The Complexity of Fixed-Height Patterned Tile Self-Assembly

We characterize the complexity of the PATS problem for patterns of fixed height and color count in variants of the model where seed glues are either chosen or fixed and identical (so-called non-uniform and uniform variants). We prove that both variants are NP-complete for patterns of height 2 or more and admit O(n)-time algorithms for patterns of height 1. We also prove that if the height and number of colors in the pattern is fixed, the non-uniform variant admits a O(n)-time algorithm while the uniform variant remains NP-complete. The NP-completeness results use a new reduction from a constrained version of a problem on finite state transducers.Comment: An abstract version appears in the proceedings of CIAA 201

Full Tilt: Universal Constructors for General Shapes with Uniform External Forces

We investigate the problem of assembling general shapes and patterns in a model in which particles move based on uniform external forces until they encounter an obstacle. In this model, corresponding particles may bond when adjacent with one another. Succinctly, this model considers a 2D grid of “open” and “blocked” spaces, along with a set of slidable polyominoes placed at open locations on the board. The board may be tilted in any of the 4 cardinal directions, causing all slidable polyominoes to move maximally in the specified direction until blocked. By successively applying a sequence of such tilts, along with allowing different polyominoes to stick when adjacent, tilt sequences provide a method to reconfigure an initial board configuration so as to assemble a collection of previous separate polyominoes into a larger shape. While previous work within this model of assembly has focused on designing a specific board configuration for the assembly of a specific given shape, we propose the problem of designing universal configurations that are capable of constructing a large class of shapes and patterns. For these constructions, we present the notions of weak and strong universality which indicate the presence of “excess” polyominoes after the shape is constructed. In particular, for given integers h, w, we show that there exists a weakly universal configuration with O(hw) 1 × 1 slidable particles that can be reconfigured to build any h × w patterned rectangle. We then expand this result to show that there exists a weakly universal configuration that can build any h × w-bounded size connected shape. Following these results, which require an admittedly relaxed assembly definition, we go on to show the existence of a strongly universal configuration (no excess particles) which can assemble any shape within a previously studied “drop” class, while using quadratically less space than previous results. Finally, we include a study of the complexity of deciding if a particle within a configuration may be relocated to another position, and deciding if a given configuration may be transformed into a second given configuration. We show both problems to be PSPACE-complete even when no particles stick to one another and movable particles are restricted to 1 × 1 tiles and a single 2 × 2 polyomino

Binary pattern tile set synthesis is NP-hard

In the field of algorithmic self-assembly, a long-standing unproven conjecture has been that of the NP-hardness of binary pattern tile set synthesis (2-PATS). The $k$-PATS problem is that of designing a tile assembly system with the smallest number of tile types which will self-assemble an input pattern of $k$ colors. Of both theoretical and practical significance, $k$-PATS has been studied in a series of papers which have shown $k$-PATS to be NP-hard for $k = 60$, $k = 29$, and then $k = 11$. In this paper, we close the fundamental conjecture that 2-PATS is NP-hard, concluding this line of study. While most of our proof relies on standard mathematical proof techniques, one crucial lemma makes use of a computer-assisted proof, which is a relatively novel but increasingly utilized paradigm for deriving proofs for complex mathematical problems. This tool is especially powerful for attacking combinatorial problems, as exemplified by the proof of the four color theorem by Appel and Haken (simplified later by Robertson, Sanders, Seymour, and Thomas) or the recent important advance on the Erd\H{o}s discrepancy problem by Konev and Lisitsa using computer programs. We utilize a massively parallel algorithm and thus turn an otherwise intractable portion of our proof into a program which requires approximately a year of computation time, bringing the use of computer-assisted proofs to a new scale. We fully detail the algorithm employed by our code, and make the code freely available online

