Self-assembly: modelling, simulation, and planning

Samoskládání je proces, při kterém se kolekce neuspořádaných částic samovolně orientuje do uspořádaného vzoru nebo funkční struktury bez působení vnější síly, pouze za pomoci lokálních interakcí mezi samotnými částicemi. Tato teze se zaměřuje na teorii dlaždicových samoskládacích systémů a jejich syntézu. Nejdříve je představena oblast výzkumu věnující se dlaždičovým samoskládacím systémům, a poté jsou důkladně popsány základní typy dlaždicových skládacích systémů, kterými jsou abstract Tile Assembly Model (aTAM ), kinetic Tile Assembly Model (kTAM ), a 2-Handed Assembly Model (2HAM ). Poté jsou představeny novější modely a modely se specifickým použitím. Dále je zahrnut stručný popis původu teorie dlaždicového samoskládání společně s krátkým popisem nedávného výzkumu. Dále jsou představeny dva obecné otevřené problémy dlaždicového samoskládání s hlavním zaměřením na problém Pattern Self-Assembly Tile Set Synthesis (PATS), což je NP-těžká kombinatorická optimalizační úloha. Nakonec je ukázán algoritmus Partition Search with Heuristics (PS-H ), který se používá k řešení problému PATS. Následovně jsou demonstrovány dvě aplikace, které byly vyvinuty pro podporu výzkumu abstraktních dlaždicových skládacích modelů a syntézy množin dlaždic pro samoskládání zadaných vzorů. První aplikace je schopná simulovat aTAM a 2HAM systémy ve 2D prostoru. Druhá aplikace je řešič PATS problému, který využívá algoritmu PS-H. Pro obě aplikace jsou popsány hlavní vlastnosti a návrhová rozhodnutí, která řídila jejich vývoj. Nakonec jsou předloženy výsledky několika experimentů. Jedna skupina experimentů byla zaměřena na ověření výpočetní náročnosti vyvinutých algoritmů pro simulátor. Druhá sada experimentů zkoumala vliv jednotlivých vlastností vzorů na vlastnosti dlaždicových systémů, které byly získány syntézou ze vzorů pomocí vyvinutého řešiče PATS problému. Bylo prokázáno, že algoritmus simulující aTAM systém má lineární časovou výpočetní náročnost, zatímco algoritmus simulující 2HAM systém má exponenciální časovou výpočetní náročnost, která navíc silně závisí na simulovaném systému. Aplikace pro řešení syntézy množiny dlaždic ze vzorů je schopna najít relativně malé řešení i pro velké zadané vzory, a to v přiměřeném čase.Self-assembly is the process in which a collection of disordered units organise themselves into ordered patterns or functional structures without any external direction, solely using local interactions among the components. This thesis focuses on the theory of tile-based self-assembly systems and their synthesis. First, an introduction to the study field of tile-based self-assembly systems are given, followed by a thorough description of common types of tile assembly systems such as abstract Tile Assembly Model (aTAM ), kinetic Tile Assembly Model (kTAM ), and 2-Handed Assembly Model (2HAM ). After that, various recently developed models and models with specific applications are listed. A brief summary of the origins of the tile-based self-assembly is also included together with a short review of recent results. Two general open problems are presented with the main focus on the Pattern Self-Assembly Tile Set Synthesis (PATS) problem, which is NP-hard combinatorial optimisation problem. Partition Search with Heuristics (PS-H ) algorithm is presented as it is used for solving the PATS problem. Next, two applications which were developed to study the abstract tile assembly models and the synthesis of tile sets for pattern self-assembly are introduced. The first application is a simulator capable of simulating aTAM and 2HAM systems in 2D. The second application is a solver of the PATS problem based around the PS-H algorithm. Main features and design decisions are described for both applications. Finally, results from several experiments are presented. One set of experiments were focused on verification of computation complexity of algorithms developed for the simulator, and the other set of experiments studied the influences of the properties of the pattern on the tile assembly system synthesised by our implementation of PATS problem solver. It was shown that the algorithm for simulating aTAM systems have linear computation time complexity, whereas the algorithm simulating 2HAM systems have exponential computation time complexity, which strongly varies based on the simulated system. The synthesiser application is capable of finding a relatively small solution even for quite large input patterns in reasonable amounts of time

Growing machines

Thesis (Ph. D.)--Massachusetts Institute of Technology, School of Architecture and Planning, Program in Media Arts and Sciences, 2004.Includes bibliographical references.construction is developed in three dimensions. It is similarly shown that right-angled tetrahedrons, when folded from an edge-connected string, can generate any three dimensional structure where the primitive pixel (or voxel) is a rhombic hexahedron. This construction also suggests a concept of 3D completeness for assembly, somewhat analogous to the concept of Turing completeness in computation. In combination, these pieces of work suggest that a manufacturing system based on four tiles, with seven states per tile, is capable of self-replication of arbitrary 3D structure by copying, then folding, bit strings of those tiles where the desired structure is encoded in the tile sequence.Biological systems are replete with examples of high complexity structures that have "self assembled," or more accurately, programmatically assembled from many smaller, simpler components. By comparison, the fabrication systems engineered by humans are typically top down, or subtractive, processes where systems of limited complexity are carved from bulk materials. Self-assembly to date has resembled crystallization more than it has the programmatic assembly of complex or useful structures--these systems are information limited. This thesis explores the programming of self-assembling systems by the introduction of small amounts of state to the sub-units of the assembly. A six-state, kinematic, conformational latching component is presented that is capable of self-replicating bit strings of two shape differentiated versions of the same component where the two variants represent the 0 and 1 bits. Individual units do not assemble until a string is introduced to the assembly environment to be copied. Electro-mechanical state machine emulators were constructed. Operating on an air table, the units demonstrated logic limited aggregation, or error-preventing assembly, as well as autonomous self-replication of bit strings. A new construction was developed that demonstrates that any two dimensional shape composed of square pixels can be deterministically folded from a linear string of vertex-connected square tiles. This non-intersecting series of folds implies a 'resolution' limit of four tiles per pixel. It is shown that four types of tiles, patterned magnetically, is sufficient to construct any shape given sequential folding. The construction was implemented to fold the letters 'M I T' from sequences of the 4 tile types. An analogousSaul Thomas Griffith.Ph.D

