## The program-size complexity of self-assembled squares (2000)

Venue: | In Proceedings of the thirty-second annual ACM symposium on Theory of computing |

Citations: | 175 - 14 self |

### Citations

2121 | An Introduction to Kolmogorov Complexity and its Applications
- Li, Vit'anyi
- 1997
(Show Context)
Citation Context ...:a:( log N log log N ). Proof. The Kolmogorov complexityofaninteger N with respect to a universal Turing machine U is KU(N) = min jpj s.t. U(p) =#N where #N is the binary string representing N. (See [=-=Li and Vitanyi, 1997-=-] for results on Kolmogorov complexity.) Recall that KU (N) < dlog Ne ; for at most 1 2 of all N, by the pidgeonhole principle. Therefore, for any >0, KU(N) > (1 ; )logN for almost all N. There exists... |

771 | Molecular Computation of Solutions to Combinatorial Problems. Science 266
- Adleman
- 1994
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Citation Context ...al mechanisms play an important role in understanding natural self-organization, at least in biological systems: algorithms control the generation of order. Research in DNA and molecular computation [=-=Adleman, 1994-=-] has established that universal computation can be performed in biochemical systems, such as enzymatic (ribosome-like) modi cation or translation of a heteropolymer [Bennett, 1982� Kurtz et al., 1997... |

370 |
The undecidability of the dominoe problem
- Berger
- 1966
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Citation Context ... to see that to produce a 1D line of N tiles requires N tiles for all T > 0. This result exactly parallels the decidability the 1D tiling problem� the 2D tiling problem, inc contrast, is undecidable [=-=Berger, 1966-=-]. At the other extreme, it seems unlikely that 3D allows for phenomena fundamentally di erent from2D,sinceuniversal computation is already possible in 2D. 2. A MODEL OF SELF-ASSEMBLY Our discussion o... |

88 |
The thermodynamics of computation – a review
- Bennett
- 1982
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Citation Context ...ecular computation [Adleman, 1994] has established that universal computation can be performed in biochemical systems, such as enzymatic (ribosome-like) modi cation or translation of a heteropolymer [=-=Bennett, 1982-=-� Kurtz et al., 1997], signal-transduction cascades [Hjelmfelt and Ross, 1995� Magnasco, 1997], and the self-assembly of protein or DNA into supramolecular structures [Radin, 1991� Winfree, 1996]. How... |

87 |
On convex functions
- RADO
- 1935
(Show Context)
Citation Context ... of a square we show that:sTHEOREM 5. K~AiN ) = O~.o.(f(N)) for f (N) any non-sdecreasing unbounded computable function.sPROOF. Our proof relies on a self-assembly version of thesBusy Beaver problem [=-=Rado, 1962-=-]. Define:sBTSAi ~) -~ max{N s.t. (N,n) • SqT}.sTo show Theorem 5, we first showsB~A(n) = f2(F(n)) for any computable function F(n).s(1)sTheorem 5 follows from (1) by contradiction: if false, thensthe... |

57 | Self-assembly of mesoscale objects into ordered two-dimensional arrays
- Bowden, Terfort, et al.
- 1997
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Citation Context ...l is obtained in the limit of strong binding domains and low monomer concentrations [Winfree, 1998]. Macroscopic systems for 2D self-assembly based on lateral capillary forces [Hosokawa et al., 1996� =-=Bowden et al., 1997-=-� Bowden et al., 1999� Rothemund, 2000] provide additional motivation for the Tile Assembly Model. In these systems millimeter-scale plastic tiles oat at an interface between hydrophobic and hydrophil... |

39 |
Chemical Kinetics is Turing Universal.
- Magnasco
- 1997
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Citation Context ...ed in biochemical systems, such as enzymatic (ribosome-like) modi cation or translation of a heteropolymer [Bennett, 1982� Kurtz et al., 1997], signal-transduction cascades [Hjelmfelt and Ross, 1995� =-=Magnasco, 1997-=-], and the self-assembly of protein or DNA into supramolecular structures [Radin, 1991� Winfree, 1996]. How widespread is in uence of computational mechanisms in the generation of order { does it spre... |

33 | Global order from local sources
- Radin
- 1991
(Show Context)
Citation Context ...f a heteropolymer [Bennett, 1982� Kurtz et al., 1997], signal-transduction cascades [Hjelmfelt and Ross, 1995� Magnasco, 1997], and the self-assembly of protein or DNA into supramolecular structures [=-=Radin, 1991-=-� Winfree, 1996]. How widespread is in uence of computational mechanisms in the generation of order { does it spread beyond the biological domain? STOC ’00 Portland, Oregon USA For most of this centur... |

