arXiv subjects

Matthew J. Patitz

Publications and source records attributed to Matthew J. Patitz.

At least 19 recordsLinked to original sources

arXiv2026

Powers and Limitations of Synchronous Self-Assembly

In abstract models of algorithmic self-assembly, synchronization between attachments has emerged as a crucial distinction between the classical asynchronous model (aTAM) and a new synchronous model, the syncTAM. This paper presents recent advances in gauging the additional power afforded by the syncTAM. While it is known that the syncTAM and the aTAM are each unable to fully simulate the other, this paper offers evidence that the syncTAM is computationally significantly more powerful than the aTAM, especially in the non-cooperative setting. The additional power of the non-cooperative syncTAM is witnessed by the following constructions, all impossible in the non-cooperative aTAM: a flagpole, a strict self-assembly of a variant of the discrete Sierpinski triangle, and the ability to build the same assemblies (modulo scale factor) as directed aTAM systems. The second topic is that of limited synchronization, wherein, when the number of attachments is smaller than some threshold $l$, they happen synchronously, but attachments in excess of that number must wait. In that context, the precise value of $l$ is crucial, and changes to that value prevent simulation and can change which shapes can be obtained.

cs.CG↗

arXiv2024

Strict Self-Assembly of Discrete Self-Similar Fractals in the abstract Tile-Assembly Model

This paper answers a long-standing open question in tile-assembly theory, namely that it is possible to strictly assemble discrete self-similar fractals (DSSFs) in the abstract Tile-Assembly Model (aTAM). We prove this in 2 separate ways, each taking advantage of a novel set of tools. One of our constructions shows that specializing the notion of a quine, a program which prints its own output, to the language of tile-assembly naturally induces a fractal structure. The other construction introduces self-describing circuits as a means to abstractly represent the information flow through a tile-assembly construction and shows that such circuits may be constructed for a relative of the Sierpinski carpet, and indeed many other DSSFs, through a process of fixed-point iteration. This later result, or more specifically the machinery used in its construction, further enable us to provide a polynomial time procedure for deciding whether any given subset of $\mathbb{Z}^2$ will generate an aTAM producible DSSF. To this end, we also introduce the Tree Pump Theorem, a result analogous to the important Window Movie Lemma, but with requirements on the set of productions rather than on the self-assembling system itself.

cs.CG↗

arXiv2024

Simulation of the abstract Tile Assembly Model Using Crisscross Slats

Tile assembly systems in the abstract Tile Assembly Model (aTAM) are computationally universal and capable of building complex shapes, but DNA-based implementations encounter formidable error rates that stifle this theoretical potential. Slat-based self-assembly is a recent development wherein DNA forms long slats that combine together in 2 layers, rather than the aTAM's square tiles in a plane. While tiles tend to bind to 2 neighboring tiles at a time, slats may bind to dozens of other slats. Large slat-based DNA constructions have been implemented in the lab with incredible resilience to many of the errors that plague tile-based constructions, but these come at a cost as slat-based systems are often more difficult to design and simulate. Also, it has not been clear if slats, with their larger sizes and different geometries, have the same theoretical capabilities as tiles. Here we show that slats do, at least at scale. We give constructions showing that any aTAM system may be simulated by a system of slats and that these can be made more efficiently, using shorter slats and a smaller scale factor, when simulating simpler classes of systems. We consider 5 classes of aTAM systems with increasing complexity, from zig-zag systems to the full class of all aTAM systems, and show how they can be converted to equivalent slat systems. Zig-zag systems can be simulated by slats at only a $2c \times 2c$ scale (where $c$ is the freely chosen cooperativity of the slats), the full class of aTAM systems at only a $5c \times 5c$ scale, and intermediate classes using scales between these. Together, these results prove that slats have the full theoretical power of aTAM tiles while providing constructions compact enough to potentially provide designs for DNA-based implementations of slat systems that are both capable of powerful algorithmic self-assembly and possessing the strong error resilience of slats.

