A general architecture of oritatami systems for simulating arbitrary finite automata
AA general architecture of oritatami systems forsimulating arbitrary finite automata
Yo-Sub Han, Hwee Kim, Yusei Masuda, and Shinnosuke SekiApril 24, 2019
Abstract
In this paper, we propose an architecture of oritatami systems withwhich one can simulate an arbitrary nondeterministic finite automaton(NFA) in a unified manner. The oritatami system is known to be Turing-universal but the simulation available so far requires 542 bead types and O ( t log t ) steps in order to simulate t steps of a Turing machine. Thearchitecture we propose employs only 329 bead types and requires just O ( t | Q | | Σ | ) steps to simulate an NFA over an input alphabet Σ with astate set Q working on a word of length t . Figure 1: RNA origami, a novel self-assembly technology by cotranscriptionalfolding [6]. RNA polymerase (orange complex) attaches to an artificial templateDNA sequence (gray spiral) and synthesizes the complementary RNA sequence(blue sequence), which folds into a rectangular tile while being synthesized.
Transcription (Figure 1) is a process in which from a template DNA se-quence, its complementary RNA sequence is synthesized by an
RNA polymerase letter by letter. The product RNA sequence ( transcript ) is folding upon itselfinto a structure while being synthesized. This phenomenon called cotranscrip-tional folding has proven programmable by Geary, Rothemund, and Andersenin [6], in which they programmed an RNA rectangular tile as a template DNAsequence in the sense that the transcript synthesized from this template folds1 a r X i v : . [ c s . D M ] A p r otranscriptionally into that specific RNA tile highly probably in vitro . As co-transcriptional folding has turned out to play significant computational roles inorganisms (see, e.g., [9]), a next step is to program computation in cotranscrip-tional folding. Oritatami is a mathematical model proposed by Geary et al. [4] to un-derstand computational aspects of cotranscriptional folding. This model hasrecently enabled them to prove that cotranscriptional folding is actually Turinguniversal [5]. Their Turing-universal oritatami system Ξ TU adopts a periodictranscript whose period consists of functional units called modules . Some ofthese modules do computation by folding into different shapes, which resemblessomehow computation by cotranscriptional folding in nature [9]. Being thusmotivated, the study of cotranscriptional folding of shapes in oritatami wasinitiated by Masuda, Seki, and Ubukata in [8] and extended independently byDomaine et al. [2] as well as by Han and Kim [7] further. In [8], an arbitraryfinite portion of the Heighway dragon fractal was folded by an oritatami systemΞ H . The Heighway dragon can be described as an automatic sequence [1], thatis, as a sequence producible by a deterministic finite automaton with output(DFAO) in an algorithmic manner. The system Ξ HD involves a module thatsimulates a 4-state DFAO A HD for the Heighway dragon. The Turing-universalsystem was not embedded into Ξ HD in place for this module primarily becauseit may fold into different shapes even on inputs of the same length and secondlybecause it employs unnecessarily many 542 types of abstract molecules ( bead )along with an intricate network of interactions (rule set) among them. Theirimplementation of the DFAO module however relies on the cycle-freeness of A H too heavily to be generalized for other DFAs; let alone for nondeterministic FAs(NFAs).In this paper, we propose an architecture of oritatami system that allows forsimulating an arbitrary NFA using 329 bead types. In order to run an NFA overan alphabet Σ with a state set Q on an input of length t , it takes O ( t | Q | | Σ | )steps (stabilization of this number of beads). In contrast, the system Ξ TU requires O ( t log t ) steps to simulate t steps of a Turing machine. A novelfeature of technical interest is that all the four modules of the architectureshare a common interface ((2) in Section 3). Let B be a set of types of abstract molecules, or beads , and B ∗ be the set offinite sequences of beads including the empty sequence λ . A bead of type b ∈ B is called a b -bead. Let w = b b · · · b n ∈ B ∗ be a sequence of length n for someinteger n and bead types b , . . . , b