Encoding Color Sequences in Active Tile Self-Assembly

Constructing patterns is a well-studied problem in both theoretical and experimental self-assembly with much of the work focused on multi-staged assembly. In this paper, we study building 1D patterns in a model of active self assembly: Tile Automata. This is a generalization of the 2-handed assembly model that borrows the concept of state changes from Cellular Automata. In this work we further develop the model by partitioning states as colors and show lower and upper bounds for building patterned assemblies based on an input pattern. Our first two sections utilize recent results to build binary strings along with Turing machine constructions to get Kolmogorov optimal state complexity for building patterns in Tile Automata, and show nearly optimal bounds for one case. For affinity strengthening Tile Automata, where transitions can only increase affinity so there is no detachment, we focus on scaled patterns based on Space Bounded Kolmogorov Complexity. Finally, we examine the affinity strengthening freezing case providing an upper bound based on the minimum context-free grammar. This system utilizes only one dimensional assemblies and has tiles that do not change color

An information-bearing seed for nucleating algorithmic self-assembly

Self-assembly creates natural mineral, chemical, and biological structures of great complexity. Often, the same starting materials have the potential to form an infinite variety of distinct structures; information in a seed molecule can determine which form is grown as well as where and when. These phenomena can be exploited to program the growth of complex supramolecular structures, as demonstrated by the algorithmic self-assembly of DNA tiles. However, the lack of effective seeds has limited the reliability and yield of algorithmic crystals. Here, we present a programmable DNA origami seed that can display up to 32 distinct binding sites and demonstrate the use of seeds to nucleate three types of algorithmic crystals. In the simplest case, the starting materials are a set of tiles that can form crystalline ribbons of any width; the seed directs assembly of a chosen width with >90% yield. Increased structural diversity is obtained by using tiles that copy a binary string from layer to layer; the seed specifies the initial string and triggers growth under near-optimal conditions where the bit copying error rate is 17 kb of sequence information. In sum, this work demonstrates how DNA origami seeds enable the easy, high-yield, low-error-rate growth of algorithmic crystals as a route toward programmable bottom-up fabrication

Self-Assembly of Tiles: Theoretical Models, the Power of Signals, and Local Computing

DNA-based self-assembly is an autonomous process whereby a disordered system of DNA sequences forms an organized structure or pattern as a consequence of Watson-Crick complementarity of DNA sequences, without external direction. Here, we propose self-assembly (SA) hypergraph automata as an automata-theoretic model for patterned self-assembly. We investigate the computational power of SA-hypergraph automata and show that for every recognizable picture language, there exists an SA-hypergraph automaton that accepts this language. Conversely, we prove that for any restricted SA-hypergraph automaton, there exists a Wang Tile System, a model for recognizable picture languages, that accepts the same language. Moreover, we investigate the computational power of some variants of the Signal-passing Tile Assembly Model (STAM), as well as propose the concept of {\it Smart Tiles}, i.e., tiles with glues that can be activated or deactivated by signals, and which possess a limited amount of local computing capability. We demonstrate the potential of smart tiles to perform some robotic tasks such as replicating complex shapes

Algorithmic Assembly of Nanoscale Structures

The development of nanotechnology has become one of the most significant endeavors of our time. A natural objective of this field is discovering how to engineer nanoscale structures. Limitations of current top-down techniques inspire investigation into bottom-up approaches to reach this objective. A fundamental precondition for a bottom-up approach is the ability to control the behavior of nanoscale particles. Many abstract representations have been developed to model systems of particles and to research methods for controlling their behavior. This thesis develops theories on two such approaches for building complex structures: the self-assembly of simple particles, and the use of simple robot swarms. The concepts for these two approaches are straightforward. Self-assembly is the process by which simple particles, following the rules of some behavior-governing system, naturally coalesce into a more complex form. The other method of bottom-up assembly involves controlling nanoscale particles through explicit directions and assembling them into a desired form. Regarding the self-assembly of nanoscale structures, we present two construction methods in a variant of a popular theoretical model known as the 2-Handed Tile Self-Assembly Model. The first technique achieves shape construction at only a constant scale factor, while the second result uses only a constant number of unique particle types. Regarding the use of robot swarms for construction, we first develop a novel technique for reconfiguring a swarm of globally-controlled robots into a desired shape even when the robots can only move maximally in a commanded direction. We then expand on this work by formally defining an entire hierarchy of shapes which can be built in this manner and we provide a technique for doing so

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