cs.CG↗

arXiv2024

Self-Assembly of Patterns in the abstract Tile Assembly Model

In the abstract Tile Assembly Model, self-assembling systems consisting of tiles of different colors can form structures on which colored patterns are ``painted.'' We explore the complexity, in terms of the numbers of unique tile types required, of assembling various patterns. We first demonstrate how to efficiently self-assemble a set of simple patterns, then show tight bounds on the tile type complexity of self-assembling 2-colored patterns on the surfaces of square assemblies. Finally, we demonstrate an exponential gap in tile type complexity of self-assembling an infinite series of patterns between systems restricted to one plane versus those allowed two planes.

cs.ET↗

arXiv2023

The Impacts of Dimensionality, Diffusion, and Directedness on Intrinsic Cross-Model Simulation in Tile-Based Self-Assembly

Algorithmic self-assembly occurs when disorganized components autonomously combine to form structures and, by their design and the dynamics of the system, are forced to follow the execution of algorithms. Motivated by applications in DNA-nanotechnology, investigations in algorithmic tile-based self-assembly have blossomed into a mature theory with research leveraging tools from computability theory, complexity theory, information theory, and graph theory to develop a wide range of models and show that many are computationally universal, while also exposing powers and limitations of each. Beyond computational universality, the abstract Tile Assembly Model (aTAM) was shown to be intrinsically universal (IU), a strong notion of completeness where a single tile set is capable of simulating all systems within the model; however, this result required non-deterministic tile attachments. This was later confirmed necessary when it was shown that the class of directed aTAM systems is not IU. Building on these results to further investigate the impacts of other dynamics, Hader et al. examined several tile-assembly models which varied across (1) the numbers of dimensions used, (2) restrictions based on diffusion of tiles through space, and (3) whether each system is directed, and showed which models are IU. Such results have shed much light on the roles of various aspects of the dynamics of tile-assembly and their effects on the intrinsic universality of each model. Here we provide direct comparisons of the various models by considering intrinsic simulations between models. We show that in some cases one model is more powerful than another, and in others, pairs of models have mutually exclusive capabilities. This comparison helps to expose the impacts of these three important aspects and further helps define a hierarchy of tile-assembly models.

cs.CG↗

arXiv2022

The Need for Seed (in the abstract Tile Assembly Model)

In the abstract Tile Assembly Model (aTAM) square tiles self-assemble, autonomously binding via glues on their edges, to form structures. Algorithmic aTAM systems can be designed in which the patterns of tile attachments are forced to follow the execution of targeted algorithms. Such systems have been proven to be computationally universal as well as intrinsically universal (IU), a notion borrowed and adapted from cellular automata showing that a single tile set exists which is capable of simulating all aTAM systems (FOCS 2012). The input to an algorithmic aTAM system can be provided in a variety of ways, with a common method being via the "seed" assembly, which is a pre-formed assembly from which all growth propagates. In this paper we present a series of results which investigate the the trade-offs of using seeds consisting of a single tile, versus those containing multiple tiles. We show that arbitrary systems with multi-tile seeds cannot be converted to functionally equivalent systems with single-tile seeds without using a scale factor > 1. We prove tight bounds on the scale factor required, and also present a construction which uses a large scale factor but an optimal number of unique tile types. That construction is then used to develop a construction that performs simultaneous simulation of all aTAM systems in parallel, as well as to display a connection to other tile-based self-assembly models via the notion of intrinsic universality.

cs.ET↗

arXiv2022

Universal Shape Replication Via Self-Assembly With Signal-Passing Tiles

In this paper, we investigate shape-assembling power of a tile-based model of self-assembly called the Signal-Passing Tile Assembly Model (STAM). In this model, the glues that bind tiles together can be turned on and off by the binding actions of other glues via "signals". Specifically, the problem we investigate is "shape replication" wherein, given a set of input assemblies of arbitrary shape, a system must construct an arbitrary number of assemblies with the same shapes and, with the exception of size-bounded junk assemblies that result from the process, no others. We provide the first fully universal shape replication result, namely a single tile set capable of performing shape replication on arbitrary sets of any 3-dimensional shapes without requiring any scaling or pre-encoded information in the input assemblies. Our result requires the input assemblies to be composed of signal-passing tiles whose glues can be deactivated to allow deconstruction of those assemblies, which we also prove is necessary by showing that there are shapes whose geometry cannot be replicated without deconstruction. Additionally, we modularize our construction to create systems capable of creating binary encodings of arbitrary shapes, and building arbitrary shapes from their encodings. Because the STAM is capable of universal computation, this then allows for arbitrary programs to be run within an STAM system, using the shape encodings as input, so that any computable transformation can be performed on the shapes.

