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

The first step in our encoding process begins by forming a bounding box assembly $\beta_{i}$ through the attachment of special tiles, called filler tiles, to $\alpha_{i}$. These filler tiles cooperatively bind to 2 diagonally adjacent tiles of our shape assembly in order to fill out any concave portions. When a filler tile attaches to an assembly, signals are fired from the newly bound glues which activate additional glues between the filler tile and shape assembly. These new glues ensure that the filler tile is bound with strength 2 on each face to the shape assembly as this will be important during the disassembly process. After the filler tile is firmly attached with 2 strength-2 bonds, signals are then fired within the filler tile which activate strength-1 glues of type $g_{f}$ on all other faces. These will be used for further filler tile attachment. Figure 8 illustrates the attachment of a filler tile to an assembly and shows how sequential signaling is used to ensure that the filler tile is attached with strength 2 on both of its input faces before activating glues on each of its output faces.

Because filler tiles must be able to cooperatively bind to both shape tiles and other previously attached filler tiles, we need 3 unique types of filler tiles: One which initially presents 2 glues of type $g_{x}^{*}$ to bind with 2 shape tiles, one which initially presents 2 glues of type $g_{f}^{*}$ to bind with 2 other filler tiles, and one which presents one of each glue to cooperatively bind with a shape tile and a filler tile. Each type of filler tile is otherwise identical. Because we’ve chosen our binding threshold $\tau=2$, the two initially present glues are sufficient for binding into any location on the assembly with at least 2 adjacent shape or filler tiles. The signals from the binding of these glues then activates additional glues on the same faces which ensures that the filler tile is attached with strength 2 on two separate faces, regardless of whether or not additional filler tiles later bind to this one. This property will be used to guarantee that the assembly stays connected during the disassembly process.

Eventually, after sufficiently many filler tiles have attached, there will be no more locations in which another filler tile can attach. There are often many ways in which this can occur for any shape assembly, but the resulting bounding box assembly will always be a minimal bounding box of our shape. It should be noted that its possible that not every location within the bounding box is filled. This can occur if the original shape had enclosed cavities, but can also occur because the attachment of filler tiles can create additional cavities as they attach. This is not a problem and it will always be possible for filler tiles to complete the outer surface of the bounding box. Additionally, this bounding box will be uniformly covered by glues of type $g_{x}$ and $g_{f}$.

In order to continue with the encoding process, we first need to verify that the bounding box is fully formed. This is done by growing a shell of tiles around our assembly. This shell, which we call the inner shell, is able to grow to completion only if the assembly is a fully formed bounding box. Figure 9 illustrates the high-level construction of the inner shell around a fully formed bounding box.

Growth of the inner shell begins with the attachment of corner gadgets to our assembly. We use 2 types of 3D corner gadgets, one which is able to bind to a corner of our assembly presenting 3 glues of type $g_{x}$ and one which is able to bind to a corner presenting 3 glues of type $g_{f}$ (note that at $\tau=2$ only two glues are needed for a corner gadget to attach, but any tile allowing a corner gadget to attach must expose all 3). That is, the corner gadgets can attach either to a shape tile or a filler tile on a corner of our assembly. Note that these gadgets exist in our system while the bounding box is being constructed; therefore, it’s possible that corner gadgets attach to tiles in our assembly before the bounding box has been fully constructed. Additionally, special care needs to be taken when the bounding box surrounding our shape assembly has at least one side of dimension 1. The details of the inner shell’s construction is described below and these various cases are addressed.

When a corner gadget attaches to our assembly, signals from the attachment cause strength-1 glues to activate on the faces of the corner gadget which point parallel to the surface of our assembly. These glues will be used to allow cooperative attachment of special edge tiles that will attach in a line along the edges of a completed bounding box. The glues activated on the corner gadget can either be of type $g_{\text{edge}}^{L}$ or $g_{\text{edge}}^{R}$ depending on which face of the corner gadget they reside. Glues of type $g_{\text{edge}}^{L}$ indicate that the edge to be grown is a left edge of the bounding prism relative to the direction of growth of the edge while glues of type $g_{\text{edge}}^{R}$ indicate a right edge.

Like filler tiles, edge tiles initially have 2 active glues on adjacent faces: one of these glues is either of type $g_{\text{edge}}^{L*}$ or $g_{\text{edge}}^{R*}$ so as to be complementary to the glue presented by the corner gadget, and one of type $g_{x}^{*}$ or $g_{f}^{*}$ so as to also be complementary to a glue on the surface of our assembly. Since any combination of these glues is necessary, there are 4 unique types of edge tiles. Once an edge tile has cooperatively attached to our assembly, signals are fired which activate another glue of type $g_{\text{edge}}^{L}$ or $g_{\text{edge}}^{R}$ to allow additional edge tiles to cooperatively attach to it and the assembly. Additionally, glues are activated on all other exposed sides of the edge tile which will be used by detector gadgets later. These glues are unique to the specific face of the edge tile so that detector gadgets can distinguish between the interior and exterior sides of an edge as well as the side of the edge tile pointing away from the assembly. Although tiles are allowed to rotate in the STAM^R and don’t have fixed orientations, this directionality can be enforced by the relative orientations of the two glues used for the initial binding of a tile. Edge tiles will continue to grow along the surface of our assembly from corner gadgets until they are either blocked by another tile, reach the end of the surface of our assembly, or it is detected that the edge is invalid.