25 |
Crystal Growth for Beginners: Fundamentals of Nucleation, Crystal Growth, and Epitaxy. World Scientific,
- Markov
- 1995
(Show Context)
Citation Context ...sen seed tile, growth occurs by addition of single tiles. (The growth of crystals by monomer addition, as opposed to merging of crystal fragments, is a common assumption in studies of crystal growth [=-=Markov, 1995-=-]� large defect-free crystals are not observed under physical conditions where growth occurs by aggregation of small fragments.) Tiles bind a growing assembly only if their binding interactions are of... |

22 |
Two-dimensional micro-selfassembly using the surface tension of water
- Hosokawa, Shimoyama, et al.
- 1996
(Show Context)
Citation Context ... the Tile Assembly Model is obtained in the limit of strong binding domains and low monomer concentrations [Winfree, 1998]. Macroscopic systems for 2D self-assembly based on lateral capillary forces [=-=Hosokawa et al., 1996-=-� Bowden et al., 1997� Bowden et al., 1999� Rothemund, 2000] provide additional motivation for the Tile Assembly Model. In these systems millimeter-scale plastic tiles oat at an interface between hydr... |

20 |
Mesoscale self-assembly of hexagonal plates using lateral capillary forces: Synthesis using the "capilary" bond.
- Bowden, Choi, et al.
- 1999
(Show Context)
Citation Context ...limit of strong binding domains and low monomer concentrations [Winfree, 1998]. Macroscopic systems for 2D self-assembly based on lateral capillary forces [Hosokawa et al., 1996� Bowden et al., 1997� =-=Bowden et al., 1999-=-� Rothemund, 2000] provide additional motivation for the Tile Assembly Model. In these systems millimeter-scale plastic tiles oat at an interface between hydrophobic and hydrophilic liquids (e.g., oil... |

19 |
A genetic switch, 2nd ed
- Ptashne
- 1992
(Show Context)
Citation Context ... one-dimensional cellular automata can be simulated� hence T = 2 self-assembly is universal [Winfree, 1996]. It is interesting to observe that cooperative e ects play a major role in gene regulation [=-=Ptashne, 1992-=-] and many other biological systems. Branched DNA molecules [Seeman, 1998] provide a direct physical motivation for the Tile Assembly Model. DNA double-crossover molecules, each bearing four \sticky e... |

10 | Biological computing
- Kurtz, Mahaney, et al.
- 1996
(Show Context)
Citation Context ...ion [Adleman, 1994] has established that universal computation can be performed in biochemical systems, such as enzymatic (ribosome-like) modi cation or translation of a heteropolymer [Bennett, 1982� =-=Kurtz et al., 1997-=-], signal-transduction cascades [Hjelmfelt and Ross, 1995� Magnasco, 1997], and the self-assembly of protein or DNA into supramolecular structures [Radin, 1991� Winfree, 1996]. How widespread is in ue... |

8 |
Implementation of logic functions and computations by chemical kinetics.
- Hjelmfelt, Ross
- 1995
(Show Context)
Citation Context ...computation can be performed in biochemical systems, such as enzymatic (ribosome-like) modi cation or translation of a heteropolymer [Bennett, 1982� Kurtz et al., 1997], signal-transduction cascades [=-=Hjelmfelt and Ross, 1995-=-� Magnasco, 1997], and the self-assembly of protein or DNA into supramolecular structures [Radin, 1991� Winfree, 1996]. How widespread is in uence of computational mechanisms in the generation of orde... |

6 |
Generalised crystallography
- Mackay
- 1975
(Show Context)
Citation Context ...overed and excludes altogether biological materials that have complex order. These concerns have led the crystallographer Alan Mackay to propose that a \generalized crystallography" might de ne order[=-=Mackay, 1995-=-] using computer programs and cellular automata. Such an algorithmic framework for studying self-assembly is attractive for two reasons. First, because of Church's thesis, we expect that computer prog... |

3 | unpublished manuscript, 2000). Toward a mathematical theory of self-assembly - Adleman |

1 | Crystal Growth for Beginners: fundamentals ofnucleation, crystal growth, and epitaxy - Markov - 1995 |