Nondeterministic Polynomial Time Factoring in the Tile Assembly Model

Transcription

1 Nondeterminitic Polynomial Time Factoring in the Tile Aembly Model Yuriy Brun Department of Computer Science Univerity of Southern California o Angele, CA ybrun@uc.edu Abtract Formalized tudy of elf-aembly ha led to the definition of the tile aembly model [Win98b]. In [Bru], I preented way to compute arithmetic function, uch a addition and multiplication, in the tile aembly model: a highly ditributed parallel model of computation that may be implemented uing molecule or a large computer network uch a the Internet. Here, I preent tile aembly model ytem that factor number nondeterminitically uing Θ() ditinct component. The computation take advantage of nondeterminim, but theoretically, each of the nondeterminitic path i executed in parallel, yielding the olution in time linear in the ize of the input, with high probability. I decribe mechanim for finding the ucceful olution among the many parallel execution and explore bound on the probability of uch a nondeterminitic ytem ucceeding and prove that probability can be made arbitrarily cloe to. Introduction Self-aembly i a proce which i ubiquitou in nature. Sytem form on all cale via elf-aembly: atom elf-aemble to form molecule, molecule to form complexe, and tar and planet to form galaxie. One manifetation of elf-aembly i crytal growth: molecule elf-aembling to form crytal. Crytal growth i an intereting area of reearch for computer cientit becaue it ha been hown that, in theory, under careful control, crytal can compute [Win98a]. The field of DNA computation demontrated that DNA can be ued to compute [Adl94], olving NP-complete problem uch a the atifiability problem [BJR +, BCJ + 2]. Thi idea of uing molecule to compute nondeterminitically i the driving motivation behind my work. Winfree howed that DNA computation i Turing-univeral [Win96]. While DNA computation uffer from relatively high error rate, the tudy of elfaembly how how to utilize redundancy to deign ytem with built-in error

2 correction [WB3, BCSY5, CG4, RSY4, Win6]. Reearcher have ued DNA to aemble crytal with pattern of binary counter [BRW5] and Sierpinki triangle [RPW4], but while thoe crytal are determinitic, generating nondeterminitic crytal may hold the power to olving complex problem quickly. Two important quetion about elf-aembling ytem that create hape or compute function are: what i a minimal tile et that can accomplih thi goal? and what i the minimum aembly time for thi ytem? Here, I tudy ytem that factor number and ak thee quetion, a well a another that i important to nondeterminitic computation: what i the probability of aembling the crytal that encode the olution? Adleman ha emphaized tudying the number of tep it take for an aembly to complete (auming maximum parallelim) and the minimal number of tile neceary to aemble a hape [Adl]. He anwered thee quetion for n-long linear polymer [ACG + ]. Previouly, I have extended thee quetion to apply to ytem that compute function, rather than aemble hape [Bru], and now I extend them to ytem that compute function nondeterminitically. Adleman propoed tudying the complexity of tile ytem that can uniquely produce n n quare. A erie of reearcher proceeded to anwer the quetion: what i a minimal tile et that can aemble uch hape? and what i the aembly time for thee ytem? They howed that the minimal tile et that aemble n n quare i of ize Θ( log n log log n ) and the optimal aembly time i Θ(n) [RW, ACG + 2, AGHdE]. A key iue related to aembling quare i the aembly of mall binary counter, which theoretically can have a few a 7 tile type [de5]. Soloveichik et al. tudied aembling all decidable hape in the tile aembly model and found that the minimal et of tile neceary to uniquely aemble a hape i directly related to the Kolmogorov complexity of that hape. Interetingly, they found that for the reult to hold, cale mut not be a factor. That i, the minimal et of tile they find build a given hape (e.g. quare, a particular approximation of the map of the world, etc.) on ome cale, but not on all cale. Thu they howed that maller verion of the ame hape might require larger et of tile to aemble [SW4]. I propoed and tudied ytem that compute the um and product of two number uing the tile aembly model [Bru]. I found that in the tile aembly model, adding and multiplying can be done uing Θ() tile (a few a 8 tile for addition and a few a 28 tile for multiplication), and that both computation can be carried out in time linear in the input ize. I alo peculated that thoe ytem could be modified to work in equence with other ytem, in effect chaining computation together. In thi paper I ue jut that technique: I utilize the multiplication ytem a a ubroutine for the factoring ytem, exploring nondeterminitic computation in the tile aembly model. Other early attempt at nondeterminitic computation include a propoal by agoudaki et al. to olve the atifiability problem [99]. They informally define a ytem that nondeterminitically compute whether or not an n-variable 2