n ∈ B . For i, j with 1 ≤ i, j ≤ n , let w [ i..j ]refer to the subsequence b i b i +1 · · · b j of w ; we simplify w [ i..i ] as w [ i ].The oritatami system folds its transcript, which is a sequence of beads, overthe triangular grid graph T = ( V, E ) cotranscriptionally based on hydrogen-bond-based interactions ( h-interaction for short) which the system allows for A periodic transcript is likely to be able to be transcribed from a circular DNA [3]. bound . The i -th point of adirected path P = p p · · · p n in T is referred to as P [ i ], that is, P [ i ] = p i . A (finite) conformation C is a triple ( P, w, H ) of a directed path P in T , w ∈ B ∗ of the same length as P , and a set of h-interactions H ⊆ (cid:8) { i, j } (cid:12)(cid:12) ≤ i, i +2 ≤ j, { P [ i ] , P [ j ] } ∈ E (cid:9) . This is to be interpreted as the sequence w being foldedin such a manner that its i -th bead is placed at the i -th point of the path P and the i -th and j -th beads are bound iff { i, j } ∈ H . A symmetric relation R ⊆ B × B called rule set governs which types of two beads can form an h-interaction between. An h-interaction { i, j } ∈ H is valid with respect to R , or R -valid , if ( w [ i ] , w [ j ]) ∈ R . A conformation is R -valid if all of its h-interactionsare R -valid. For α ≥
1, a conformation is of arity α if it contains a bead thatforms α h-interactions and none of its beads forms more. By C ≤ α , we denotethe set of all conformations of arity at most α .An oritatami system grows conformations by elongating them according toits own rule set R . Given an R -valid finite conformation C = ( P, w, H ), wesay that another conformation C is its elongation by a bead of type b ∈ B ,written as C R −→ b C , if C = ( P p, wb, H ∪ H (cid:48) ) for some point p not alongthe path P and possibly-empty set of h-interactions H (cid:48) ⊆ (cid:8) { i, | w | + 1 } (cid:12)(cid:12) ≤ i < | w | , { P [ i ] , p } ∈ E, ( w [ i ] , b ) ∈ R (cid:9) . Observe that C is also R -valid. Thisoperation is recursively extended to the elongation by a finite sequence of beadsas: C R −→ ∗ λ C for any conformation C ; and C R −→ ∗ wb C for conformations C , C ,a finite sequence of beads w ∈ Σ ∗ , and a bead b ∈ Σ if there is a conformation C (cid:48) such that C R −→ ∗ w C (cid:48) and C (cid:48) R −→ b C .A finite oritatami system is a tuple Ξ = ( R, α, δ, σ, w ), where R is a ruleset, α is an arity, δ ≥ delay , σ is an R -valid initialconformation of arity at most α called seed , upon which its finite transcript w ∈ B ∗ is to be folded by stabilizing beads of w one at a time so as to minimizeenergy collaboratively with its succeeding δ − energy of aconformation C = ( P, w, H ), denoted by ∆ G ( C ), is defined to be −| H | ; thatis, more h-interactions make a conformation more stable. The set F (Ξ) ofconformations foldable by this system is recursively defined as: the seed σ is in F (Ξ); and provided that an elongation C i of σ by the prefix w [1 ..i ] be foldable(i.e., C = σ ), its further elongation C i +1 by the next bead w [ i +1] is foldable if C i +1 ∈ arg min C ∈C ≤ α s.t.C i R −→ w [ i +1] C min (cid:110) ∆ G ( C (cid:48) ) (cid:12)(cid:12)(cid:12) C R −→ ∗ w [ i +2 ...i + k ] C (cid:48) , k ≤ δ, C (cid:48) ∈ C ≤ α (cid:111) . (1)We say that the bead w [ i +1] and the h-interactions it forms are stabilized (notnascent any more) according to C i +1 . Note that an arity- α oritatami systemcannot fold any conformation of arity larger than α . The system Ξ is deter-ministic if for all i ≥
0, there exists at most one C i +1 that satisfies (1). Anoritatami system is cyclic if its transcript admits a period shorter than the halfof itself. 3
587 588
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581 579 ⇒ ⇒ ⇒
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Figure 2: Growth (folding) of a spacer of glider shape(g-spacr). The rule set R to fold this is { ( , ),( , ) , ( , ) , ( , ) , ( , ) , ( , ) , ( , ) } . Example . Let us provide an example of deterministic oritatamisystem that folds into a glider motif, which will be used as a component called g-spacer in Section 3. Consider a delay-3 oritatami system whose transcript w is a repetition of − − · · · − and rule set R is as captioned in Figure 2.Its seed, colored in red, can be elongated by the first three beads w [1 ..