For an edge to be valid, there must be no shape or filler tiles adjacent to any edge tiles except for those underneath the edge tiles to which the edge tiles cooperatively attached; additionally, if an edge is a right (respectively, left) edge, then there must not be a shape or filler tile occupying a location diagonally adjacent to the right (resp., left) of the edge tiles making up the edge with respect to the forward growth direction of the edge. Edge tiles which violate these validity conditions can be easily detected using detector gadgets specific to the particular situation as illustrated in Figure 10. Following the attachment of such a detector gadget, a signal is propagated along the edge causing all connected edge tiles and corner gadgets to dissolve. Before this signal is propagated though, signals from the detector gadget ensure that a new filler tile is effectively added to the assembly in a safe location (that is without causing the eventual bounding box to be bigger than necessary). This is done using signals from the detector gadget to convert one of its own tiles or the detected invalid tile into a filler tile. This conversion is done so that we don’t risk infinite assembly sequences wherein a corner gadget attaches, attempts to grow an invalid edge, and dissolves repeatedly. Because a filler tile is always effectively added upon detection of an invalid edge, eventually it will be impossible for invalid edges to occur.

In the case where a valid edge is blocked by another tile, then there are 2 possibilities: (1) the edge is blocked by a shape or filler tile, or (2) the edge is blocked by another edge or corner gadget. If a filler tile blocks the path, then like with invalid edges, a detector gadget can cooperatively bind to the blocking tile and the edge tile, convert the edge tile into a filler tile, and propagate a dissolve signal down the remaining edge tiles. If another edge tile or corner gadget blocks the edge, then we need to determine if the blocking tile is part of a valid edge. If the edge is invalid, then it will eventually dissolve and nothing needs to be done. Otherwise, the tile blocking our edge belongs to another valid edge. In this case the meeting point can either be at a corner of our assembly or along the edge of our assembly. Because of the unique glues presented on all sides of an edge tile, these situations can easily be differentiated by detector gadgets. If the meeting point is a corner, then signals from the corresponding detector gadget will cause the corner to convert to a piece of a corner gadget. The remaining corner gadget pieces can then attach and the result will be a corner gadget connected to two incoming edges. If the meeting point is an edge, the detector gadget will fire signals to activate glues between the colliding edge tiles connecting the edge tiles and allowing future signals to propagate between them.

Care must be taken when dissolving edge tiles and corner gadgets to avoid erroneous attachments of tiles which have dissolved, but on which not all of the glues have yet deactivated. When dissolving edge and corner tiles, we use a procedure called careful dissolving to guarantee safe detachment. To understand this procedure first note that, we make a distinction between those glues which were initially active on a tile before it attached to an assembly, which we call prior glues, and those which activated after the initial attachment, called posterior glues. Here we make one exception regarding the strength 2 glues present between the outermost corner tile of a corner gadget and its 3 neighboring tiles, these are classified as corner glues and will be handled differently. Also, in addition to all of the functional glues present on an edge tile or corner gadget tile, when two edge tiles bind to each other, a strength 1 pair of glues of type $g_{d}$ and $g_{d}^{*}$, called dissolve helper glues, are activated between them. Corner gadgets also have these glues activated between their tiles, but this is done in a tree-like structure with the root being the outermost corner tile. This tile shares dissolve helper glues with the 3 corner gadget tiles adjacent to it, and these share dissolve helper glues with the 3 corner gadget tiles which cooperatively bound in between, though only on 1 face each so as to form a tree.

Careful dissolving begins when a detector gadget binds to an edge or corner gadget tile. This binding initiates a dissolve signal that propagates along the edge and corner gadget tiles, deactivating all prior glues. Now suppose $\gamma$ is a group of connected edge tiles which have detached from the assembly as a result of these deactivations. By our tile design, prior glues always only bind with with either posterior glues or bounding box glues ($g_{x}$ or $g_{f}$), and posterior glues, which are always strength 1, only bind with prior glues. Notice that $\gamma$ can be presenting at most 1 prior glue of strength 1, otherwise it would not have detached from the assembly, though it may have any number of posterior glues and some dissolve helper glues. Because attachment to an assembly requires either a prior glue of strength 2 or two prior glues of strength 2 to bind with posterior glues exposed by a bounding box assembly, $\gamma$ is effectively inert. It is possible that two detached junk assemblies have dissolve helper tiles exposed, but any cooperation between these junk assemblies would require the cooperation of a dissolve helper glue and a prior/posterior glue pair to occur. This may happen, but eventually the prior glue will deactivate and the combined junk will dissolve.

By the connectivity offered by the dissolve helper tiles, even as $\gamma$ further breaks up into smaller assemblies or individual tiles, this property is preserved, since in addition to the dissolve helper glues between each pair of tiles in $\gamma$, any glues holding tiles together form a prior/posterior pair. For a strength 1 cut to exist in $\gamma$, allowing it to break apart, it must be the case that the prior glue deactivates between the tiles, otherwise they will still be held together with at least strength 2. Eventually, we will be left with only individual inert tiles or the 4 tiles that make up the corner of a corner gadget which will also be inert by the same argument. Thus we have a maximum junk size of 4. Careful dissolving is possible so long as the above conditions regarding prior and posterior glues are met. This is true for all gadgets and tiles used during the leader election process, so during the leader election process, when we say that a dissolve signal is propagated, we mean that careful dissolving occurs between those tiles.

When the edges growing from 2 corner gadgets meet via edge tiles between them along the surface of a bounding prism, signals between them through the edge tiles activate glues which allow a filler verification process to begin. This process proceeds in iterations growing inwards towards the surface’s center and verifies that there are no gaps or protrusions in the surface. If gaps are found, nothing happens until those gaps are filled with filler tiles, after which the verification continues. If protrusions are found, then as illustrated in Figure 11, detector gadgets are able to cooperatively bind with a verification tile and a shape/filler tile of the protrusion. Signals from this attachment cause the verification tile to become a filler tile and cause all other involved verification tiles, edge tiles, and corner gadgets to dissolve.

