Tight Bounds for Active Self-assembly Using an Insertion Primitive

Malchik, Caleb; Winslow, Andrew

doi:10.1007/978-3-662-44777-2_56

Cited by 5 publications

(7 citation statements)

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“…We follow these positive results with a set of lower bounds proving that these are the best possible polymer lengths and expected construction times. * Some of the results in this work have been published as [18].types and Ω(n) expected assembly time [1] and any shape with n tiles requires Ω( √ n) expected time to assemble [13].Such a limitation may not seem so significant, except that a wide range of biological systems form complex assemblies in time polylogarithmic in the assembly size, as Dabby and Chen [6] and Woods et al [24] observe. These biological systems are capable of such growth because their particles (e.g.…”

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Tight Bounds for Active Self-Assembly Using an Insertion Primitive

2015

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We prove two limits on the behavior of a model of self-assembling particles introduced by Dabby and Chen (Proceedings of 24th ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 1526-1536, 2013), called insertion systems, where monomers insert themselves into the middle of a growing linear polymer. First, we prove that the expressive power of these systems is equal to context-free grammars, answering a question posed by Dabby and Chen. Second, we prove that systems of k monomer types can deterministically construct polymers of length n = 2 Θ(k 3/2 ) in O(log 5/3 (n)) expected time, and that this is optimal in both the number of monomer types and expected time.

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confidence: 99%

Tight Bounds for Active Self-Assembly Using an Insertion Primitive

2015

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“…They also show that the set of 1D polymers produced by any instance of their model is a context-free language, and give a design for implementation with DNA molecules. Malchik and Winslow (2014) show that any context-free language can be expressed as an instance of this model, and give an asymptotically tight bound of 2 Hðk 3=2 Þ on the length of polymers produced using k monomer types, thus characterising two aspects of the model.…”

Section: Previous Work On Active Self-assembly With Movementmentioning

confidence: 99%

Parallel computation using active self-assembly

2014

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We study the computational complexity of the recently proposed nubots model of molecular-scale selfassembly. The model generalises asynchronous cellular automata to have non-local movement where large assemblies of molecules can be moved around, analogous to millions of molecular motors in animal muscle effecting the rapid movement of macroscale arms and legs. We show that nubots is capable of simulating Boolean circuits of polylogarithmic depth and polynomial size, in only polylogarithmic expected time. In computational complexity terms, we show that any problem from the complexity class NC is solved in polylogarithmic expected time on nubots that use a polynomial amount of workspace. Along the way, we give fast parallel algorithms for a number of problems including line growth, sorting, Boolean matrix multiplication and space-bounded Turing machine simulation, all using a constant number of nubot states (monomer types). Circuit depth is a well-studied notion of parallel time, and our result implies that nubots is a highly parallel model of computation in a formal sense. Asynchronous cellular automata are not capable of such parallelism, and our result shows that adding a movement primitive to such a model, to get the nubots model, drastically increases parallel processing abilities.

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“…Examples include computational power [8,39], efficiency in terms of the number of tile types needed to build shapes or patterns [40,41] and computational complexity of verification of properties of tile assembly systems [42]. See other surveys for more details [43,44].…”

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confidence: 99%

“…It is important to point out that besides simulation there are a number of other methods to compare self-assembly models that also lead to interesting results and proof techniques. Examples include computational power [8,39], efficiency in terms of the number of tile types needed to build shapes or patterns [40,41] and computational complexity of verification of properties of tile assembly systems [42]. See other surveys for more details [43,44].…”

Section: (A) Introduction To the Use Of Simulation In Self-assemblymentioning

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Intrinsic universality and the computational power of self-assembly

Woods

2015

Phil. Trans. R. Soc. A.

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Molecular self-assembly, the formation of large structures by small pieces of matter sticking together according to simple local interactions, is a ubiquitous phenomenon. A challenging engineering goal is to design a few molecules so that large numbers of them can self-assemble into desired complicated target objects. Indeed, we would like to understand the ultimate capabilities and limitations of this bottom-up fabrication process. We look to theoretical models of algorithmic self-assembly, where small square tiles stick together according to simple local rules in order to carry out a crystal growth process. In this survey, we focus on the use of simulation between such models to classify and separate their computational and expressive powers. Roughly speaking, one model simulates another if they grow the same structures, via the same dynamical growth processes. Our journey begins with the result that there is a single intrinsically universal tile set that, with appropriate initialization and spatial scaling, simulates any instance of Winfree's abstract Tile Assembly Model. This universal tile set exhibits something stronger than Turing universality: it captures the geometry and dynamics of any simulated system in a very direct way. From there we find that there is no such tile set in the more restrictive non-cooperative model, proving it weaker than the full Tile Assembly Model. In the two-handed model, where large structures can bind together in one step, we encounter an infinite set of infinite hierarchies of strictly increasing simulation power. Towards the end of our trip, we find one tile to rule them all: a single rotatable flipable polygonal tile that simulates any tile assembly system. We find another tile that aperiodically tiles the plane (but with small gaps). These and other recent results show that simulation is giving rise to a kind of computational complexity theory for self-assembly. It seems this could be the beginning of a much longer journey, so directions for future work are suggested.

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Tight Bounds for Active Self-assembly Using an Insertion Primitive

Cited by 5 publications

References 23 publications

Tight Bounds for Active Self-Assembly Using an Insertion Primitive

Tight Bounds for Active Self-Assembly Using an Insertion Primitive

Parallel computation using active self-assembly

Intrinsic universality and the computational power of self-assembly

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