cs.FL↗

arXiv2022

Replication of arbitrary hole-free shapes via self-assembly with signal-passing tiles

In this paper, we investigate the abilities of systems of self-assembling tiles which can each pass a constant number of signals to their immediate neighbors to create replicas of input shapes. Namely, we work within the Signal-passing Tile Assembly Model (STAM), and we provide a universal STAM tile set which is capable of creating unbounded numbers of assemblies of shapes identical to those of input assemblies. The shapes of the input assemblies can be arbitrary 2-dimensional hole-free shapes. This improves previous shape replication results in self-assembly that required models in which multiple assembly stages and/or bins were required, and the shapes which could be replicated were more constrained, as well as a previous version of this result that required input shapes to be represented at scale factor 2.

cs.ET↗

arXiv2022

Self-Replication via Tile Self-Assembly

In this paper we present a model containing modifications to the Signal-passing Tile Assembly Model (STAM), a tile-based self-assembly model whose tiles are capable of activating and deactivating glues based on the binding of other glues. These modifications consist of an extension to 3D, the ability of tiles to form "flexible" bonds that allow bound tiles to rotate relative to each other, and allowing tiles of multiple shapes within the same system. We call this new model the STAM*, and we present a series of constructions within it that are capable of self-replicating behavior. Namely, the input seed assemblies to our STAM* systems can encode either "genomes" specifying the instructions for building a target shape, or can be copies of the target shape with instructions built in. A universal tile set exists for any target shape (at scale factor 2), and from a genome assembly creates infinite copies of the genome as well as the target shape. An input target structure, on the other hand, can be "deconstructed" by the universal tile set to form a genome encoding it, which will then replicate and also initiate the growth of copies of assemblies of the target shape. Since the lengths of the genomes for these constructions are proportional to the number of points in the target shape, we also present a replicator which utilizes hierarchical self-assembly to greatly reduce the size of the genomes required. The main goals of this work are to examine minimal requirements of self-assembling systems capable of self-replicating behavior, with the aim of better understanding self-replication in nature as well as understanding the complexity of mimicking it.

cs.ET↗

arXiv2019

The Impacts of Dimensionality, Diffusion, and Directedness on Intrinsic Universality in the abstract Tile Assembly Model

We present a series of results related to mathematical models of self-assembling tiles and the impacts that three diverse properties have on their dynamics. We expand upon a series of prior results which showed that (1) the abstract Tile Assembly Model (aTAM) is intrinsically universal (IU) [FOCS 2012], and (2) the class of directed aTAM systems is not IU [FOCS 2016]. IU for a model (or class of systems within a model) means that there is a universal tile set which can be used to simulate an arbitrary system within that model (or class). Furthermore, the simulation must not only produce the same resultant structures, it must also maintain the full dynamics of the systems being simulated modulo only a scale factor. While the FOCS 2012 result showed the standard, two-dimensional (2D) aTAM is IU, here we show this is also the case for the 3D version. Conversely, the FOCS 2016 result showed the class of aTAM systems which are directed (a.k.a. deterministic, or confluent) is not IU, implying that nondeterminism is fundamentally required for such simulations. Here, however, we show that in 3D the class of directed aTAM systems is actually IU, i.e. there is a universal directed simulator for them. We then consider the influence of more rigid notions of dimensionality. Namely, we introduce the Planar aTAM, where tiles are not only restricted to binding in the plane, but also to traveling in the plane, and prove that the Planar aTAM is not IU, and that the class of directed systems within the Planar aTAM also is not IU. Finally, analogous to the Planar aTAM, we introduce the Spatial aTAM, its 3D counterpart, and prove that it is IU. To prove our positive results we have not only designed, but also implemented what we believe to be the first IU tile set ever implemented and simulated in any tile assembly model. We've made it and a simulator which can demonstrate it freely available.