The filler verification procedure is as follows. When the edge between two corner gadgets is filled with edge tiles, a signal is able to propagate between the corner gadgets. Once a corner gadget has received signals from its 2 neighboring corner gadgets, glues are activated on the adjacent edge tiles allowing the cooperative binding of a tile called a verification corner tile. This verification corner tile attaches diagonally adjacent to the corner gadget within the region bounded by the edge tiles. Additionally, signals from the corner gadgets activate glues on the other edge tiles which allow special verification edge tiles to cooperatively bind with the edge tile and surface of the bounding prism. If there is a gap preventing such a binding, it will simply not occur until filler tiles attach to fill the gap. If there is a protrusion, a detector gadget will be able to cooperatively bind with a filler/shape tile on the protrusion and a verification tile. That verification tile will then convert to a filler tile through signals fired from the detector gadget and all other involved edge tiles, verification tiles, and corner gadgets will dissolve. If no protrusion is found, the process repeats with the verification corner tiles acting as the corner gadgets and verification edge tiles acting as the edge tiles. A new iteration of the verification process will begin in the next inner layer of the surface.

This process will continue until the center is reached. This can happen in 2 different ways depending on whether the shortest side of the surface is of even length or odd length. (See Figure 12.) If the shortest side of the surface is of even length, then eventually 2 verification corner tiles will be adjacent to each other. A duple detector gadget will be able to cooperatively bind with those tiles indicating that the center has been reached. This will happen on both pairs of adjacent corner verification tiles and once the verification edge tiles attach between them, signals will be able to propagate between the pairs of corner verification tiles. These signals will propagate back along the iterations of the verification tiles and activate glues on the corner gadgets which will allow for the growth of the outer shell to begin on this face of the bounding prism. If the shortest side of the surface is of odd length, the process is similar, but instead of 2 verification corner tiles being adjacent, there will be a single verification corner which is adjacent to either 2 verification corner tiles from the previous iteration, or all 4 if the surface of the bounding prism was a square. In either case, detection gadgets will be able to initiate signals which inform the corner gadgets that verification of this face is complete. Additionally, upon completion, a dissolve signal causes all glues on the verification tiles to turn off and the verification tiles themselves to dissolve.

The process described above assumes thick shapes, those whose minimum bounding box has no sides of length 1. To handle thin shapes (i.e. those shapes that are not thick), first note that for every corner gadget attached to a thin shape, there will be at least one direction where no edge tile can cooperatively attach to the corner gadget and shape assembly. This can be detected by a detector gadget and upon detection signals will be fired accomplishing 2 tasks: (1) glues will be activated on the corner gadget which allow other corner gadget tiles to attach as if two mirrored corner gadgets were overlapping along the thin edge, and (2) edge tiles running along the thin edge of the assembly from the corner gadget will be dissolved and the outgoing $g_{\text{edge}}^{L}$ or $g_{\text{edge}}^{R}$ glue from the corner gadget will be deactivated and replaced by a newly activated glue of type $g_{\text{edge}}^{T}$. We call corner gadgets that have been modified in this way extended corner gadgets. To the glue of type $g_{\text{edge}}^{T}$, a different type of tile, called a thin edge tile, can cooperatively attach to the assembly and corner gadget. Thin edge tiles behave similarly to regular edge tiles and grow sequentially along the assembly. Upon meeting another thin edge tile, like with normal edge tiles, a detector gadget cooperatively binds and activates glues on the thin edge tiles allowing them to bind with each other if they meet along a thin edge or converting the thin edge tiles into corner gadget tiles if they met at a corner. If the path of the thin edge tiles is blocked by a shape or filler tile, a detector gadget can cooperatively bind and the last thin edge tile is converted to a filler tile and a dissolve signal is propagated down the remaining edge tiles.

In the case where our initial shape assembly is a thin rod, having dimensions $1\times 1\times m$, the corner gadgets which bind to the ends of the ends of the rod will be extended twice (or 3 times if $m=1$). Detector gadgets can be used to determine that a corner gadget has been extended more than once and signals from the attachment of these detector gadgets will activate the same glues on the corner gadgets indicating that filler verification is complete for the corresponding $1\times 1$ side of the assembly.

Whenever the filler verification process is completed on a surface of the bounding prism, signals activate glues on the corner gadgets of that surface which initiate the growth of an outer shell. The glues activated on the corner gadgets exist on the outward pointing faces of the tiles between edge tiles and allow tiles called outer shell tiles to bind with strength 2 to these locations as illustrated in Figure 13. Once attached, these outer shell tiles present strength-1 glues of type $g_{\text{out}}$ on all sides except the one that points away from the assembly. Another type of tile, called an outer edge tile, is then able to cooperatively bind to these outer shell tiles and the edge tiles from the inner shell. These outer edge tiles also present $g_{\text{out}}$ glues which further allow other outer edge tiles to cooperatively bind on top of the edge tiles from the inner shell. When two outer edge tiles meet along an edge, detector gadgets can cooperatively bind to the pair causing them to activate glues between each other and bind.

Additionally, special corner gadgets called outer corner gadgets bind with 3 outer shell tiles on the corners of the assembly. (Because in our construction $\tau=2$, outer corner gadgets really only cooperatively bind with 2 of the outer shell tiles to attach, but by using sequential signaling, we can ensure that they do not propagate their signals to other outer corner gadgets until they are bound to all 3 outer shell tiles on their respective corner of the assembly.) These outer corner gadgets are different from normal corner gadgets in that they have 12 tiles as illustrated in Figure 13. Once an outer corner gadget attaches, signals are propagated along outer shell and outer edge tiles to adjacent outer corner gadgets.

