Matthew J. Patitz

SODA Conference 2025 Conference Paper

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

• Florent Becker
• Daniel Hader
• Matthew J. Patitz

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 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.

SODA Conference 2023 Conference Paper

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

• Andrew Alseth
• Matthew J. Patitz

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. Arbitrary amounts of information can be encoded into seed assemblies by both (1) the types and patterns of glues exposed on their exteriors, and (2) their shapes. Since a common metric by which aTAM systems are measured is their tile complexity (i. e. the number of unique types of tiles they utilize), in order to provide a fair basis for comparison, systems are often designed with seed assemblies consisting of only a single seed tile, a. k. a. single-tile seeds. (For instance, in STOC 2000 and 2001 information theoretically optimal tile complexity was shown possible for the self-assembly of squares.) This requires the transferring of any information that may be encoded in a multi-tile seed assembly into tile complexity. In this paper, we explore this process to show when and how such transformations are possible while ensuring that a derived system with a single-tile seed faithfully replicates the behaviors of the original system. We first show that a trivial transformation, in which the locations of a multi-tile seed are tiled by “hard- coded” tiles that can grow to represent that seed from a single tile, can succeed only if (1) there are not tile locations in the seed such that there exist growth sequences where those locations could block future growth, or (2) an ordering of growth can be enforced for the growth of the seed from a single tile to ensure that such blocking locations are tiled before collisions are possible. However, we show that knowing if this is the case is uncomputable. Therefore, we examine what is possible if the scale factor of the original system is increased and show that all systems with multi-tile seeds can be transformed into systems with single-tile seeds at scale factor 3 (i. e. each tile of the original system is replaced by a 3 × 3 square of tiles), such that the transformed systems faithfully replicate the dynamics of the original systems. We also prove that this scale factor is optimal, and that in fact there exist systems with multi-tile seeds for which no systems at scale factors 1 or 2 (or scale factor 3 when a more restrictive form of simulation is required) with single-tile seeds exist that can even produce the same sets of terminal output shapes. Since the scale 3 transformation results in a tile complexity which is proportional to the size of the original tile set plus the size of the multi-tile seed multiplied by the scale factor, we then also provide a transformation that yields an asymptotically optimal tile complexity proportional to the Kolmogorov complexity of the original system and which is based on the IU construction from FOCS 2012. Additionally, we are able to make simple modifications to that construction to provide a single aTAM system which simultaneously and in parallel simulates all aTAM systems, and provide a connection between that system and the existence of systems within models other than the aTAM which are IU for the aTAM. This set of results provides a full characterization of the tradeoff's between systems with multi-tile seeds and those with single-tile seeds, which is fundamental to the measure of complexity of aTAM systems.

TCS Journal 2021 Journal Article

On the effects of hierarchical self-assembly for reducing program-size complexity

• Sarah Cannon
• Erik D. Demaine
• Martin L. Demaine
• Sarah Eisenstat
• David Furcy
• Matthew J. Patitz
• Robert Schweller
• Scott M. Summers

In this paper we present a series of results which show separations between the standard seeded model of self-assembly, Winfree's abstract Tile Assembly Model (aTAM), and the “seedless” 2-Handed Assembly Model (2HAM), which incorporates the dynamics of hierarchical self-assembly. In particular, we focus on the problem of self-assembling various shapes while minimizing the sizes of tile sets, or “programs”, in each of these models in order to compare and contrast the models. A high-level overview of a subset of these results was presented in a paper by the authors in STACS 2013, but in this version we expand and improve the set of results related to showing separations between the two models according to their abilities to self-assemble various shapes. We exhibit classes of finite shapes that can be self-assembled more efficiently in each model. We also demonstrate infinite shapes that can self-assemble in one model but not in the other, as well as a shape which cannot self-assemble in either model.

