Optimal Program-Size Complexity for Self-Assembly at Temperature 1 in 3D

Furcy, David; Micka, Samuel; Summers, Scott M.

doi:10.1007/978-3-319-21999-8_5

Cited by 5 publications

(11 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In this type of selfassembly, a tile may non-cooperatively bind to an assembly via (at least) one of its sides, unlike in cooperative self-assembly, in which some tiles may be required to bind on two or more sides. It is worth noting that Summers [10] and Cook, Fu and Schweller [3], we show how the optimal construction of Soloveichik and Winfree can be simulated in 3D using non-cooperative self-assembly with optimal tile complexity. Thus, our main result represents a Turing-universal way of guiding the self-assembly of a scaled-up, just-barely 3D version of an arbitrary finite shape X at temperature 1 with optimal tile complexity.…”

Section: Introductionmentioning

confidence: 88%

“…In the decoding phase, the bits of p are decoded from a O(log | p |)-bits-per-tile representation to a 1-bit-per-tile representation (actually, we end up with a 1-bitper-gadget representation, which is sufficient to maintain the optimality of our construction). To accomplish this, we use the 3D, temperature 1 optimal encoding scheme of Furcy, Micka and Summers [10]. When the decoding phase completes, the decoded bits of p are advertised in a one-bit-per-gadget representation along the top of the optimal encoding region (the rectangle with the vertical lines in Figure 2).…”

Section: Seed Blockmentioning

confidence: 99%

“…Let W decode and H decode be the maximum horizontal and vertical extent, respectively, of the seed block after the decoding phase (the rectangle with vertical lines pattern in Figure 2) completes. From [10], we know that W decode > H decode . Next, in the simulation phase, the tape grows to the right by two tape cells for each transition.…”

Section: Seed Blockmentioning

confidence: 99%

“…Let W sim and H sim be the maximum horizontal and vertical extent, respectively, of the seed block after the simulation phase (the zig-zag pattern in Figure 2 Figure 2: Self-assembly of the upper portion of the seed block consists of three logical phases. In the first phase (the region filled with vertical lines), the bits of p are decoded using the 3D, temperature 1 optimal encoding scheme of Furcy, Micka and Summers [10] (the encoded bits of p are depicted as the shorter binary string). The decoded bits of p (the longer binary string) are input to a fixed universal Turing machine U .…”

Section: Seed Blockmentioning

confidence: 99%

See 3 more Smart Citations

Optimal Self-Assembly of Finite Shapes at Temperature 1 in 3D

Furcy

Summers

2016

Algorithmica

Self Cite

View full text Add to dashboard Cite

Working in a three-dimensional variant of Winfree's abstract Tile Assembly Model, we show that, for an arbitrary finite, connected shape X ⊂ Z 2 , there is a tile set that uniquely self-assembles into a 3D representation of X at temperature 1 in 3D with optimal program-size complexity (the program-size complexity, also known as tile complexity, of a shape is the minimum number of tile types required to uniquely self-assemble it). Moreover, our construction is "just barely" 3D in the sense that it only places tiles in the z = 0 and z = 1 planes. Our result is essentially a just-barely 3D temperature 1 simulation of a similar 2D temperature 2 result by Soloveichik and Winfree (SICOMP 2007).cooperative self-assembly leads to highly non-trivial theoretical behavior, e.g., Turing universality [25] and the efficient self-assembly of N × N squares [1,18] and other algorithmically specified shapes [23].Despite its theoretical algorithmic capabilities, when cooperative self-assembly is implemented using DNA tiles in the laboratory [2,14,20,21,26], tiles may (and do) erroneously bind in a non-cooperative fashion, which usually results in the production of undesired final structures. In order to completely avoid the erroneous effects of tiles unexpectedly binding in a non-cooperative fashion, the experimenter should only build nanoscale structures using constructions that are guaranteed to work correctly in non-cooperative self-assembly. Thus, characterizing the theoretical power of non-cooperative self-assembly has significant practical implications.Although no characterization of the power of non-cooperative self-assembly exists at the time of this writing, Doty, Patitz and Summers conjecture [7] that 2D non-cooperative self-assembly is weaker than 2D cooperative self-assembly because a certain technical condition, known as "pumpability", is true for any 2D non-cooperative tile set. If the pumpability conjecture is true, then non-cooperative 2D self-assembly can only produce simple, highly-regular shapes and patterns, which are too simple and regular to be the result of complex computation.In addition to the pumpability conjecture, there are a number of results that study the suspected weakness of non-cooperative self-assembly. For example, Rothemund and Winfree [18] proved that, if the final assembly must be fully connected, then the minimum number of unique tile types required to self-assemble an N × N square (i.e., its tile complexity) is exactly 2N − 1. Manuch, Stacho and Stoll [13] showed that the previous tile complexity is also true when the final assembly cannot contain even any glue mismatches. Moreover, at the time of this writing, the only way in which non-cooperative self-assembly has been shown to be unconditionally weaker than cooperative self-assembly is in the sense of intrinsic universality [5,6]. First, Doty, Lutz, Patitz, Schweller, Summers and Woods [5] proved the existence of a universal cooperative tile set that can be programmed to simulate the behavior of any tile set (i.e., the aTAM is intrinsical...

show abstract

Section: Introductionmentioning

confidence: 88%

Section: Seed Blockmentioning

confidence: 99%

Section: Seed Blockmentioning

confidence: 99%

Section: Seed Blockmentioning

confidence: 99%

See 2 more Smart Citations

Optimal Self-Assembly of Finite Shapes at Temperature 1 in 3D

Furcy

Summers

2016

Algorithmica

Self Cite

View full text Add to dashboard Cite

show abstract

“…Claim F. 10. Any configuration of c that includes a polymer p seedless that does not contain the seed but does contain two or more elements from the set off all computation monomers, tape extension supertiles, and end monomers is unstable.…”

Section: F4 Seedless Polymersmentioning

confidence: 99%

Thermodynamically Favorable Computation via Tile Self-assembly

Chalk

Hendricks

Patitz

et al. 2018

Unconventional Computation and Natural Computation

View full text Add to dashboard Cite

The recently introduced Thermodynamic Binding Networks (TBN) model was developed with the purpose of studying self-assembling systems by focusing on their thermodynamically favorable final states, and ignoring the kinetic pathways through which they evolve. The model was intentionally developed to abstract away not only the notion of time, but also the constraints of geometry. Collections of monomers with binding domains which allow them to form polymers via complementary bonds are analyzed to determine their final, stable configurations, which are those which maximize the number of bonds formed (i.e. enthalpy) and the number of independent components (i.e. entropy). In this paper, we first develop a definition of what it means for a TBN to perform a computation, and then present a set of constructions which are capable of performing computations by simulating the behaviors of space-bounded Turing machines and boolean circuits. In contrast to previous TBN results, these constructions are robust to great variability in the counts of monomers existing in the systems and the numbers of polymers that form in parallel. Although the Turing machine simulating TBNs are inefficient in terms of the numbers of unique monomer types required, as compared to algorithmic self-assembling systems in the abstract Tile Assembly Model (aTAM), we then show that a general strategy of porting those aTAM system designs to TBNs produces TBNs which incorrectly simulate computations. Finally, we present a refinement of the TBN model which we call the Geometric Thermodynamic Binding Networks (GTBN) model in which monomers are defined with rigid geometries and form rigid bonds.

show abstract