When an outer corner gadget has received this signal from all 3 of its neighbor outer corner gadgets, a dissolve signal is propagated to the inner shell corner gadget below. This signal prompts that corner gadget and its edge tiles (but not any other corner gadgets) to dissolve and additionally causes glues, called candidate glues, of type $g_{\text{cand}}$ to activate on the corners of the bounding box assembly underneath and glues of a complementary type $g_{\text{cand}}^{*}$ to activate on the interior corners of the outer corner gadgets. Because of the condition under which these signals are fired, an outer corner gadget will not signal its underlying inner shell corner gadget to dissolve until all of the outer shell corner gadgets neighbors are bound to the assembly. Consequently, even though the outer shell gadgets cause the inner shell between them and the assembly to dissolve, the outer shell will remain attached to the assembly on at least one corner until all outer corner gadgets have attached. Once the final outer corner gadget attaches however, the inner shell underneath will be able to fully dissolve and we will be left with our bounding box enclosed within but not attached to the outer shell. While the bounding box will be free to move (slightly) within the outer shell, it will be trapped inside of it due to their relative sizes.

Because the corners of the bounding box and interior corners of the outer shell have complementary glues, the corners of the bounding box assembly are able to bind to the interior corners of the outer shell; however, because the interior of the outer shell is larger than the bounding box itself, only 1 corner will be able to touch the outer shell at any given time, and thus to bind. The corner of the bounding box which happens to bind is elected leader and a special glue $g_{\text{lead}}$ on that corner is activated. Additionally, the binding of the bounding box assembly to the outer shell causes signals to propagate which cause the $g_{\text{cand}}^{*}$ glues on the outer shell to deactivate and then cause the outer shell to dissolve. We are then left with a bounding box with 1 corner “elected as leader” and containing a $g_{\text{lead}}$ glue from which the disassembly process can begin.

The first step of the encoding process is for an encoding corner gadget (Figure 15), similar in structure to the corner gadgets utilized in the leader election process, to bind to the leader corner. We then treat that corner as the origin of our shape, where the directions of the $x$, $y$, and $z$ axes are shown in Figure 16. This reference point and orientation allows us to assign coordinates to each voxel of the bounding box. Of key importance during the deconstruction process is that the deconstructing supertile remains connected with strength 2 at all times. It is given that the shape tiles are connected with strength 2, and filler tiles similarly connect to both shape tiles and each other with strength 2. However, filler tiles are connected to only the 2 tiles which caused their cooperative placement and exterior filler tiles expose only strength 1 $g_{f}$ glues. To ensure that during the deconstruction process no tiles prematurely disconnect from the bounding box (and to provide additional functionality during the deconstruction process), shell tiles are added which create a shell around the bounding box and utilize the signals demonstrated in Figure 8 to enable strength 2 connections with the exterior-most fill tiles. At the end of the creation of the deconstruction shell (which we will also simply refer to as the ‘shell’), the bounding box will have all tiles on its faces covered, aside from those that are part of the first slice of the bounding box to be encoded. The shell consists of three parts corresponding to tile types: (1) the shell base, tiles which cover one face of the bounding box, allow for communication between tiles in the shell and allow for cooperative binding of recognizer tiles (to be described), (2) shell slices, which cover 3 faces of the bounding box (aside from the tiles that are part of the first slice of the bounding box) and are removed after each slice of the bounding box is encoded, and (3) a cap, which covers the remaining face and allows for the encoding process to sense when it has completed the decoding process.

With the deconstruction shell created around the bounding box, we are now able to begin the process of building the encoding structure ($\phi$) by deconstruction. Before continuing into the details of the encoding process, we provide a description of how the information provided by the location of tiles in a bounding box is encoded into binary values. Beginning with the origin point $(0,0,0)$, we read the tile type information for each tile in the first row sequentially by incrementing the $x$-coordinate; for example, the second tile read is in the voxel with coordinates $(1,0,0)$. Once all tiles in the current row have been read, we jump to the next row up. For example, in a $3\times 4\times 5\>\>(x\times y\times z)$ bounding box shown by Figure 21, the final location in the first layer is $(2,0,0)$. The next tile encoded is at coordinates $(2,1,0)$. We then encode tiles heading towards the origin; the next voxel encoded in our example encoding would be $(1,1,0)$. Upon arriving at the coordinate $(0,1,0)$ (the last of the row moving in this direction) we jump to the next row up, then encoding $(0,2,0)$. By this process of visiting every tile in a slice in a ‘zig-zag’ pattern, we are able to encode the information regarding any slice of a bounding box sequentially.

The very first row of the encoding subassembly contains additional information regarding the direction of the growth in our zig-zag pattern, and as a byproduct we also are able to easily retrieve the width of the rows of tiles. We compare the $x$ values in the coordinates $(x,y,z)$ between the first tile of a row and the last tile of a row by subtracting the $x$ value between the two such that $\Delta x=x_{\text{last}}-x_{\text{first}}$. If a tile is contained in a row where $\Delta x>0$ we denote this growth in the positive (‘+’) direction. Alternatively, if $\Delta x<0$ we denote this growth in the negative (‘-’) direction. We encode ‘+’ direction growth as a ‘0’, and ‘-’ direction growth as a ‘1’. For example, in Figure 21, the first row begins growth at tile 1, the origin $(0,0,0)$ and ends at $(2,0,0)$, leading to $\Delta x=2-0=2$. In contrast, the second row begins at $(2,1,0)$ and ends at $(0,1,0)$, leading to $\Delta x=0-2=-2$. We can see that the direction tiles placed in front of row 1 are encoded as 0, and encoded as 1 for row 2. All further slices only add a single tile for each voxel, as the direction for all tiles which have the same $x,\>y$ value in their tuple $(x,y,z)$ is the same (e.g., the tile in $(1,0,0)$ which is the second tile placed in the first slice; the tile in $(1,0,1)$ is the second tile placed in the second slice).

