Winfree,
Algorithmic self-assembly of DNA Sierpinskitriangles , PLoS Biology (2004), no. 12.[21] David Soloveichik and Erik Winfree, Complexity of compact proofreading for self-assembled patterns , Theeleventh International Meeting on DNA Computing, 2005.[22] ,
Complexity of self-assembled shapes , SIAM Journal on Computing 36, 2007, pp. 1544–1569.[23] Thomas LaBean Urmi Majumder, Sudheer Sahu and John H. Reif,
Design and simulation of self-repairing .J.Patitzand S.M.Summers 225
DNA lattices , DNA Computing: DNA12, Lecture Notes in Computer Science, vol. 4287, Springer-Verlag,2006.[24] Hao Wang,
Proving theorems by pattern recognition – II , The Bell System Technical Journal XL (1961),no. 1, 1–41.[25] , Dominoes and the AEA case of the decision problem , Proceedings of the Symposium on Mathemat-ical Theory of Automata (New York, 1962), Polytechnic Press of Polytechnic Inst. of Brooklyn, Brooklyn,N.Y., 1963, pp. 23–55.[26] Erik Winfree,
Simulations of computing by self-assembly , Tech. Report CaltechCSTR:1998.22, CaliforniaInstitute of Technology.[27] ,
Algorithmic self-assembly of DNA , Ph.D. thesis, California Institute of Technology, June 1998.[28] Erik Winfree and Renat Bekbolatov,