The two-handed tile assembly model is not intrinsically universal

The two-handed tile assembly model is not intrinsically universal

Abstract

The well-studied Two-Handed Tile Assembly Model (2HAM) is a model of tile assembly in which pairs of large assemblies can bind, or self-assemble, together. In order to bind, two assemblies must have matching glues that can simultaneously touch each other, and stick together with strength that is at least the temperature , where is some fixed positive integer. We ask whether the 2HAM is intrinsically universal, in other words we ask: is there a single universal 2HAM tile set which can be used to simulate any instance of the model? Our main result is a negative answer to this question. We show that for all , each temperature- 2HAM tile system does not simulate at least one temperature- 2HAM tile system. This impossibility result proves that the 2HAM is not intrinsically universal, in stark contrast to the simpler (single-tile addition only) abstract Tile Assembly Model which is intrinsically universal (The tile assembly model is intrinsically universal, FOCS 2012). However, on the positive side, we prove that, for every fixed temperature , temperature- 2HAM tile systems are indeed intrinsically universal: in other words, for each  there is a single universal 2HAM tile set that, when appropriately initialized, is capable of simulating the behavior of any temperature- 2HAM tile system. As a corollary of these results we find an infinite set of infinite hierarchies of 2HAM systems with strictly increasing simulation power within each hierarchy. Finally, we show that for each , there is a temperature- 2HAM system that simultaneously simulates all temperature- 2HAM systems.

1 Introduction

Self-assembly is the process through which unorganized, simple, components automatically coalesce according to simple local rules to form some kind of target structure. It sounds simple, but the end result can be extraordinary. For example, researchers have been able to self-assemble a wide variety of structures experimentally at the nanoscale, such as regular arrays [33], fractal structures [28, 15], smiling faces [27], DNA tweezers [34], logic circuits [25], neural networks [26], and molecular robots[21]. These examples are fundamental because they demonstrate that self-assembly can, in principle, be used to manufacture specialized geometrical, mechanical and computational objects at the nanoscale. Potential future applications of nanoscale self-assembly include the production of smaller, more efficient microprocessors and medical technologies that are capable of diagnosing and even treating disease at the cellular level.

Controlling nanoscale self-assembly for the purposes of manufacturing atomically precise components will require a bottom-up, hands-off strategy. In other words, the self-assembling units themselves will have to be “programmed” to direct themselves to do the right thing—efficiently and correctly. Thus, it is necessary to study the extent to which the process of self-assembly can be controlled in an algorithmic sense.

In 1998, Erik Winfree [32] introduced the abstract Tile Assembly Model (aTAM), an over-simplified discrete mathematical model of nanoscale DNA self-assembly pioneered by Seeman [30]. The aTAM essentially augments classical Wang tiling [31] with a mechanism for sequential “growth” of a tiling (in Wang tiling, only the existence of a valid, mismatch-free tiling is considered and not the order of tile placement). In the aTAM, the fundamental components are un-rotatable, but translatable square “tile types” whose sides are labeled with (alpha-numeric) glue “colors” and (integer) “strengths”. Two tiles that are placed next to each other interact if the glue colors on their abutting sides match, and they bind if the strengths on their abutting sides match and sum to at least a certain (integer) “temperature”. Self-assembly starts from a “seed” tile type and proceeds nondeterministically and asynchronously as tiles bind to the seed-containing-assembly. Despite its deliberate over-simplification, the aTAM is a computationally expressive model. For example, Winfree [32] proved that it is Turing universal, which implies that self-assembly can be directed by a computer program.

In this paper, we work in a generalization of the aTAM, called the two-handed [3] (a.k.a., hierarchical [5], q-tile [6], polyomino [20]) abstract Tile Assembly Model (2HAM). A central feature of the 2HAM is that, unlike the aTAM, it allows two “supertile” assemblies, each consisting of one or more tiles, to fuse together. For two such assemblies to bind, they should not “sterically hinder” each other, and they should have a sufficient number of matching glues distributed along the interface where they meet. Hence the model includes notions of local interactions (individual glues) and non-local interactions (large assemblies coming together). In the 2HAM, an assembly of tiles is producible if it is either a single tile, or if it results from the stable combination of two other producible assemblies.