Negative Interactions in Irreversible Self-Assembly

This paper explores the use of negative (i.e., repulsive) interaction the abstract Tile Assembly Model defined by Winfree. Winfree postulated negative interactions to be physically plausible in his Ph.D. thesis, and Reif, Sahu, and Yin explored their power in the context of reversible attachment operations. We explore the power of negative interactions with irreversible attachments, and we achieve two main results. Our first result is an impossibility theorem: after t steps of assembly, Omega(t) tiles will be forever bound to an assembly, unable to detach. Thus negative glue strengths do not afford unlimited power to reuse tiles. Our second result is a positive one: we construct a set of tiles that can simulate a Turing machine with space bound s and time bound t, while ensuring that no intermediate assembly grows larger than O(s), rather than O(s * t) as required by the standard Turing machine simulation with tiles

Intrinsic universality and the computational power of self-assembly

This short survey of recent work in tile self-assembly discusses the use of simulation to classify and separate the computational and expressive power of self-assembly models. The journey begins with the result that there is a single universal tile set that, with proper initialization and scaling, simulates any tile assembly system. This universal tile set exhibits something stronger than Turing universality: it captures the geometry and dynamics of any simulated system. From there we find that there is no such tile set in the noncooperative, or temperature 1, model, proving it weaker than the full tile assembly model. In the two-handed or hierarchal model, where large assemblies can bind together on one step, we encounter an infinite set, of infinite hierarchies, each with 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 can simulate any tile assembly system. It seems this could be the beginning of a much longer journey, so directions for future work are suggested.Comment: In Proceedings MCU 2013, arXiv:1309.104

Combinatorial optimization problems in self-assembly

Self-assembly is the ubiquitous process by which simple objects autonomously assemble into intricate complexes. It has been suggested that intricate self-assembly processes will ultimately be used in circuit fabrication, nano-robotics, DNA computation, and amorphous computing. In this paper, we study two combinatorial optimization problems related to efficient self-assembly of shapes in the Tile Assembly Model of self-assembly proposed by Rothemund and Winfree [18]. The first is the Minimum Tile Set Problem, where the goal is to find the smallest tile system that uniquely produces a given shape. The second is the Tile Concentrations Problem, where the goal is to decide on the relative concentrations of different types of tiles so that a tile system assembles as quickly as possible. The first problem is akin to finding optimum program size, and the second to finding optimum running time for a "program" to assemble the shape.Self-assembly is the ubiquitous process by which simple objects autonomously assemble into intricate complexes. It has been suggested that intricate self-assembly processes will ultimately be used in circuit fabrication, nano-robotics, DNA computation, and amorphous computing. In this paper, we study two combinatorial optimization problems related to efficient self-assembly of shapes in the Tile Assembly Model of self-assembly proposed by Rothemund and Winfree [18]. The first is the Minimum Tile Set Problem, where the goal is to find the smallest tile system that uniquely produces a given shape. The second is the Tile Concentrations Problem, where the goal is to decide on the relative concentrations of different types of tiles so that a tile system assembles as quickly as possible. The first problem is akin to finding optimum program size, and the second to finding optimum running time for a "program" to assemble the shape. We prove that the first problem is NP-complete in general, and polynomial time solvable on trees and squares. In order to prove that the problem is in NP, we present a polynomial time algorithm to verify whether a given tile system uniquely produces a given shape. This algorithm is analogous to a program verifier for traditional computational systems, and may well be of independent interest. For the second problem, we present a polynomial time $O(\log n)$-approximation algorithm that works for a large class of tile systems that we call partial order systems

The Power of Duples (in Self-Assembly): It's Not So Hip To Be Square

In this paper we define the Dupled abstract Tile Assembly Model (DaTAM), which is a slight extension to the abstract Tile Assembly Model (aTAM) that allows for not only the standard square tiles, but also "duple" tiles which are rectangles pre-formed by the joining of two square tiles. We show that the addition of duples allows for powerful behaviors of self-assembling systems at temperature 1, meaning systems which exclude the requirement of cooperative binding by tiles (i.e., the requirement that a tile must be able to bind to at least 2 tiles in an existing assembly if it is to attach). Cooperative binding is conjectured to be required in the standard aTAM for Turing universal computation and the efficient self-assembly of shapes, but we show that in the DaTAM these behaviors can in fact be exhibited at temperature 1. We then show that the DaTAM doesn't provide asymptotic improvements over the aTAM in its ability to efficiently build thin rectangles. Finally, we present a series of results which prove that the temperature-2 aTAM and temperature-1 DaTAM have mutually exclusive powers. That is, each is able to self-assemble shapes that the other can't, and each has systems which cannot be simulated by the other. Beyond being of purely theoretical interest, these results have practical motivation as duples have already proven to be useful in laboratory implementations of DNA-based tiles