3] = − − in various ways, only three of which are shown in Figure 2 (left).The rule set R allows w [1] to be bound to , w [2] to , and w [3] to , but -bead is not around. In order for both w [2] and w [3] to be thus bound, thenascent fragment w [1 ..
3] must be folded as bolded in Figure 2 (left). Accordingto this most stable elongation, the bead w [1] = is stabilized to the eastof the previous -bead. Then w [4] = is transcribed. It is capable ofbinding to a -bead but no such bead is reachable, and hence, this newly-transcribed bead cannot override the “bolded” decision. Therefore, w [2] is alsostabilized according to this decision along with its bond with the -bead. Thenext bead w [5] = cannot override the decision, either, and hence, w [3] isstabilized along with its bond with the -bead as shown in Figure 2 (right). We shall propose an architecture of a nondeterministic cyclic oritatami system Ξthat simulates at delay 3 an NFA A = ( Q, Σ , q , Acc, f ), where Q is a finite set ofstates, Σ is an alphabet, q ∈ Q is the initial state, Acc ⊆ Q is a set of acceptingstates, and f : Q × Σ → Q is a nondeterministic transition function. What thearchitecture actually simulates is rather its modification A $ = ( Q ∪ { q Acc } , Σ ∪{ $ } , q , { q Acc } , f ∪ f $ ), where f $ : ( Q ∪{ q Acc } ) × (Σ ∪{ $ } ) → Q ∪{ q Acc } is definedover Σ exactly same as f , and moreover, f $ ( q, $) = { q Acc } for all q ∈ Acc . Notethat w ∈ L ( A ) iff w $ ∈ L ( A $ ). For the sake of upcoming arguments, we regardthe transition function f ∪ f $ rather as a set of transitions { f , f , . . . , f n } , where f k is a triple ( o k , a k , t k ), meaning that reading a k in the state o k (origin) causesa transition to t k (target). Note that n = O ( | Q | | Σ | ).The architecture assumes that each state q is uniquely assigned with an n -bitbinary sequence, whose i -th bit from most significant bit (MSB) is referred to4igure 3: Encoding of the initial state q = 01 and the first letter b = 10 of aninput word on the seed of Γ-shape.as q [ i ]. It also assumes a unique m -bit binary sequence for each letter a ∈ Σ,whose (cid:96) -th bit from MSB is referred to as a [ (cid:96) ], where m = (cid:100) log | Σ |(cid:101) . Seed.