We study the intrinsic universality in the 2HAM. Intrinsic universality uses a special notion of simulation, where the simulator preserves the dynamics of the simulated system. For tile assembly systems this means that, modulo spatial rescaling, a simulator self-assembles the same assemblies as any simulated system, and even does this in the same way (via the same assembly sequences). In the field of cellular automata, the topic of intrinsic universality has given rise to a rich theory [14, 8, 9, 2, 4, 23, 24] and indeed has also been studied in Wang tiling [16, 17, 18] and tile self-assembly [13, 12, 22, 11]. The aTAM has been shown to be intrinsically universal [12], meaning that there is a single set of tiles  that works at temperature 2, and when appropriately initialized, is capable of simulating the behavior of an arbitrary aTAM tile assembly system. Modulo rescaling, this single tile set represents the full power and expressivity of the entire aTAM model, at any temperature. On the other hand, it has been shown that at temperature-1, there is no tile set that can simulate the aTAM [22]. Interestingly, the latter negative result holds for 3D temperature-1 systems, which are known to be Turing universal [7]. Here, we ask whether there is such a universal tile set for the 2HAM.

The theoretical power of non-local interaction in the 2HAM has been the subject of recent research. For example, Doty and Chen [5] proved that, surprisingly, squares do not self-assemble any faster in so-called partial order 2HAM systems than they do in the aTAM, despite being able to exploit massive parallelism. More recently, Cannon, et al. [3], while comparing the abilities of the 2HAM and the aTAM, proved three main results, which seem to suggest that the 2HAM is at least as powerful as the aTAM: (1) non-local binding in the 2HAM can dramatically reduce the tile complexity (i.e., minimum number of unique tile types required to self-assemble a shape) for certain classes of shapes; (2) the 2HAM can simulate the aTAM in the following sense: for any aTAM tile system , there is a corresponding 2HAM tile system , which simulates the exact behavior—modulo connectivity—of , at scale factor 5; (3) the problem of verifying whether a 2HAM system uniquely produces a given assembly is coNP-complete (for the aTAM this problem is decidable in polynomial time [1]).

Main results.  In this paper, we ask if the 2HAM is intrinsically universal: does there exist a “universal” 2HAM tile set that, when appropriately initialized, is capable of simulating the behavior of an arbitrary 2HAM tile system? A positive answer would imply that such a tile set has the ability to model the capabilities of all 2HAM systems.1 Our first main result, Theorem 3.1, says that the 2HAM is not intrinsically universal, which means that the 2HAM is incapable of simulating itself. This statement stands in stark contrast to the case of the aTAM, which was recently shown to be intrinsically universal by Doty, Lutz, Patitz, Schweller, Summers and Woods [12]. Specifically, we show that for any sufficiently large temperature , there is a temperature 2HAM system that cannot be simulated by any temperature 2HAM system. It is worthy of note that, in order to prove this result, we use a simple, yet novel combinatorial argument, which as far as we are aware of, is the first lower bound proof in the 2HAM that does not use an information-theoretic argument. In our proof of Theorem 3.1 we show that the 2HAM cannot simulate massively cooperative binding, where the number of cooperative bindings is larger than the temperature of the simulator.

Our second main result, Theorem 4.1, is positive: we show, via constructions, that the 2HAM is intrinsically universal for fixed temperature, that is, the temperature  2HAM can simulate the temperature  2HAM. So although our impossibility result tells us that the 2HAM can not simulate “too much” cooperative binding, our positive result tells us it can indeed simulate some cooperative binding: an amount exactly equal to the temperature of the simulator.

As a corollary of these results, we get a separation between classes of 2HAM tile systems based on their temperatures. That is, we exhibit an infinite hierarchy of 2HAM systems, of strictly-increasing temperature, that cannot be simulated by lesser temperature systems but can downward simulate lower temperature systems. Moreover, we exhibit an infinite number of such hierarchies in Theorem 4.2. Thus, as was suggested as future work in [12], and as has been shown in the theory of cellular automata [9], we use the notion of intrinsic universality to classify, and separate, these tile assembly systems via their simulation ability.

As noted above, we show that temperature 2HAM systems are intrinsically universal. We actually show this for two different, seemingly natural, notions of simulation (called simulation and strong simulation), showing trade-offs between, and even within, these notions of simulation. For both notions of simulation, we show tradeoffs between scale factor, number of tile types, and complexity of the initial configuration. Finally, we show how to construct, for each , a temperature- 2HAM system that simultaneously simulates all temperature- 2HAM systems. We finish with a conjecture:

Conjecture 1.

There exists , such that for each , temperature 2HAM systems do not strongly simulate temperature 2HAM systems.