SODA Conference 2020 Conference Paper

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

• Daniel Hader
• Aaron Koch
• Matthew J. Patitz
• Michael Sharp

In this paper we present a series of results related to mathematical models of self-assembling systems of tiles and the impacts that three diverse properties have on their dynamics. In these self-assembling systems, initially unorganized collections of tiles undergo random motion and can bind together, if they collide and enough of their incident glues match, to form assemblies. Here we greatly expand upon a series of prior results which showed that (1) the abstract Tile Assembly Model (aTAM) is intrinsically universal (FOCS 2012), and (2) the class of directed aTAM systems is not intrinsically universal (FOCS 2016). Intrinsic universality (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 and display the same behaviors modulo a scale factor. While the FOCS 2012 result showed that the standard, two-dimensional (2D) aTAM is IU, here we show that this is also the case for the three-dimensional (3D) version. Conversely, the FOCS 2016 result showed that the class of aTAM systems which are directed (a. k. a. deterministic, or confluent) is not IU, meaning that there is no universal simulator which can simulate directed aTAM systems while itself always remaining directed, 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. This implies that the constraint of tiles binding only in the plane forced the necessity of nondeterminism for the simulation of 2D directed systems. This then leads us to continue to explore the impacts of dimensionality and directedness on simulation of tile-based self-assembling systems by considering 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 they are also restricted to traveling within the plane, and we prove that the Planar aTAM is not IU, and prove 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 the Spatial aTAM is IU. This paper adds to a broad set of results which have been used to classify and compare the relative powers of differing models and classes of self-assembling systems, and also helps to further the understanding of the roles of dimension and nondeterminism on the dynamics of self-assembling systems. Furthermore, 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, and have made it, along with a simulator which can demonstrate it, freely available.

TCS Journal 2018 Journal Article

The power of duples (in self-assembly): It's not so hip to be square

• Jacob Hendricks
• Matthew J. Patitz
• Trent A. Rogers
• Scott M. Summers

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 give a lower bound on the tile complexity for building thin rectangles in the DaTAM that matches the analogous lower bound in the aTAM. 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.

TCS Journal 2017 Journal Article

TCS Special Issue on Computational Self-Assembly

• Matthew J. Patitz

SODA Conference 2016 Conference Paper

Computing in continuous space with self-assembling polygonal tiles (extended abstract)

• Oscar Gilbert
• Jacob Hendricks
• Matthew J. Patitz
• Trent A. Rogers

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 which are arbitrary 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 for each n > 6, the class of systems whose tiles are the shape of the regular polygon P with n 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 sides 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.

FOCS Conference 2016 Conference Paper

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

• Jacob Hendricks
• Matthew J. Patitz
• Trent A. Rogers

As a mathematical model of tile-based self-assembling systems, Winfree's abstract Tile Assembly Model (aTAM) has proven to be a remarkable platform for studying and understanding the behaviors and powers of self-assembling systems. Furthermore, as it is 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. This power has yielded a wide variety of theoretical results in the aTAM utilizing algorithmic self-assembly to design systems capable of performing complex computations and forming extremely intricate structures. Adding to the completeness of the model, in FOCS 2012 the aTAM was shown to also be intrinsically universal, which means that there exists one 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" of that result makes use of nondeterminism in terms of the tiles placed in several key locations when different assembly sequences are followed. This nondeterminism remains even when the simulator is simulating a system which is directed, meaning that it has exactly one unique terminal assembly and for any given location, no matter which assembly sequence is followed, the same tile type is always placed there. The question which then arose was whether or not that nondeterminism is fundamentally required, and if any universal simulator must in fact utilize more nondeterminism than directed systems when simulating them. In this paper, we answer that question 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 result provides a powerful insight into the role of nondeterminism in self-assembly, which is itself a fundamentally nondeterministic process occurring via unguided local interactions. Furthermore, 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 tailor design systems with precisely controllable space resources available to computations embedded within them. We also develop novel techniques for designing systems containing subsystems with disjoint, mutually exclusive computational powers. The main result will be important in the development of future simulation systems, and the supporting design techniques and lemmas will provide powerful tools for the development of future aTAM systems as well as proofs of their computational abilities.

SODA Conference 2015 Conference Paper

Universal Computation with Arbitrary Polyomino Tiles in Non-Cooperative Self-Assembly

• Sándor P. Fekete
• Jacob Hendricks
• Matthew J. Patitz
• Trent A. Rogers
• Robert Schweller

SODA Conference 2014 Conference Paper

Intrinsic universality in tile self-assembly requires cooperation

• Pierre-Étienne Meunier
• Matthew J. Patitz
• Scott M. Summers
• Guillaume Theyssier
• Andrew Winslow
• Damien Woods

We prove a negative result on the power of a model of algorithmic self-assembly for which finding general techniques and results has been notoriously difficult. Specifically, we prove that Winfree's abstract Tile Assembly Model is not intrinsically universal when restricted to use noncooperative tile binding. This stands in stark contrast to the recent result that the abstract Tile Assembly Model is indeed intrinsically universal when cooperative binding is used (FOCS 2012). Noncooperative self-assembly, also known as “temperature 1”, is where all tiles bind to each other if they match on at least one side. On the other hand, cooperative self-assembly requires that some tiles bind on at least two sides. Our result shows that the change from non-cooperative to cooperative binding qualitatively improves the range of dynamics and behaviors found in these models of nanoscale self-assembly. The result holds in both two and three dimensions; the latter being quite surprising given that three-dimensional noncooperative tile assembly systems simulate Turing machines. This shows that Turing universal behavior in self-assembly does not imply the ability to simulate all algorithmic self-assembly processes. In addition to the negative result, we exhibit a three-dimensional noncooperative self-assembly tile set capable of simulating any two-dimensional noncooperative self-assembly system. This tile set implies that, in a restricted sense, non-cooperative self-assembly is intrinsically universal for itself.