The seed of Ξ is of Γ shape as shown in Figure 3. Below its horizontalarm is encoded the initial state q in the following format: (cid:74) nk =1 (cid:0) x f k → ( → → ) z q [ k ] → ( → → ) (cid:1) → , (2)where z = → → → → → , z = → → → → → ,and for some bead types b, c ∈ B , the arrow b → c (resp. (cid:37) , (cid:45) , ← , (cid:46) , (cid:38) ) im-plies that a c -bead is located to the eastern (resp. north-eastern, north-western,western, south-western, and south-eastern) neighbor of a b -bead. Note thatthe seed and Module 4, which we shall explain soon, initialize all the variables f , . . . , f n to N by having a sequence x N of bead types be exposed to the cor-responding positions x f , · · · , x f n , where x N = z while x Y = z , which is notused here but shall be used later. An input word u = b b · · · is encoded on theright side of its vertical arm as: | u | (cid:75) j =1 (cid:18) ( y sp (cid:46) ) | f |− (cid:74) m(cid:96) =1 (cid:0) y b i [ (cid:96) ] (cid:46) y sp (cid:46) (cid:1) ( y sp (cid:46) ) | f | (cid:19) , (3)where y sp = y = (cid:46) (cid:46) (cid:46) (cid:46) (cid:46) and y = (cid:46) (cid:46) (cid:46) (cid:46) (cid:46) . 5he first period of the transcript of Ξ starts folding at the top left cornerof the seed, or more precisely, as to succeed its last bead circled in blue inFigure 3. It folds into a parallelogram macroscopically by folding in a zigzagmanner microscopically (zig ( (cid:44) → ) and zag ( ← (cid:45) ) are both of height 3) while read-ing the current (initial) state q from above and the first letter b from left,and outputs one of the states in f ( q , b ), say q , nondeterministically belowin the format (2). All the states in f ( q , b ) are chosen equally probably. Thefolding ends at the bottom left corner of the parallelogram. The next periodlikewise reads the state q and next letter b , and outputs a state in f ( q , b )nondeterministically below the parallelogram it has folded. Generally speaking,for i ≥
2, the i -th period simulates a nondeterministic transition from the state q i − , output by the previous period, on the letter b i . Modules.
One period of the transcript is semantically factorized into fourfunctional subsequences called modules. All these modules fold into a parallel-ogram of width Θ( n ) and of respective height 6 × n , 6 × m , 6 ×
2, and 6 × n (recall one zigzag is of height 6); that is, the first module makes 2 n zigzags, forexample. These parallelograms pile down one after another. One period thusresults in a parallelogram of width and height both Θ( n ). Their roles are asfollows: Module 1 extracts all the transitions that originate at the current state;
Module 2 extracts all the transitions that read the current letter among thosechosen by Module 1;
Module 3 nondeterministically chooses one state among those chosen by Mod-ule 2, if any, or halts the system otherwise;
Module 4 outputs the chosen state downward.In this way, these modules filter candidates for the next transition, and impor-tantly, through a common interface, which is the format (2).
Let us assume that the transcript has been folded up to its ( i − q i − below, and thenext period reads the letter b i . See Figures 4, 12, 15, and 17 for an examplerun. Bricks.
All the modules (more precisely, their transcripts) consist of func-tional submodules. Each submodule expects several surrounding environments.A conformation that the submodule takes in such an expected environment iscalled a brick [5]. On the inductive assumption that all the previous submoduleshave folded into a brick, a submodule never encounters an unexpected environ-ment, and hence, folds into a brick. Any expected environment exposes 1-bit60110011 10001111 f / f / f / f /
101 1 0 1 1Current state q i − f f f f f = Nf = Y f = Y f = Y Figure 4: Example run of the proposed architecture. (Left) A 4-state FA with4 transitions f , f , f , f to be simulated, which is obtained from a 3-state FAwith the 2 transitions f and f in the way explained in the text, that is, byadding a new accepting sink state 0011 and transitions f , f on the specialletter $, encoded as 101. (Right) Outline of the flow of 1-bit information ofwhether each of the transitions originates from the current state 1011 or notthrough Module 1.information b below and a submodule enters it in such a manner that its firstbead is placed either 1-bead below y ( starting at the top ) or 3-beads below y ( atthe bottom ). Hence, in this paper, a brick of a module X is denoted as X − hy ,where h ∈ { t , b } indicates whether this brick starts at the top or bottom and yis the input from above. Spacers and Turners.
The transcript of a zig or a zag is a chain of submod-ules interleaved by a structural sequence called a spacer . A spacer keeps twocontinuous submodules far enough horizontally so as to prevent their undesir-able interaction. We employ spacers that fold into a parallelogram (p-spacer)or glider (g-spacer) of height 3. For a g-spacer, see Figure 2. They start andend folding at the same height (top or bottom) in order to propagate 1-bit ofinformation. The spacer and its 1-bit carrying capability are classical, foundalready in the first oritatami system [4].After a zig is transcribed a structural submodule called turner . Its role is tofold so as to guide the transcript to the point where the next zag is supposedto start. A zag is also followed by a turner for the next zig. Some turners playalso a functional role.