2 Definitions

2.1 Informal definition of the 2HAM

The 2HAM [6, 10] is a generalization of the aTAM in that it allows for two assemblies, both possibly consisting of more than one tile, to attach to each other. Since we must allow that the assemblies might require translation before they can bind, we define a supertile to be the set of all translations of a -stable assembly, and speak of the attachment of supertiles to each other, modeling that the assemblies attach, if possible, after appropriate translation. We now give a brief, informal, sketch of the 2HAM.

A tile type is a unit square with four sides, each having a glue consisting of a label (a finite string) and strength (a non-negative integer). We assume a finite set of tile types, but an infinite number of copies of each tile type, each copy referred to as a tile. A supertile is (the set of all translations of) a positioning of tiles on the integer lattice . Two adjacent tiles in a supertile interact if the glues on their abutting sides are equal and have positive strength. Each supertile induces a binding graph, a grid graph whose vertices are tiles, with an edge between two tiles if they interact. The supertile is -stable if every cut of its binding graph has strength at least , where the weight of an edge is the strength of the glue it represents. That is, the supertile is stable if at least energy is required to separate the supertile into two parts. A 2HAM tile assembly system (TAS) is a pair , where is a finite tile set and is the temperature, usually 1 or 2. Given a TAS , a supertile is producible, written as if either it is a single tile from , or it is the -stable result of translating two producible assemblies without overlap. A supertile is terminal, written as if for every producible supertile , and cannot be -stably attached. A TAS is directed if it has only one terminal, producible supertile.2

2.2 Formal definition of the 2HAM

We now formally define the 2HAM.

Two assemblies and are disjoint if For two assemblies and , define the union to be the assembly defined for all by if is defined, and otherwise. Say that this union is disjoint if and are disjoint.

The binding graph of an assembly is the grid graph , where , and if and only if (1) , (2) , and (3) . Given , an assembly is -stable (or simply stable if is understood from context), if it cannot be broken up into smaller assemblies without breaking bonds of total strength at least ; i.e., if every cut of has weight at least , where the weight of an edge is the strength of the glue it represents. In contrast to the model of Wang tiling, the nonnegativity of the strength function implies that glue mismatches between adjacent tiles do not prevent a tile from binding to an assembly, so long as sufficient binding strength is received from the (other) sides of the tile at which the glues match.

For assemblies and , we write to denote the assembly defined for all by , and write if there exists such that ; i.e., if is a translation of . Given two assemblies , we say is a subassembly of , and we write , if and, for all points , . Define the supertile of to be the set . A supertile is -stable (or simply stable) if all of the assemblies it contains are -stable; equivalently, is stable if it contains a stable assembly, since translation preserves the property of stability. Note also that the notation is the size of the supertile (i.e., number of tiles in the supertile) is well-defined, since translation preserves cardinality (and note in particular that even though we define as a set, does not denote the cardinality of this set, which is always ).

For two supertiles and , and temperature , define the combination set to be the set of all supertiles such that there exist and such that (1) and are disjoint (steric protection), (2) is -stable, and (3) . That is, is the set of all -stable supertiles that can be obtained by “attaching” to stably, with if there is more than one position at which could attach stably to .

It is common with seeded assembly to stipulate an infinite number of copies of each tile, but our definition allows for a finite number of tiles as well. Our definition also allows for the growth of infinite assemblies and finite assemblies to be captured by a single definition, similar to the definitions of [19] for seeded assembly.

Given a set of tiles , define a state of to be a multiset of supertiles, or equivalently, is a function mapping supertiles of to , indicating the multiplicity of each supertile in the state. We therefore write if and only if .

A (two-handed) tile assembly system (TAS) is an ordered triple , where is a finite set of tile types, is the initial state, and is the temperature. If not stated otherwise, assume that the initial state is defined for all supertiles such that , and for all other supertiles . That is, is the state consisting of a countably infinite number of copies of each individual tile type from , and no other supertiles. In such a case we write to indicate that uses the default initial state. For notational convenience we sometimes describe as a set of supertiles, in which case we actually mean that is a multiset of supertiles with infinite count of each supertile. We also assume that, in general, unless stated otherwise, the count for any supertile in the initial state is infinite.

Given a TAS , define an assembly sequence of to be a sequence of states (where if is an infinite assembly sequence), and is constrained based on in the following way: There exist supertiles such that (1) , (2) ,3 (3) if , then , , otherwise if , then , and (4) for all . That is, is obtained from by picking two supertiles from that can attach to each other, and attaching them, thereby decreasing the count of the two reactant supertiles and increasing the count of the product supertile. If , we say that is nascent.