For simplicity, the differentiation between shape and fill tiles is excluded in remaining figures in this section.

In this section, we demonstrate a template for tiles which allows for the decoding process to be carried out, allowing for connections between all shape tiles and their neighbors within a slice. Additionally, we provide examples of gadgets which allow for the growth of consecutive slices of a shape encoding without causing filler tiles (which are not part of the final shape, but may be temporarily required to ensure a strength 2 connection between portions of shapes where a cut may exist in the binding graph of the partially decoded shape) to be stuck in an enclosed volume of a shape. At a high level, these tiles ensure three properties: (1) each tile is, at a minimum, connected to its neighbor in an encoding, (2) shape tiles are connected to all adjacent shape tiles with strength 2, and (3) before any tile is added to a new slice, if the tile in the same $x,y$ coordinates of the prior slice is a filler tile, that filler tile must be removed before placement of the next tile occurs. While we demonstrate how these properties are carried out in the current section, we prove their correctness in Section 3.3.8.

Before continuing, we first provide an example shape (and its encoding) which will be used throughout the remainder of this section. Figure 45 demonstrates the encoding of the shape provided in Figure 46. This shape and the encoding of the shape are used throughout the section as an example.

We demonstrate the set of tiles which create a base, the initial set of tiles which, when combined with an encoding of a shape, nucleate growth and serve as a foundation for the remainder of the growth process. We note that this encoding in a rectangular plane is convenient for our purposes (and prevents a massive increase in the number of tiles and signals required), however this entire process could be completed with only ‘0’ and ‘1’ tiles encoded in a line.

This encoding supertile begins with a strength 2 $g_{0}$ glue exposed, allowing for the tile $t_{0}$ to bind (Figure 47). Once $t_{0}$ is bound, it begins the process of growing the base by activating signals which cause uniquely mapped tiles to bind with the purpose of finding the width of the shape, demonstrated in Figure 48.

We first determine the width of the shape. Since each row alternates direction, we can utilize this information to construct a set of tiles which are able to identify the width of the base required for decoding. A set of counting tiles are able to add tiles to the existing supertile which define a base the width of the shape. This counting process operates by cooperatively adding one tile to attach to the width-detection tiles. The first row is able to utilize signals passed through the unique tiles which initiated growth to cause the addition of at least a width-1 base (Figure 49). We note that the encoding will be guaranteed to contain more than two tiles in any row due to the tiles added in the process of leader election. The tile encoding the second location of the base then activates a strength 2 glue which allows for the binding of a counting tile (Figure 50). This tile enables cooperative growth along the edge of the currently exposed counting tiles.

Once the counting tiles reach the end of the existing growth, one of two possible counting detectors is able to bind to the new growth of counting tiles and the encoding structure (Figure 51). The two counting detectors have glues which sense either a ‘0’ direction tile or a ‘1’ direction tile. Since the initial row is of direction ‘0’, the counting process will be sent a signal along the new growth to both extend the width of the base by 1 tile and dissolve the prior placed counting detectors into size 1 junk in order to allow for the counting process to repeat (Figure 52). Otherwise, if a direction ‘1’ tile is sensed, we have found the beginning of the second row and can terminate the counting process. Once this counting process is completed, we activate glues on the initial base tiles to cooperatively fill in the remainder of the tiles (Figure 53). The cooperative filling is determined to be complete by the binding of a base completion gadget (Figure 54), returning a signal to $t_{0}$ that causes another set of signals to be propagated that enable the placement of a base tile in the $+z$ direction.

Once the base is complete, a signal is sent to begin the decoding process of the first row. Figure 55 demonstrates how this signal allows for a strength 2 glue to be exposed in the $+y$ axis, allowing for a base tile to generate cooperative binding on top of the first directional tile. Unlike other directional tiles, the directional tile of first tile of the first row encodes the information that a row change tile is to be utilized, without the need for sensing the directional tile prior (as there is no prior directional tile). Once the directional tile binds, it then activates a glue allowing for the cooperative binding of a decoder tile that determines if the origin tile is a shape or fill tile. Additionally, this binding causes a signal to be passed backwards through the base tile most recently placed such that it initiates the growth of a backing tile. Backing tiles serve two main purposes; first, to indicate to tiles of the first slice that they are adjacent to an exterior edge, and any shape tile must encode exterior glues on its $-z$ face. Second, backing tiles allow for the tiles in the topmost row of a slice to bind along their top edge with strength 2 connections. The process by which this second item proceeds is outlined in Section 3.3.6.

Once the decoder tile determines which type is to be placed, a glue is exposed in the $+x$ direction to enable growth of the decoding tiles. Due to the current decoding tile being the first tile of the row, we can guarantee that at this point a neighbor detection gadget must bind to the recently placed backing tile and the decoding tile (Figure 56). This binding of the neighbor detection gadget with the backing tile additionally causes the backing tile to activate a glue allowing for cooperative binding of another backing tile with the base in the $+x$ direction. The decoding tile now contains all the information regarding the tile type to place after binding with the neighbor detection gadget. A strength 2 glue allows for the growth of an additional decoder tile (mapping to the tile type indicated in the encoding assembly); this enables cooperative binding of the tile type mapped between itself and the base tiles (Figure 57). After the base tile and decoded tile of the shape are connected with strength 2, signals are sent back through the decoder tiles towards the directional tile which initiated growth. Upon passing this signal to the decoder tile’s predecessor, all decoder tiles not bound to the directional tile dissolve into size 1 junk (Figure 58). The decoder tile adjacent to the directional tile activates a glue indicating for the next directional tile to be placed, thus allowing for the placement of an additional decoding tile.