Module 1 (origin state checker) folds into 2 n zigzags. Recall that all the n variables f , f , . . . , f n have been set to N by the seed or Module 4 in theprevious period. The (2 k − o k of the k -th7igure 5: (Left) The two bricks of P zig , that is, P zig − and P zig − . (Right)The two bricks of P zag , that is, P zag − and P zag − .Figure 6: The four bricks of A (cid:48) , that is, A (cid:48) Nb , A (cid:48) Yb , A (cid:48) Nt , and A (cid:48) Yt .transition f k is equal to q i − or not, and if so, it sets the variable f k to Y .Every other zigzag (2nd, 4th, and so on) just formats these variables as wellas the z -variables (for the current state in (2)) using two submodules P zig and P zag (see Figure 5 for their bricks); this is a common feature among all the fourmodules. The transcript for such a formatting zig (resp. zag) is a chain of 2 n instances of P zig (resp. P zag ), unless otherwise noted.The transcript for the (2 k − (cid:12) nj =1 ( A (cid:48) A o k [ j ] )for submodules A (cid:48) , A , A . See Figures 6, 7, and 8 for the four bricks of thesesubmodules, respectively. The zig starts folding at the bottom. The n instancesof A (cid:48) propagate f , . . . , f n downward using the four bricks, all of which endfolding at the same height as they start. A o k [ j ] checks whether o k [ j ] = q i − [ j ]or not when it starts at the bottom; it ends at the bottom if these bits are equal,or top otherwise. Starting at the top, it certainly ends at the top. In any case,it propagates q i − [ j ] downward. The zig thus ends at the bottom iff o k = q i − .Figure 7: The four bricks of A , that is, A − , A − , A − , and A − A , that is, A − , A − , A − , and A − .Figure 9: The four bricks of B , that is, B , B , B , and B .The succeeding turner admits two conformations to let the next zag start eitherat the bottom if o k = q i − , or top otherwise.The transcript for the next zag is B n − k +1 B (cid:48) B k − for submodules B (seeFigure 9 for its bricks) and B (cid:48) . It is transcribed from right to left so that B (cid:48) can read f k . B (cid:48) is in fact just a g-spacer shown in Figure 2. This glider exposesbelow the bead-type sequence - - - if it starts folding at the bottom,or - - - otherwise; the former and latter shall be formatted into x Y and x N , respectively, by the next zigzag.The variables f , . . . , f n are updated in this way and output below in theformat (2) along with the current state q i − , which is not used any more though.Figure 10: The three bricks of C , that is, C −∗ b , C − Nt , and C − Yt .9igure 11: The three bricks of C , that is, C − Nb , C − Yb , and C −∗ t . f = Nf = Y f = Y f = Y b i = f = Nf = Y f = Nf = Y Figure 12: Example run of the oritatami system constructed according to theproposed architecture in order to simulate the FA in Figure 4. Here Module 2filters transitions f , f , f chosen by Module 1 further depending on whethereach of them reads the letter 100 or not; thus f is out. Module 2 (input letter checker) folds into 2 m zigzags; recall m = (cid:100) log | Σ |(cid:101) .The (cid:96) -th bit of the input letter b i is read by the turner between the (2 (cid:96) − (cid:96) − f k reads a k for all 1 ≤ k ≤ n . The (cid:96) -th bit of theseletters is encoded in the transcript for the (2 (cid:96) − C a [ (cid:96) ] C a [ (cid:96) ] · · · C a n [ (cid:96) ] using submodules C and C . All the bricks of C and C start and end at thesame height, as shown in Figures 10 and 11; thus propagating b i [ (cid:96) ] throughoutthe zig. Starting at the top (i.e., b i [ (cid:96) ] = 0), C takes the brick C − Nt if it reads N from above or C − Yt if it reads Y ; these bricks output N and Y downward,respectively; thus propagating the x -variables downward. On the