cs.CG↗

arXiv2019

Simulation of Programmable Matter Systems Using Active Tile-Based Self-Assembly

Self-assembly refers to the process by which small, simple components mix and combine to form complex structures using only local interactions. Designed as a hybrid between tile assembly models and cellular automata, the Tile Automata (TA) model was recently introduced as a platform to help study connections between various models of self-assembly. However, in this paper we present a result in which we use TA to simulate arbitrary systems within the amoebot model, a theoretical model of programmable matter in which the individual components are relatively simple state machines that are able to sense the states of their neighbors and to move via series of expansions and contractions. We show that for every amoebot system, there is a TA system capable of simulating the local information transmission built into amoebot particles, and that the TA "macrotiles" used to simulate its particles are capable of simulating movement (via attachment and detachment operations) while maintaining the necessary properties of amoebot particle systems. The TA systems are able to utilize only the local interactions of state changes and binding and unbinding along tile edges, but are able to fully simulate the dynamics of these programmable matter systems.

cs.ET↗

arXiv2019

Geometric Tiles and Powers and Limitations of Geometric Hindrance in Self-Assembly

Tile-based self-assembly systems are capable of universal computation and algorithmically-directed growth. Systems capable of such behavior typically make use of "glue cooperation" in which the glues on at least $2$ sides of a tile must match and bind to those exposed on the perimeter of an assembly for that tile to attach. However, several models have been developed which utilize "weak cooperation", where only a single glue needs to bind but other preventative forces (such as geometric, or steric, hindrance) provide additional selection for which tiles may attach, and where this allows for algorithmic behavior. In this paper we first work in a model where tiles are allowed to have geometric bumps and dents on their edges. We show how such tiles can simulate systems of square tiles with complex glue functions (using asymptotically optimal sizes of bumps and dents), and also how they can simulate weakly cooperative systems in a model which allows for duples (i.e. tiles either twice as long or twice as tall as square tiles). We then show that with only weak cooperation via geometric hindrance, no system in any model can simulate even a class of tightly constrained, deterministic cooperative systems, further defining the boundary of what is possible using this tool.

cs.ET↗

arXiv2018

Self-Assembly of 3-D Structures Using 2-D Folding Tiles

Self-assembly is a process which is ubiquitous in natural, especially biological systems. It occurs when groups of relatively simple components spontaneously combine to form more complex structures. While such systems have inspired a large amount of research into designing theoretical models of self-assembling systems, and even laboratory-based implementations of them, these artificial models and systems often tend to be lacking in one of the powerful features of natural systems (e.g. the assembly and folding of proteins), namely the dynamic reconfigurability of structures. In this paper, we present a new mathematical model of self-assembly, based on the abstract Tile Assembly Model (aTAM), called the Flexible Tile Assembly Model (FTAM). In the FTAM, the individual components are 2-dimensional square tiles as in the aTAM, but in the FTAM, bonds between the edges of tiles can be flexible, allowing bonds to flex and entire structures to reconfigure, thus allowing 2-dimensional components to form 3-dimensional structures. We analyze the powers and limitations of FTAM systems by (1) demonstrating how flexibility can be controlled to carefully build desired structures, and (2) showing how flexibility can be beneficially harnessed to form structures which can "efficiently" reconfigure into many different configurations and/or greatly varying configurations. We also show that with such power comes a heavy burden in terms of computational complexity of simulation and prediction by proving that, for important properties of FTAM systems, determining their existence is intractable, even for properties which are easily computed for systems in less dynamic models.

cs.CG↗

arXiv2018

Know When to Fold 'Em: Self-Assembly of Shapes by Folding in Oritatami

An oritatami system (OS) is a theoretical model of self-assembly via co-transcriptional folding. It consists of a growing chain of beads which can form bonds with each other as they are transcribed. During the transcription process, the $δ$ most recently produced beads dynamically fold so as to maximize the number of bonds formed, self-assemblying into a shape incrementally. The parameter $δ$ is called the delay and is related to the transcription rate in nature. This article initiates the study of shape self-assembly using oritatami. A shape is a connected set of points in the triangular lattice. We first show that oritatami systems differ fundamentally from tile-assembly systems by exhibiting a family of infinite shapes that can be tile-assembled but cannot be folded by any OS. As it is NP-hard in general to determine whether there is an OS that folds into (self-assembles) a given finite shape, we explore the folding of upscaled versions of finite shapes. We show that any shape can be folded from a constant size seed, at any scale n >= 3, by an OS with delay 1. We also show that any shape can be folded at the smaller scale 2 by an OS with unbounded delay. This leads us to investigate the influence of delay and to prove that, for all δ > 2, there are shapes that can be folded (at scale 1) with delay δ but not with delay δ'<δ. These results serve as a foundation for the study of shape-building in this new model of self-assembly, and have the potential to provide better understanding of cotranscriptional folding in biology, as well as improved abilities of experimentalists to design artificial systems that self-assemble via this complex dynamical process.