Given an assembly sequence of and a supertile for some , define the predecessors of in to be the multiset if and and attached to create at step of the assembly sequence, and define otherwise. Define the successor of in to be if is one of the predecessors of in , and define otherwise. A sequence of supertiles is a supertile assembly sequence of if there is an assembly sequence of such that, for all , , and is nascent if is nascent.

The result of a supertile assembly sequence is the unique supertile such that there exist an assembly and, for each , assemblies such that and, for each , . For all supertiles , we write (or when is clear from context) to denote that there is a supertile assembly sequence such that and . It can be shown using the techniques of [29] for seeded systems that for all two-handed tile assembly systems supplying an infinite number of each tile type, is a transitive, reflexive relation on supertiles of . We write () to denote an assembly sequence of length 1 from to and () to denote an assembly sequence of length 1 from to if and an assembly sequence of length 0 otherwise.

A supertile is producible, and we write , if it is the result of a nascent supertile assembly sequence. A supertile is terminal if, for all producible supertiles , .4 Define to be the set of terminal and producible supertiles of . is directed (a.k.a., deterministic, confluent) if .

2.3 Definitions for simulation

In this subsection, we formally define what it means for one 2HAM TAS to “simulate” another 2HAM TAS. For a tileset , let and denote the set of all assemblies over and all supertiles over respectively. Let and denote the set of all finite assemblies over and all finite supertiles over respectively.

In what follows, let be a tile set. An -block assembly, or macrotile, over tile set is a partial function , where . Let be the set of all -block assemblies over . The -block with no domain is said to be empty. For an arbitrary assembly define to be the -block defined by for .

For a partial function , define the assembly representation function such that if and only if for all . Further, is said to map cleanly to under if either (1) for all non empty blocks , for some such that , or (2) has at most one non-empty -block . In other words, we allow for the existence of simulator “fuzz” directly north, south, east or west of a simulator macrotile, but we exclude the possibility of diagonal fuzz.

For a given assembly representation function , define the supertile representation function such that . is said to map cleanly to if and maps cleanly to for all .

In the following definitions, let be a 2HAM TAS and, for some initial configuration , that depends on , let be a 2HAM TAS, and let be an -block representation function .

Definition 2.1.

We say that and have equivalent productions (at scale factor ), and we write if the following conditions hold:

  1. .

  2. For all , maps cleanly to

Definition 2.2.

We say that follows (at scale factor ), and we write if, for any such that , .

Definition 2.3.

We say that weakly models (at scale factor ), and we write if, for any such that , for all such that , there exists an such that , , and for some with .

Definition 2.4.

We say that strongly models (at scale factor ), and we write if for any , such that , then for all such that and , it must be that there exist , such that , , , , , and .

Definition 2.5.

Let and .

  1. simulates (at scale factor ) if .

  2. strongly simulates (at scale factor ) if .

For simulation, we require that when a simulated supertile may grow, via one combination attachment, into a second supertile , then any simulator supertile that maps to must also grow into a simulator supertile that maps to . The converse should also be true.

For strong simulation, in addition to requiring that all supertiles mapping to must be capable of growing into a supertile mapping to when can grow into in the simulated system, we further require that this growth can take place by the attachment of any supertile mapping to , where is the supertile that attaches to to get .

2.4 Intrinsic universality

Let denote the set of all -block (or macrotile) representation functions. Let be a class of tile assembly systems, and let be a tile set. We say is intrinsically universal for if there are computable functions and , and a such that, for each , there is a constant such that, letting , , and , simulates at scale and using macrotile representation function . That is, gives a representation function that interprets macrotiles (or -blocks) of as assemblies of , and gives the initial state used to create the necessary macrotiles from to represent subject to the constraint that no macrotile in can be larger than a single square.

3 The 2HAM is not intrinsically universal

In this section, we prove the main result of this paper: there is no universal 2HAM tile set that, when appropriately initialized, is capable of simulating an arbitrary 2HAM system. That is, we prove that the 2HAM, unlike the aTAM, is not intrinsically universal.

Theorem 3.1.

The 2HAM is not intrinsically universal.

We first prove Theorem 3.2, which says that, for any claimed 2HAM simulator , that runs at temperature , there exists a 2HAM system, with temperature , that cannot be simulated by . We we use this as the main tool to prove Theorem 3.3, a restatement of Theorem 3.1; our main result.

Theorem 3.2.

Let . For every tile set , there exists a 2HAM TAS such that for any initial configuration over and , the 2HAM TAS does not simulate .