other hand, ifit starts at the bottom (i.e., b i [ (cid:96) ] = 1), C certainly takes C −∗ b and outputs N downward. C propagates what it reads downward by the bricks C − Nb , C − Yb in Figure 11 if b i [ (cid:96) ] = 1 while it outputs N downward by C −∗ t if b i [ (cid:96) ] = 0.Functioning in this way, the submodules C a [ j ] , . . . , C a n [ j ] compare the lettersthat f , . . . , f n read with b i and filter those with unmatching j -th bit out. Thenext zag propagates the result of this filtering downward using B ’s.10igure 13: The four bricks of D : (Top) D − Nb and D − Yb ; (Bottom) D − Nt and D − Yt . Module 3 (nondeterministic choice of the next transition) folds intojust 2 zigzags. Each transition f k = ( o k , a k , t k ) has been checked whether o k = q i − in Module 1 and whether a k = b i in Module 2, and the variable f k isset to Y iff f k passed both the checks, that is, proved valid . The first zig marksthe valid transition with smallest subscript by setting its variable to Y (cid:48) using asubmodule D . This submodule was invented in [8] for the same purpose, andhence, we just mention a property that its four bricks (Figure 13) ensure thiszig to end at the bottom if none of the transition has proven valid. In that case,the succeeding turner is geometrically trapped as shown in Figure 14 and thesystem halts.The transcript for the first zag consists of n instances of a submodule E . TheFigure 14: Turner from the first zig of Module 3 to the first zag. (Left) It istrapped geometrically in the pocket of the previous turner and halts the systemif it starts folding at the bottom. (Right) It is not trapped and lets the nextzag be transcribed. 11 = N f = Y f = N f = YD D D Df = N f = Y (cid:48) f = N f = Y Nondeterministic choice f = N f = Y (cid:48) f = N f = YE E E EP zig P zig P zig P zig P zig P zig P zig P zig P zag P zag P zag P zag P zag P zag P zag P zag N Y N N f = N f = Y (cid:48) f = N f = YE E E EP zig P zig P zig P zig P zig P zig P zig P zig P zag P zag P zag P zag P zag P zag P zag P zag N N N Y Figure 15: Example run of Module 3, in which the transitions that have provedvalid in Modules 1 and 2 ( f and f here) are chosen nondeterministically.five bricks of E are shown in Figure 16. The zag starts folding at the bottom.When an E starts at the bottom and reads Y from above, it folds into the brick E − YbY or E − YbN in Figure 16 nondeterministically, which amounts to choosingthe corresponding valid transition or not. Observe that E − YbY ends at the top,notifying the succeeding E ’s that the decision has been already made. Whenstarting at the top, E takes no brick but E −∗ t , which outputs N no matter whatit reads. The brick E − Y (cid:48) b and Y (cid:48) (marked Y ) prevent the oritatami system fromnot choosing any valid transition; that is, if an E starts at the bottom (meaningthat none of the valid transitions has been chosen yet) and reads Y (cid:48) from above,it deterministically folds into E − Y (cid:48) b , which outputs Y .The transcript of the next zig differs from that of normal formatting zig inthat every other instance of P zig is replaced by a spacer. This replacement allowsthe n instances of P zag responsible for the z -variables to take their “default”brick P zag − , which outputs 0. This is a preprocess for Module 4 to set thesevariables to the target state of the transition chosen. Module 4 (outputting the target state of the transition chosen) foldsinto 2 n zigzags. Its (2 k − f k was chosen or not, and ifit was, the next zag sets z q i [ j ] to t k [ j ] (recall t k is the target of f k ).The transcript for the (2 k − A (cid:48) A (cid:48) ) k − A A (cid:48) ( A (cid:48) A (cid:48) ) n − k . (4)Observe that the sole A is positioned so as to read