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...: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...

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...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...

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... 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...

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...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...

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... 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...

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...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...

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...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...

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...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...

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...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...

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... 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...

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...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...

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... 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...

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...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...

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...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...

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...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...