The basic idea of the proof of Theorem 3.2 is to use Definitions 2.3 and 2.1 in order to exhibit two producible supertiles in , that do not combine in because of a lack of total binding strength, and show that the supertiles that simulate them in do combine in the (lower temperature) simulator . Then we argue that Definition 2.2 says that, because the simulating supertiles can combine in the simulator , then so too can the supertiles being simulated in the simulated system , which contradicts the fact that the two originally chosen supertiles from do not combine in .

Proof.

Our proof is by contradiction. Therefore, suppose, for the sake of obtaining a contradiction, that there exists a universal tile set such that, for any 2HAM TAS , there exists an initial configuration and , such that simulates . Define where is the tile set defined in Figure 1, the default initial state is used, and . Let be the temperature 2HAM system, which uses tile set and initial configuration (depending on ) to simulate at scale factor . Let denote the assembly replacement function that testifies to the fact that simulates .

Figure 1: The tile set for the proof of Theorem 3.2. Black rectangles represent strength- glues (labeled 1-8), and black squares represent the strength-1 glue (labeled 0).

We say that a supertile is a left half-ladder of height if it contains tiles of the type A2 and tiles of type A3, arranged in a vertical column, plus tiles of each of the types A1 and A0. (An example of a left half-ladder is shown on the left in Figure 2. The dotted lines show positions at which tiles of type A1 and A0 could potentially attach, but since a half-ladder has exactly of each, only such locations have tiles.) Essentially, a left half-ladder consists of a single-tile-wide vertical column of height with an A2 tile at the bottom and top, and those in between alternating between A3 and A2 tiles. To the east of exactly of the A2 tiles, an A1 tile is attached and to the east of each A1 tile, an A0 tile type is attached. These A1-A0 pairs, collectively, form the rungs of the left half-ladder. We can define right half-ladders similarly. A right half-ladder of height is defined exactly the same way but using the tile types B3, B2, B1, and B0 and with rungs growing to the left of the vertical column. The east glue of A0 is a strength- glue matching the west glue of B0.

Figure 2: Example half-ladders with rungs.

Let and be the set of all left and right half-ladders of height , respectively. Note that there are half-ladders of height in (). Define, for each , the mirror image of as the supertile such that has rungs at the same positions as .

For some , we say that is a simulator left half-ladder of height if . Note that need not be unique. (One could even imagine and satisfying and but and only differ by a single tile!) The notation is defined as the set of all supertiles that result in the -stable combination of the supertiles and .

For some , we say that is a mate of if , where , (they combine in ), and (they combine in ). For a simulator left half-ladder , we say that is combinable if has a mate. Part 1 of Definition 2.5 guarantees the existence of at least one combinable simulator left half-ladder for each left half-ladder. It is easy to see from Part 1 of Definition 2.5 that an arbitrary simulator left half-ladder need not be combinable, since by Definition 2.3, it may be a half-ladder , which must first “grow into” a combinable left half-ladder (analogous to in Definition 2.3).

Denote as some set that contains exactly one combinable simulator left half-ladder for each . Note that, by Definitions 2.1 and 2.3, there must be at least one combinable simulator left half-ladder for each , but that there also may be more than one, so the set , while certainly not empty, need not be unique. By the definition of , it is easy to see that . We know that each combinable simulator left half-ladder has exactly rungs, and furthermore, since glue strengths in the 2HAM cannot be fractional, it is the case that of these rungs bind to (the corresponding rungs of) a mate with a combined total strength of at least . (Note that some, but not all, of these rungs may be redundant in the sense that they do not interact with positive strength.)

There are ways to position/choose rungs on a (simulator) half-ladder of height . (Note that a rung on a simulator half-ladder need not be a block of tiles but merely a collection of rung-like blocks that map to rungs in the input system via .) Now consider the size set of all possible rung positions, each denoted by a subset , and the size set . For each simulated half-ladder , there must exist a set of rungs such that binds to a mate via the rungs specified by , with total strength at least . As there are elements of and only choices for , the Generalized Pigeonhole Principle implies that there must be some set with such that every simulator left half-ladder in binds to a mate via the rungs specified by a single choice of , with total strength at least . In the case that , we have that .