Tile additions continue also utilizing the direction change decoding demonstrated in Section 3.3.1, until the final tile of the row is reached. At this point growth continues by the standard process of directional tiles allowing cooperative binding with the encoding structure, however switched to direction ‘1’ growth. In order to enable the placement of encoding tiles via direction ‘1’ growth, the backing tiles must be present in the new row to allow for binding of neighbor detection gadgets. A backing growth detector (see Figure 60)binds to the most recently placed backing tile and the base (or backing) tile in the row prior. Binding of the backing growth detector allows for a strength 2 glue to be turned on to enable the growth of a backing tile in the $+y$ direction (Figure 61).

For each even numbered row, tiles grow in the ‘-’ direction; that is, the first tile in the some row 2n is placed above the last tile of the prior row (2n - 1) For row 0 growth, each additional tile placed took us further away from the origin point (e.g., incrementing the $x$ value in the ($x,\>y,\>z$) position tuple). In the case of ‘-’ direction growth, tiles of the slice are placed at the furthest-most $x$ value of the slice and decrement to 0. While the decoding tiles of the first row bind initially to decoding tiles, the most recently placed tile and directional tiles, the decoding tiles of ‘-’ direction growth cooperatively bind with the prior decoding tile and a base tile. Growth occurs in two cases; in the case of the first tile of a row of direction 1 growth, tiles bind until they reach the furthest-most base tile. When reaching the outermost base tile, a direction ‘1’ detection gadget binds with the outermost base tile and the furthest placed decoding tile (Figure 62). At this point, a glue is activated on the decoding tile’s $+y$ face, allowing for cooperative growth to continue. This allows for cooperative growth along the previously placed tiles until no longer possible, at which point a neighbor detection gadget is able to bind to the decoding tile and the neighbor tile (in this case a backing tile, see Figure 63).

Similarly, this allows for both the placement of the encoded tile and the extension of the backing tiles; upon the placement of the encoded tiles, a signal is sent to dissolve all decoding tiles not involved in growth in the $+y$ direction into size 1 junk. The next directional tile is added, allowing for the binding of the next decoding tile and the growth to place the tile dictated by the encoding structure. To sense when the growth of the decoding tiles in the $+x$ direction has reached its furthest-most point, the remaining decoding tile which originally redirected growth in the $+y$ direction enables a glue similar to that present on the direction ‘1’ detector gadget. We note this does not cause interactions between multiple encoding processes going on in parallel, as the presence of the base tiles and the directional row offset any possible growing decoding tile (Figure 64). Once the neighbor detector gadget binds, it grows in the $+y$ direction and places its encoded tile (Figure 65). This repeats until all tiles of the row have been added.

At the end of this row, the backing tiles must grow in the $+y$ direction again. For row 2, the current backing gadget will not work as there exists a base tile hindering growth (which is necessary for future signals to be sent). A modified, one-tile gadget is utilized for this specific case. Additionally, once the row is complete after the placement of a direction change tile, all remaining decoding tiles are dissolved into size 1 junk allow for growth of direction ‘0’ tiles of the following layer.

While growth of row 1 was in direction ‘0’, it is a special case due to the fact that it placed tiles in voxels with the same coordinate in the $y$ axis as the decoding tiles by a set of tiles unique to the first row. For remaining odd-numbered rows, we must carry out a similar growth in the $+y$ direction before as placing the encoded tile as demonstrated by the row 2 growth example, but incrementing $x$ values. We note that the example figures in this section do not directly correspond to the encoding provided in Figure 45, however these are presented to provide the reader with examples of how this process would occur in an encoding which does contain at least 3 rows. Decoding tiles of some odd valued row grow by cooperatively binding with the decoding tile and previously placed directional tiles, as with the row 1 tiles. However, upon binding with a shape or a fill tile they activate a glue in the $+y$ direction. This glue attempts to allow for growth of decoding tiles in the $+y$ direction, leading to the binding of a neighbor detection gadget and the placement of the encoded tile (Figures 67, 68). Similarly to even numbered direction ‘1’ row growth, decoding tiles are dissolved into size 1 junk to allow for reuse of voxels. In contrast, all but the bottom-most decoding tile are removed, and glues are activated allowing remaining decoding tiles to sense that a tile has already been placed in the current location (Figure 69). In the case when the decoding tile activates its glue in the $+y$ direction and binds to a tile, it continues growth in the $+x$ direction until finding an open location to grow (Figure 70).

Once the directional tiles reach the end of the encoding of the final row within the structure, a slice completion gadget binds to the end of the encoding and the directional tile. At this point, a message is returned through the current row of directional tiles which enabled growth of the slice (Figure 71). Once the message is received by the first directional tile, it carries out two operations - the first being unique to the first slice. In order for the growth of the next slice, we must be able to guarantee the shape tiles in the slice are connected to either the shape which has grown, or are connected to the newly growing slice. To guarantee connection of all tiles of the first slice persist even after filler tile removal, we must create strength 2 connections between the encoding structure and all tiles of the first slice. This is accomplished by extending the growth of the backing tiles, which allows for all tiles to be connected via strength 2 to the encoding structure. The message is sent through the base tile which initiated growth of the first slice, into the adjacent backing tiles. After backing tiles receiving the message, strength 2 glues are activated on all the $+y$ direction faces of the currently placed backing tiles. Only the topmost layer of backing tiles will allow for cooperative placement of the new backing tiles on top of the newly created slice. The newly placed backing tiles opens up cooperative binding locations for the backing tiles to then bind with the top row of the slice (Figure 72). This allows for the tiles in the topmost row of the slice to activate glues for binding to their neighbor in the $-y$ direction. Once bound to the neighbor in the $-y$ direction, the tiles are then able to activate glues which allow for neighbor detection gadgets to bind, allowing for the growth of a new slice.