cs.DS↗

arXiv2018

Thermodynamically Favorable Computation via Tile Self-assembly

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. Utilizing the constraints imposed by geometry, we then provide a GTBN construction capable of simulating Turing machines as efficiently as in the aTAM.

cs.ET↗

arXiv2016

Hierarchical Self-Assembly of Fractals with Signal-Passing Tiles

In this paper, we present high-level overviews of tile-based self-assembling systems capable of producing complex, infinite, aperiodic structures known as discrete self-similar fractals. Fractals have a variety of interesting mathematical and structural properties, and by utilizing the bottom-up growth paradigm of self-assembly to create them we not only learn important techniques for building such complex structures, we also gain insight into how similar structural complexity arises in natural self-assembling systems. Our results fundamentally leverage hierarchical assembly processes, and use as our building blocks square "tile" components which are capable of activating and deactivating their binding "glues" a constant number of times each, based only on local interactions. We provide the first constructions capable of building arbitrary discrete self-similar fractals at scale factor 1, and many at temperature 1 (i.e. "non-cooperatively"), including the Sierpinski triangle.

cs.ET↗

arXiv2016

Universal Simulation of Directed Systems in the abstract Tile Assembly Model Requires Undirectedness

As a mathematical model of self-assembling systems, Winfree's abstract Tile Assembly Model (aTAM) is a remarkable platform for studying the behaviors and powers of self-assembling systems. Capable of Turing universal computation, the aTAM allows algorithmic self-assembly, in which the components can be designed so that the rules governing their behaviors force them to inherently execute prescribed algorithms as they combine. Adding to its completeness, the aTAM was shown to also be intrinsically universal, which means that there exists a single tile set such that for any arbitrary input aTAM system, that tile set can be configured into a seed structure which will then cause self-assembly using that tile set to simulate the input system, capturing its full dynamics modulo only a scale factor. However, the universal simulator previously given makes use of nondeterminism in terms of tile types placed in several key locations when different assembly sequences are followed, even when simulating a directed system, meaning one that has exactly one unique terminal assembly. The question then became whether or not that nondeterminism is fundamentally required. Here, we answer that in the affirmative: the class of directed systems in the aTAM is not intrinsically universal, meaning there is no universal simulator for directed systems which itself is always directed. This provides insight into the role of nondeterminism in self-assembly, which is itself a fundamentally nondeterministic process. To achieve this result we leverage powerful results of computational complexity hierarchies, including tight bounds on both best and worst-case complexities of decidable languages, to design systems with precisely controllable space resources available to embedded computations. We also develop novel techniques for designing systems containing subsystems with disjoint, mutually exclusive computational powers.

cs.CG↗

arXiv2015

Computing in continuous space with self-assembling polygonal tiles

In this paper we investigate the computational power of the polygonal tile assembly model (polygonal TAM) at temperature 1, i.e. in non-cooperative systems. The polygonal TAM is an extension of Winfree's abstract tile assembly model (aTAM) which not only allows for square tiles (as in the aTAM) but also allows for tile shapes that are polygons. Although a number of self-assembly results have shown computational universality at temperature 1, these are the first results to do so by fundamentally relying on tile placements in continuous, rather than discrete, space. With the square tiles of the aTAM, it is conjectured that the class of temperature 1 systems is not computationally universal. Here we show that the class of systems whose tiles are composed of a regular polygon P with n > 6 sides is computationally universal. On the other hand, we show that the class of systems whose tiles consist of a regular polygon P with n <= 6 cannot compute using any known techniques. In addition, we show a number of classes of systems whose tiles consist of a non-regular polygon with n >= 3 sides are computationally universal.

cs.CG↗