Let , which is the number of ways to tile a neighborhood of four squares from a set of distinct tile types. If , then . There are ways to tile neighborhoods that map to tiles of type A0 (plus any additional simulator fuzz that connects to simulated A0 tiles), under , at the ends of the rungs of a simulator left half-ladder. This tells us that there are at least two (combinable) simulator left half-ladders such that binds to a mate via the rungs specified by , with total strength at least , binds to a mate via the rungs specified by , with total strength at least and the rungs (along with any surrounding fuzz) specified by of are tiled exactly the same as the rungs specified by of are tiled. Thus, we can conclude that , a mate of , is a mate of . We can conclude this because, while and agree exactly along of their rungs, they also each have one rung in a unique position and since consecutive rungs in have at least two empty spaces between then, the offset simulator rungs (and even their fuzz) cannot prevent from matching up with the mate of .

However, , but because and differ from each other in one rung location and therefore interact in with total strength at most . This is a contradiction to Definition 2.2, which implies . ∎

We now have the main tool needed to prove our main Theorem 3.1, which we restate as follows.

Theorem 3.3.

There is no universal tile set for the 2HAM, i.e., there is no  such that, for all 2HAM tile assembly systems , there exists an initial configuration over and temperature such that simulates .

Proof.

Our proof is by contradiction, so assume that is a universal tile set. Denote as the strength of the strongest glue on any tile type in . Let be a modified version of the TAS from the proof of Theorem 3.2 with each -strength glue in converted to a strength glue in (all other glues and labels are unmodified). By the proof of Theorem 3.2, we know that for any initial configuration over , does not simulate for any . If , then the size of the largest supertile in is 1 (since is the maximum glue strength in , the supertiles in the initial state (input) are not -stable and indeed no tile can bind to any assembly with strength ), whence is not a universal tile set.∎

4 The temperature- 2HAM is intrinsically universal

In this section we state our second main result, which states that for fixed temperature the class of 2HAM systems at temperature is intrinsically universal. In other words, for such there is a tile set that, when appropriately initialized, simulates any temperature 2HAM system. Denote as the set of all 2HAM systems at temperature .

Theorem 4.1.

For all , 2HAM() is intrinsically universal.

We prove this theorem for two different, but seemingly natural notions of simulation. The first, simply called simulation, is where we require that when a simulated supertile may grow, via one attachment, into a second supertile , then any simulator supertile that maps to must also grow into a simulator supertile that maps to . The converse should also be true. Results for simulation are given in Section 6.

The second notion, called strong simulation, is a stricter definition where in addition to requiring that all supertiles mapping to must be capable of growing into a supertile mapping to when can grow into in the simulated system, we further require that this growth can take place by the attachment of any supertile mapping to , where is the supertile that attaches to to get . Results for strong simulation are given in Section 5.

For each of the two notions of simulation we provide three results, and in all cases we provide lower scale factor for simulation relative to strong simulation. Specifically, strong simulation achieves a modest linear scale factor simulation, but a compact single input assembly is sufficient to encode the entire simulated system. In contrast, for simulation (i.e. not strong), we are able to achieve a logarithmic scale factor in the size of the simulated system. However, such small scale requires that the simulated system be encoded in a larger (linear) number of input assemblies, in order to describe the simulated system without loss of information.

For strong simulation, Theorems 5.15.2, and 5.3 provide three different proofs of our main positive result (Theorem 4.1) with each of the three providing different trade-offs between number tile types, scale factor, and complexity of initial configuration for the simulator. For simulation, Theorems 6.16.2 and 6.3, provide similar trade-offs.

When we combine our negative and positive results, we get a separation between classes of 2HAM tile systems based on their temperatures.

Theorem 4.2.

There exists an infinite number of infinite hierarchies of 2HAM systems with strictly-increasing power (and temperature) that can simulate downward within their own hierarchy.

Proof.

Our first main result (Theorem 3.2) tells us that the temperature- 2HAM cannot be simulated by any temperature 2HAM. Hence we have, for all , , where is the relation “cannot be simulated by”. Moreover, Theorem 4.1 tells us that temperature 2HAM is intrinsically universal for fixed temperature . Suppose that such that . Then the temperature- 2HAM can simulate temperature (by simulating strength attachments in the temperature system with strength attachments in the temperature system). Thus, for all , can simulate, via Theorem 4.1, . The theorem follows by noting that our choice of was arbitrary. ∎

We have shown that for each there exists a single set of tile types , and a set of input supertiles over , such that the 2HAM system strongly simulates any 2HAM TAS . A related question is: does there exist a tile set that can simulate, or strongly simulate, all temperature 2HAM TASs simultaneously? Surprisingly, the answer is yes! The proof of the following theorem is given in Section 7.

Theorem 4.3.

For each , there exists a 2HAM system which simultaneously strongly simulates all 2HAM systems .