the 1-bit of whether f k waschosen or not. The zig starts at the bottom. Since A (cid:48) always start and end at12igure 16: The five bricks of E : (Top) E − Nb and E − Y (cid:48) b ; (Bottom) E −∗ t , E − YbY ,and E − YbN . Note that E − YbY and E − YbN are chosen nondeterministically andequally probably.the same height (Figure 6), the A starts at the bottom. It ends at the bottomif it reads Y , or top otherwise (Figure 8). The succeeding turner is functional,which lets the next zag start at the bottom if the previous zig has ended at thebottom, or at the top otherwise. In this way, the (2 k − f k .The transcript for the (2 k − (cid:0)(cid:74) k +1 j = n ( G t k [ j ] B ) (cid:1) G t k [ k ] G (cid:0)(cid:74) j = k − ( G t k [ j ] B ) (cid:1) . (5)All the bricks of submodules G and G (see Figures 18 and 19) start andend at the same height; thus propagating the 1-bit of whether f k was chosenor not (bottom means chosen) through this zag. Note that this transcript istranscribed from right to left so that these G ’s and G ’s read z -variables. G and G just copy what they read downward if they start at the top, that is, inall zags but the one corresponding to the chosen transition. In the “chosen” zag,they rather output 0 and 1 downward, respectively. Comparing (4) with (5), wecan observe that below the sole instance of A is transcribed an instance of G .This G plays a different role from other G ’s in the zag. The A above outputs Y or N depending on whether f k was chosen or not. If it outputs Y , then the(2 k − G that starts at the bottom outputs 0 = N . Otherwise, G just propagates itsoutput 0 = N downward. In this way, all the x -variables are initialized to N for the sake of succeeding period. Using a simulator developed for [8], we have checked for each submodule that itfolds as expected in all the expected environments. An expected environment13 Y N N A A (cid:48) A (cid:48) A (cid:48) A (cid:48) A (cid:48) A (cid:48) A (cid:48) G G B G B G B G A (cid:48) A (cid:48) A A (cid:48) A (cid:48) A (cid:48) A (cid:48) A (cid:48) B G G G B G B G A (cid:48) A (cid:48) A (cid:48) A (cid:48) A A (cid:48) A (cid:48) A (cid:48) B G B G G G B G P zig P zig P zig P zig P zig P zig P zig P zig P zig P zig P zig P zig P zig P zig P zig P zig P zag P zag P zag P zag P zag P zag P zag P zag P zag P zag P zag P zag P zag P zag P zag P zag N N N N Figure 17: Example run of Module 4 (due to the space shortage, the last 3zigzags are omitted). Here the transition f has been chosen so that only thecorresponding (2 × − T or B , which means that the previous brick has ended folding at the topor bottom. Module 1.
The parts of the brick automaton for Module 1 are illustratedin Figures 20, 21, and 22. Recall that the (2 k − B (cid:48) . The parts for the (2 k − k -th zigtherefore get so large that they are split into two, respectively in Figure 21 and14igure 18: The three bricks of G , that is, G −∗ b , G − , and G − .Figure 19: The three bricks of G , that is, G − , G − , and G − .Figure 22 (Top, Middle). Module 2.
The parts of the brick automaton for Module 2 are illustratedin Figures 23 and 24. As mentioned previously, this module does not have topropagate the n -bits to identify the current state q n − . Thus, in the formattingzigs and zags, the corresponding n instances of P zig and of P zag are in factreplaced by a glider. As a result, the 1st zig and the other (2 k − k ≥
2) encounter different environments. This difference is illustrated inFigure 23 (Top, Middle).
Module 3.
The parts of the brick automaton for Module 3 are illustrated inFigures 25 and 26.
Module 4.
The parts of the brick automaton for Module 4 are illustrated inFigures 27, 28, 29, and 30.
References [1] Jean-Paul Allouche and Jeffrey Shallit.