FOCS Conference 2012 Conference Paper

The Tile Assembly Model is Intrinsically Universal

• David Doty
• Jack H. Lutz
• Matthew J. Patitz
• Robert Schweller
• Scott M. Summers
• Damien Woods

We prove that the abstract Tile Assembly Model (aTAM) of nanoscale self-assembly is intrinsically universal. This means that there is a single tile assembly system U that, with proper initialization, simulates any tile assembly system T. The simulation is "intrinsic" in the sense that the self-assembly process carried out by U is exactly that carried out by T, with each tile of T represented by an m × m "super tile" of U. Our construction works for the full aTAM at any temperature, and it faithfully simulates the deterministic or nondeterministic behavior of each T. Our construction succeeds by solving an analog of the cell differentiation problem in developmental biology: Each super tile of U, starting with those in the seed assembly, carries the "genome" of the simulated system T. At each location of a potential super tile in the self-assembly of U, a decision is made whether and how to express this genome, i. e. , whether to generate a super tile and, if so, which tile of T it will represent. This decision must be achieved using asynchronous communication under incomplete information, but it achieves the correct global outcome(s).

TCS Journal 2011 Journal Article

Limitations of self-assembly at temperature 1

• David Doty
• Matthew J. Patitz
• Scott M. Summers

We prove that if a set X ⊆ Z 2 weakly self-assembles at temperature 1 in a deterministic (Winfree) tile assembly system satisfying a natural condition known as pumpability, then X is a semilinear set. This shows that only the most simple of infinite shapes and patterns can be constructed using pumpable temperature 1 tile assembly systems, and gives evidence for the thesis that temperature 2 or higher is required to carry out general-purpose computation in a deterministic two-dimensional tile assembly system. We employ this result to show that, unlike the case of temperature 2 self-assembly, no discrete self-similar fractal weakly self-assembles at temperature 1 in a pumpable tile assembly system.

TCS Journal 2011 Journal Article

Self-assembly of infinite structures: A survey

• Matthew J. Patitz
• Scott M. Summers

We survey some recent results related to the self-assembly of infinite structures in Winfree’s abstract Tile Assembly Model. These results include impossibility results, as well as the construction of novel tile assembly systems that produce computationally interesting shapes and patterns. Several open questions are also presented and motivated.

FOCS Conference 2010 Conference Paper

Strong Fault-Tolerance for Self-Assembly with Fuzzy Temperature

• David Doty
• Matthew J. Patitz
• Dustin Reishus
• Robert Schweller
• Scott M. Summers

We consider the problem of fault-tolerance in nanoscale algorithmic self-assembly. We employ a standard variant of Winfree's abstract Tile Assembly Model (aTAM), the two-handed aTAM, in which square “tiles” - a model of molecules constructed from DNA for the purpose of engineering self-assembled nanostructures - aggregate according to specific binding sites of varying strengths, and in which large aggregations of tiles may attach to each other, in contrast to the seeded aTAM, in which tiles aggregate one at a time to a single specially designated “seed” assembly. We focus on a major cause of errors in tile-based self-assembly: that of unintended growth due to “weak” strength-1 bonds, which if allowed to persist, may be stabilized by subsequent attachment of neighboring tiles in the sense that at least energy 2 is now required to break apart the resulting assembly, i. e. , the errant assembly is stable at temperature 2. We study a common self-assembly benchmark problem, that of assembling an n×n square using O(log n) unique tile types, under the two-handed model of self-assembly. Our main result achieves a much stronger notion of fault-tolerance than those achieved previously. Arbitrary strength-1 growth is allowed, however, any assembly that grows sufficiently to become stable at temperature 2 is guaranteed to assemble into the correct final assembly of an n×n square. In other words, errors due to insufficient attachment, which is the cause of errors studied in earlier papers on fault-tolerance, are prevented absolutely in our main construction, rather than only with high probability and for sufficiently small structures, as in previous fault tolerance studies.