5 The temperature- 2HAM is intrinsically universal: strong simulation

We present a total of six simulation results, three in this section and three in Section 6. In this section the three simulation results exhibit for any integer , a single set of tiles that at temperature  strongly simulates any temperature  2HAM system, given a proper configuration of initial assemblies over . Each of the three results in this section depict different tradeoffs between the number of encoded input supertiles and the scale of the simulation. Theorem 5.3 achieves this while having the simulated tile set encoded as a single input assembly.

For the aTAM it is known [12] that there is a single tile set that simulates any aTAM tile assembly system , when initialized with a single seed assembly  that encodes . Assembly proceeds by additions of single tiles to this seed. In this paper, where we study the 2HAM, it makes sense to allow the simulator to be programmed with multiple copies of the seed (input), rather than a single copy. In particular, this is the case in Theorem 5.3 for strong simulation (and thus, also the weaker notion of simulation) where the simulator’s input consists of infinitely many copies of both a single seed supertile, as well as the simulator’s tiles. However, the definition of input configuration allows fancier input configurations: it permits us to have numerous distinct seed assemblies. By exploiting this we achieve better scaling in Theorems 5.1, 5.26.1, 6.2, and 6.3 than in the single-seed case of Theorem 5.3 (the six results also give trade-offs in numbers of tile types in the simulator). However, since these improvements in scaling (and possibly number of tile types) come at the expense of having many seed assemblies in the simulator, there is an intuitive sense in which “less” self-assembly, or at least a different form of self-assembly, is happening as fewer, and larger, assemblies themselves can act as large polyomino jigsaw pieces that come together to simulate tiles. It is worth pointing out that our main result, an impossibility result, holds despite the fact that the simulator may try to use a large number of complicated-looking input assemblies.

Let denote the number of distinct elements in the multiset , i.e.  is the cardinality of the set defined by ignoring multiplicities in the multiset . Let denote the set of supertiles induced by a tile set . By this, we simply mean is the set of supertiles formed by taking all tiles in and translating them to all locations in .

5.1 Strong simulation with small scale and few tile types

Theorem 5.1.

For every there exists a single set of tile types , with (i.e. independent of ), such that for all 2HAM systems , there is a set of input supertiles such that the 2HAM system strongly simulates at scale , where is the set of glues in .

(a)
(b)
Figure 3: Strong simulation. (a) Input assembly design. For each tile type in the simulated TAS there is a unique macrotile . Glue pads and binding pads are shown in green and red, respectively. North arms project from grey regions so that they are staggered relative to south arms, west and east arms act similarly. For each unique glue in the simulated TAS, there is a unique pair of coordinates for the glue/binding pads, one denoting the position along the side of the macrotile (i.e. the particular grey/white location), and one denoting the distance that the arm projects outward from the macrotile. (b) Detail showing two arms with matching glues (encoded as binary bumps and dents using green tiles) of strength 4 (encoded using 4 red tiles), from a temperature system. Each red tile provides unit binding strength, and so macrotiles bind with strength equal to the number of matching red tiles. Only red tiles provide non-zero binding strength.

Construction. For a given simulated TAS , the TAS represents the initial state of as follows.

Each singleton tile type that is used in the initial state is represented as a macrotile of the form shown in Figure a (larger input supertiles from are described below). Tile type is mapped to a unique macrotile as follows. First, each glue , from the tile set , is uniquely encoded as a pair where (using the inverse of some simple pairing function). Assume that tile type has the four glues . Next, these glues are encoded as the four coordinates using the above encoding. The macrotile is composed of five parts: a square body (of size , where ), and four arms so that one arm is placed in each of four square regions, each of size , and adjacent to the body. The location of the red and green glue-binding pad on an arm uniquely encodes the relevant glue of tile type : for example is encoded by the glue-binding pad location , where . (The arm length is and its position along the supertile side is .) Arm lengths on east sides are “complimentary” to those on west sides, in the following sense. Let be some glue that appears on the east side of one tile and west side of another. As described above, the arm with the pad has length such that the glue pad appears at location . However, the arm with the pad is defined to have length such that the glue pad appears at location . The same complementarity trick is used for north and south arms. Finally, as can be seen in Figure a, arms are staggered relative to each other: north and west arms sit on grey patches, south and east arms sit on white patches.