Automatic Sequences: Theory, Ap-plications, Generalizations . Cambridge University Press, 2003.15igure 20: The part
M1 zig of the brick automaton for the (2 k − P zag in the previous formattingzag.[2] Erik D. Demaine, Jacob Hendricks, Meagan Olsen, Matthew J. Patitz,Trent A. Rogers, Nicolas Schabanel, Shinnosuke Seki, and Hadley Thomas.Know when to fold ’em: Self-assembly of shapes by folding in oritatami. In Proc. DNA24 , volume 11145 of
LNCS , pages 19–36. Springer, 2018.[3] C. Geary and E. S. Andersen. Design principles for single-stranded RNAorigami structures. In
Proc. DNA20 , volume 8727 of
LNCS , pages 1–19.Springer, 2014.[4] Cody Geary, Pierre-´Etienne Meunier, Nicolas Schabanel, and ShinonsukeSeki. Programming biomolecules that fold greedily during transcription. In
Proc. MFCS 2016 , volume 58 of
LIPIcs , pages 43:1–43:14, 2016.[5] Cody Geary, Pierre-´Etienne Meunier, Nicolas Schabanel, and ShinonsukeSeki. Proving the Turing universality of oritatami cotranscriptional folding.In
Proc. ISAAC 2018 , volume 123 of
LIPIcs , pages 23:1–23:13, 2018.[6] Cody Geary, Paul W. K. Rothemund, and Ebbe S. Andersen. A single-stranded architecture for cotranscriptional folding of RNA nanostructures.
Science , 345:799–804, 2014.[7] Yo-Sub Han and Hwee Kim. Construction of geometric structure by ori-tatami system. In
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LNCS , pages 173–188.Springer, 2018. 16igure 21: Two parts of the brick automaton for the (2 k − M1 zag-before , which describes transitions among bricks for this zaguntil the zag encounters the g-spacer B (cid:48) (depicted as a cyan rectangle like theother spacers), and (Bottom) M1 zag-after for describing transitions after theencounter. 17igure 22: Three parts of the brick automaton: (Top, Middle)
M1-P zig and
M1-P zig-after for the 2 k -th (formatting) zig until B (cid:48) and after B (cid:48) , respec-tively, and (Bottom) M1-P zag for the 2 k -th zag of Module 1.18igure 23: Three parts of the brick automaton: (Top) M2 zig-first for the 1stzig, (Middle)
M2 zig for the (2 k − k ≥
2, and (Bottom)
M2 zag forthe (2 k − k ≥ M2-P zig for the 2 k -th (formatting) zig and (Bottom) M2-P zag for the 2 k -th (formatting) zag ofModule 2. 20igure 25: Two parts of the brick automaton: (Top) M3 zig for the 1st zig and(Bottom)
M3 zag for the 1st zag of Module 3.21igure 26: Two parts of the brick automaton: (Top)
M3-P zig for the 2nd(formatting) zig and (Bottom)
M3-P zag for the 2nd zag of Module 3.22igure 27: Two parts of the brick automaton: (Top)
M4 zig-before , whichdescribes transitions until the sole instance of A is transcribed, and (Bottom) M4 zig-after for the transitions after that A .[8] Yusei Masuda, Shinnosuke Seki, and Yuki Ubukata. Towards the algorithmicmolecular self-assembly of fractals by cotranscriptional folding. In Proc.CIAA 2018 , volume 10977 of
LNCS , pages 261–273. Springer, 2018.[9] Kyle E. Watters, Eric J. Strobel, Angela M. Yu, John T. Lis, and Julius B.Lucks. Cotranscriptional folding of a riboswitch at nucleotide resolution.
Nat. Struct. Mol. Biol. , 23(12):1124–1133, 2016.23igure 28: Two parts
M4 zag of the brick automaton for the (2 k − M4 zag-before , which describes transitions until the soleinstance of A in the previous zig is encountered, and (Bottom) M4 zag-after for the transitions after the encounter.24igure 29: Two parts of the brick automaton: (Top)
M4-P zig-before for the2 k -th (formatting) zig of Module 4 until the zig encounters the sole instance of A , and (Bottom) M4-P zig-after after of A .Figure 30: Two part M4-P zag of the brick automaton for the 2 kk