In addition to the growth of the backing tiles, a signal is sent to place a new directional tile. This directional tile takes the information of the first row of directional tiles and cooperatively binds with both 0 and 1 tiles on the encoding structure; its purpose is to simply pass forward the directional information and allow for the tile placement process to continue in the next slice. In addition to the directional tile exposing a directional glue, we also expose a terminating glue ($g_{t}$) which is used in the detection of the completion of the final slice. Once the growth of the new directional tile occurs alongside the creation of the top row of backing tiles, growth of the new slice can begin with starting conditions shown in Figure 73.

Slice growth proceeds via the previously described process until reaching the final slice. Once the final slice is placed, a slice completion gadget binds allowing for the placement of a directional tile, as per any other row. However, the exposed terminating glue allows for the attachment of the decoder completion detector with the outermost edge of the encoding structure (Figure 74). Upon binding of the decoder completion detector, a glue is activated to allow for the growth of decoder completion tiles which cooperatively bind to the outermost slice layer. Binding of the decoder completion tiles occurs such that only attachments between shape tiles activate glues for cooperative growth, and filler tiles must form a strength 2 duple with the decoder completion tiles. Once bound as a duple, the filler tiles send glue deactivation signals to their remaining active glues.

Once a decoder completion tile binds with the outermost backing tile above the top row of a slice, it sends a dissolve message to all the base and backing tiles in the same $yz$ plane (Figure 75) to turn them into size 1 junk. The base tiles, upon receiving this dissolve message, also initiate a message to dissolve the remaining tiles placed as part of the assembly sequence into size 1 junk, including the initial binding tile $t_{0}$. The initial binding tile then signals to the encoding structure to dissolve into size 1 junk, and the only terminal assembly remaining is the shape assembly produced by the decoding process.

Here we briefly summarize the decoding process and show that during this process, the shapes which were encoded in the set of input encoding assemblies $\Phi$ are correctly assembled. We first consider the decoding process of a single encoding assembly $\phi\in\Phi$ and note that a similar process happens for all encoding assemblies simultaneously without interfering with one another.

Our decoding process begins by building a base of tiles connected to $\phi$. This base holds the shape as it’s being constructed and is used to help ensure the connectivity of the shape as it’s being constructed. The decoding process is performed in iterations, where during each iteration a row of $\phi$ is scanned tile-by-tile and a corresponding 2D slice of the shape is constructed. Each slice is constructed starting from the bottom (smallest $y$ coordinate) to the top (largest $y$ coordinate), with tiles attaching in a zig-zag manner, as illustrated in Figure 21. Each slice of the assembled shape corresponds to a unique $z$ coordinate so for convenience we call the slice whose $z$ coordinate is $i$, $\sigma_{i}$. As each slice is assembled, tiles are placed in each location of the slice, even those locations that will not be part of the final shape, though these will be removed during the assembly of the next slice.

The first slice $\sigma_{1}$ can be assembled naively, but during the assembly of each following slice, tiles which will not be part of the final shape on the previous slice must be removed. This is done as follows. Suppose that slice $\sigma_{i}$ ($i>1$) is currently being assembled. Before a tile $t_{i}$ is placed in a location $(x,y,i)$, a gadget is used to determine the type of the tile $t_{i-1}$ at location $(x,y,i-1)$ (i.e. the tile with the same $x$ and $y$ coordinates on the previous slice). If this $t_{i-1}$ is part of the final shape, then $t_{i}$ is placed and signals are used to activate strength 2 glues between $t_{i}$ and $t_{i-1}$; otherwise, if $t_{i-1}$ is not part of the final shape, it is removed before $t_{i}$ is placed. Regardless of the type of tile $t_{i-1}$, when $t_{i}$ is placed, glues are activated which connect $t_{i}$ to all adjacent tiles on the same slice. Once the final slice is assembled, a final zig-zag pass is made in the next $z$ coordinate which removes all tiles from the last slice which are not part of the final shape.

It is also important to note that the base, on which the shape is being assembled, also forms a ceiling above the slices being assembled. This ceiling helps ensure that tiles on the top row of each slice are able to remain attached to the assembly during construction. It should be clear that during this decoding process (1) each tile that belongs to the final shape is placed in its correct location, and (2) that those tiles of a slice which are not part of the final shape will be removed from the assembly during the assembly of the next slice. However, because tiles are removed during the process, we must show that none of these removals can cause parts of the assembly to unintentionally detach. We state this as Lemma 3.4.

From here, it’s clear that the assembly of the slices of the shape can complete without erroneous detachment. Since all tiles that are part of the final shape assembly have been added during the slice construction and since all tiles which are not part of the final shape assembly have been removed from their respective slices, it’s clear that the decoding process successfully assembles our final shape assembly.