Figure b shows the individual glue-binding pads at the end of two arms. Each glue is uniquely represented as a bit sequence, which in turn is represented using bumps and dents in a region shown in green on the west arm in Figure b. The same bump-dent pattern would be used on a north arm. For east and south arms, the complementary bump-dent pattern is used. To illustrate this, Figure b shows a west and east arm that share the same glue type. It can be seen that the arms are able to be translated so that the green regions fit together. A glue-binding pad also encodes the binding strength of its represented glue , in a straightforward way as a sequence of white tiles followed by red tiles. Each red tile, on the west arm, exposes a single strength 1 glue to its south. The red tiles are the only tiles on the entire macrotile that expose positive strength glues. A matching east arm, as shown in Figure b, exposes the same number of matching strength 1 glues from its red tiles.5

Due to their complementary green regions, and their matching sequence of exactly red tiles, the two arms shown in Figure b can be translated so that they bind together with strength .

This completes the description of the encoding of the singleton tile types that are used in the initial state . The remaining supertiles of (i.e. of size ) are encoded as in the following paragraph.

Consider 2 macrotiles , that represent tiles . From the above description, it can be seen that and can be positioned so that their bodies’ centers lie on the same horizontal line, at exactly distance apart, such that do not intersect. Furthermore, if and have a matching glue on their east and west sides, respectively, then the glue-binding pads of the east arm of  and the west arm of  will be positioned so that their matching bump-dent patterns interlock, and their arms bind with whatever strength and bind. Finally, if and do not have matching glues on their east and west sides, respectively, then it is the case the glue-binding pads of the two arms do not touch, and indeed their “mismatching arms” do not intersect. This holds for the only other potential binding position (i.e. north-south) of two arbitrary tiles and . Figure b shows six macrotiles translated into position to simulate an assembly of 6 tiles in some simulated TAS. The figure shows 5 matching arms (simulating matching glues) and two mismatched arms (simulating mismatched glues). Supertiles of size in are encoded in this manner.

We have now completely specified the initial state from which the self-assembly process proceeds in .

Figure 4: Strong simulation. Example assembly with 6 macrotiles and 2 simulated mismatches. The square “body” of each tile is of size , and the assembly sits on a lattice.

Scale. Each glue of the simulated TAS is encoded using a glue-binding pad consisting of (green) tiles and (red) tiles. There are such glue-binding pads, which are rasterized into each of four regions, where . To see that the tile set size is a constant (i.e. independent of the simulated TAS ), note that all tiles on the outside of the macrotile expose strength 0 glues, except for the red tiles, which each expose the same strength 1 glue. Hence, the interior of each macrotile can be filled in using a single filler tile, with a constant size set of tile types used to fill the exterior.

Correctness of Simulation. Via the following two cases, macrotiles stay “on-grid.” (1) Due to their arm lengths, if two matching green glue pads bind (causing the binding of two macrotiles) then the combined (horizontal or vertical) arm length is exactly . Thus matching macrotiles only bind in a way that their centers are exactly distance apart. (2) Due to the green glue pad design, if two glue-binding pads mismatch (i.e. represent two mismatching glues in ) then they can not bind, since the two green pads sterically hinder each other. Taken together, this means that whenever two macrotiles (that encode two tile types ) bind, they are always positioned on a square grid.

This immediately implies that whenever larger assemblies, with multiple macrotiles, bind, they have all of their tiles positioned on a square grid.

It remains to show that, due to our macrotile design, strongly simulates  (i.e. point 2 of Definition 2.5). Firstly, due the glue-binding pad design and the fact that and work at the same temperature , a pair of supertiles in bind if and only if their corresponding encoded pair of supertiles bind in . This, taken together with the fact that the initial state of is an encoding of the initial state of , implies that the two systems have the same dynamics (in the strong sense), thus satisfying Definitions 2.2 and 2.4. Definition 2.1(1) (equivalent production) is satisfied since equivalent dynamics implies equivalent production, and Definition 2.1(2) is satisfied as our choice of macrotile design directly implies that each supertile produced in maps cleanly to a supertile in . Taken together, these facts are sufficient to satisfy Definition 2.5(2).

This completes the proof of Theorem 5.1.

5.2 Strong simulation with smaller scale but more tile types

Theorem 5.2.

For each integer there exists a single set of tile types , , such that for any 2HAM system , there exists a set of input supertiles such that the 2HAM system strongly simulates at scale , where is the set of glues in .

Theorem 5.2 simply is a trade-off in tile types for scale factor with Theorem 5.1. The construction for Theorem 5.1 is modified so that where before each arm had red tiles each exposing a strength 1 glue, now each arm has a single red tile that exposes one strength