Given the set of input encoding structures $\Phi=\{\phi_{1},\ldots,\phi_{n}\}$, the STAM^R system $\mathcal{D}_{\Phi}=\{D,\Sigma_{\Phi},\tau=2\}$ produces a set of terminal supertiles $S=\{s_{1},\ldots,s_{n}\}$ in parallel with a maximum junk size of 3. $\mathcal{D}_{\Phi}$ finitely completes, as for the production of the set of shapes $s\in S$ from input encoding structures $\Phi$ a finite number of tiles are required for each encoding structure to produces a terminal assembly. We can guarantee this as each encoding produce a single terminal shape, as the encoding of the shape dissolves into size 1 junk after the terminal shape has decoded. By our construction, there are never exposed glues on the surfaces of any pair of assemblies that each contain an input encoding that would allow them to bind to each other. Since junk assemblies produced by any assembly sequence are also unable to negatively interact with other assemblies, a system whose input assemblies have multiple shapes will behave simply as the union of individual systems which each have one input assembly shape, creating terminal assemblies of all of (and only) the correct shapes. This proves Lemma 3.3.

Now that we have shown the existence of universal encoding and universal decoding tilesets, we have the basis to demonstrate a universal shape replicator. We generate a new STAM^R tileset $R=E\cup D$ and STAM^R system $\mathcal{R}_{S}=\{R,\Sigma_{S},\tau=2\}$, where $\Sigma_{S}$ consists of an infinite number of copies of each tile type from $R$ and an infinite number of copies of each uniformly covered assembly from the set $S=\{s_{1},\ldots,s_{n}\}$, whose shapes are any arbitrary set of shapes.

Recall that during the encoding process, the encoding corner gadget is bound to the encoding structure while it is being built. Once the entire encoding process finishes and the corner gadget receives a ’dissolve’ signal, it first activates a glue to signal to the first tile placed in the encoding structure that it should turn on the initiator glue which is the glue initially bound to by the tiles of $D$. Thus, exactly when an encoding of some $s_{i}$, $\phi_{i}$, is completed by the tiles of $E$, decoding that $\phi_{i}$ will begin by the tiles of $D$, resulting in a terminal assembly with the same shape as $s_{i}$. We make a slight modification to the tile of the encoding structure that exposes the initiator glue, and the initial decoding tile which attach to it, the initiator tile. We make two copies of the initiator tile, which we will call $t_{1}$ and $t_{2}$. The first, $t_{1}$, will bind to the initiator glue and cause the decoding process to proceed exactly as before. However, when the original initiator tile would have detected completion of the decoding process and sent a ‘dissolve’ signal to the first tile of the encoding structure, $t_{1}$ instead sends a signal that tells that tile to activate a glue that will allow $t_{2}$ to attach, and then $t_{1}$ will detach. This will effectively cause the encoding to produce a decoded structure and then have all of the ‘helper’ tiles dissolve, leaving the encoding structure able to bind to $t_{2}$ which then initiates the regular decoding process, and when it receives the signal telling it that has completed, $t_{2}$ does pass the ‘dissolve’ signal to the first tile of the encoding structure. In this way, each encoding structure causes two copies of the decoded assembly to be produced, and then dissolves.

By our construction, the only glues required to be shared between the two tilesets are the glues encoding 1 and 0 on the encoding structure, and the previously mentioned glues on the encoded assembly which initiate the decoding process. The glues for 0/1 are shared by multiple tiles in both $E$ and $D$. All tiles in $D$ which have the the 0/1 glue (or its complement) are required to be placed by cooperation with a non 0/1 glue. Additionally, each tile in $D$ has at most one face which contains strength 1 0/1 glue. Since no other glues are shared between $E$ and $D$ it is not possible for strength 2 binding to occur between (super)tiles in $E$ and $D$ aside from the binding of $\phi$ with the initiator tiles of $D$. Since junk assemblies produced by any assembly sequence are also unable to negatively interact with other assemblies, a system whose input assemblies have multiple shapes will behave simply as the union of individual systems which each have one input assembly shape, creating terminal assemblies of all of (and only) the correct shapes.

The maximal junk size of $R$ is 4, driven by the junk size of $E$. We can say that $\mathcal{R}_{S}$ finitely completes with respect to the set of assemblies created from the shape tiles of $D$ in the shape of each assembly in $S$, as the tileset $R$ operates such that any input shape $s_{i}$ is encoded into an intermediate structure $\phi_{i}$, $\phi_{i}$ is then decoded into two copies of $s^{\prime}_{i}$, an assembly which contains tiles in the exact same locations as $s$ (up to rotation and translation). As deconstruction leads to the production of a single structure $\phi_{i}$, and $\phi_{i}$ is only able to be decoded to $s^{\prime}_{i}$ two times, we can place a finite bound on the number of each tile type required to produce each terminal assembly $s^{\prime}$. (This largely follows from the fact that encoding systems using $E$ finitely complete with respect to the set of encoding assemblies, and that decoding systems using $D$ finitely complete with respect to the set of assemblies whose shapes are encoded.) Therefore, $R$ also finitely completes, with respect to the set of assemblies with the same shape as the input assemblies, and Theorem 3.1 is proven.

Note that the condition that a single encoding structure $\phi_{i}$ leads to the production of exactly two target assemblies $s^{\prime}_{i}$ is imposed to allow for the universal shape replicator to technically be able to replicate shapes from an arbitrarily large set of input assembly shapes without the potential to ‘starve’ the encodings of one shape so that they never produce decoded copies (and thus the replicator would not finitely complete with respect to the full set of terminal assembly shapes). If only one input assembly shape was provided as input, it would instead be possible to just remove the dissolve signals from the encoding structure and allow each to initiate the production of an unbounded number of decoded copies. It would also be trivial to add tiles that make copies of the encoded structures that can each initiate the decoding process, leading to exponential replication.