3 boolean formula i atifiable uing Θ(n 2 ) ditinct tile. In contract, all the ytem I preent in thi paper ue Θ() ditinct tile. While the contruction in thi paper are in ome way analogou to traditional computer program, and their running time are polynomially related to the running time of Turing machine and nondeterminitic Turing machine, Baryhnikov et al. began the tudy of fundamental limit on the time required for a elf-aembly ytem to compute function [BCM5]. They conider model of molecular elf-aembly and apply Markov model to how lower limit on aembly time. Reearcher have alo tudied variation of the traditional tile aembly model. Aggarwal et al. and Kao et al. have hown that changing the temperature of aembly from a contant throughout the aembly proce to a dicrete function reduce the minimal tile et that can build an n n quare to a ize Θ() tile et [ACG + 5, KS6]. Barih et al. have demontrated DNA implementation of tile ytem, one that copie an input and another that count in binary [BRW5]. Similarly, Rothemund et al. have demontrated a DNA implementation of a tile ytem that compute the xor function, reulting in a Sierpinki triangle [RPW4]. Thee ytem grow crytal uing double-croover complexe [FS93] a tile. The theoretical underpinning of thee ytem are cloely related to the work preented here becaue thee ytem compute function. Rothemund ha demontrated what i currently the tate-of-the-art of DNA nanotructure deign and implementation with DNA origami, a concept of folding a ingle long caffold trand into an arbitrary hape by uing mall helper trand [Rot6a, Rot5, Rot6b]. Similar concept may be the key to threedimenional elf-aembly, more powerful error-correction technique, and elfaembly uing biological molecule. Cook et al. have explored uing the tile aembly model to implement arbitrary circuit [CRW3]. Their model allow for tile that contain gate, counter, and even more complex logic component, a oppoed to the imple tatic tile ued in the traditional tile aembly model and in thi paper. While they peculate that the tile aembly model logic may be ued to aemble logic component attached to DNA, my aemblie require no additional logic component and encode the computation themelve. It i likely that their approach will require fewer tile type and perhap aemble fater, but at the diadvantage of having to not only only aemble crytal but alo attach component to thoe crytal and create connection among thoe component. Neverthele, Rothemund work with uing DNA a a caffold may be ueful in attaching and aembling uch component [Rot6b]. Some experimental work [Adl94, BCJ + 2] ha hown that it i poible to work with an exponential number of component and even to olve NP-complete problem. I explore the poibility of nondeterminitic computation uing the tile aembly model and prove bound on the probability of ucceful computation. The probability of uccefully factoring a number can be made arbitrarily cloe to by increaing the number of elf-aembling component and eed in the computation. 3

4 The ret of thi paper i tructured a follow: ection. will decribe in detail the tile aembly model, ection 2 will dicu what it mean for a tile aembly model ytem to compute function determinitically and nondeterminitically, ection 3 will introduce, define, and prove the correctne of two factoring ytem, and ection 4 will ummarize the contribution of thi work.. Tile Aembly Model The tile aembly model [Win98b, Win98a, RW] i a formal model of crytal growth. It wa deigned to model elf-aembly of molecule uch a DNA. It i an extenion of a model propoed by Wang [Wan6]. The model wa fully defined by Rothemund and Winfree [RW], and the definition here are imilar to thoe, and identical to the one in [Bru], but I retate them here for completene and to ait the reader. Intuitively, the model ha tile or quare that tick or do not tick together baed on variou binding domain on their four ide. Each tile ha a binding domain on it north, eat, outh, and wet ide, and may tick to another tile when the binding domain on the abutting ide of thoe tile match and the total trength of all the binding domain on that tile exceed the current temperature. The four binding domain define the type of the tile. While thi definition doe not allow tile to rotate, it i eentially equivalent to a ytem with rotating tile. Formally, let Σ be a finite alphabet of binding domain uch that null Σ. I will alway aume null Σ even when I do not pecify o explicitly. A tile over a et of binding domain Σ i a 4-tuple σ N, σ E, σ S, σ W Σ 4. A poition i an element of Z 2. The et of direction D = {N, E, S, W } i a et of 4 function from poition to poition, i.e. Z 2 to Z 2, uch that for all poition (x, y), N(x, y) = (x, y + ), E(x, y) = (x +, y), S(x, y) = (x, y ), W (x, y) = (x, y). The poition (x, y) and (x, y ) are neighbor iff d D uch that d(x, y) = (x, y ). For a tile t, for d D, I will refer to bd d (t) a the binding domain of tile t on d ide. A pecial tile empty = null, null, null, null repreent the abence of all other tile. A trength function g : Σ Σ R, where g i commutative and σ Σ g(null, σ) =, denote the trength of the binding domain. It i common to aume that g(σ, σ ) = σ σ. Thi implification of the model implie that the abutting binding domain of two tile have to match to bind. et T be a et of tile containing the empty tile. A configuration of T i a function A : Z Z T. I write (x, y) A iff A(x, y) empty. A i finite iff there i only a finite number of ditinct poition (x, y) A. Finally, a tile ytem S i a triple T, g, τ, where T i a finite et of tile containing empty, g i a trength function, and τ N i the temperature. If A i a configuration, then within ytem S, a tile t can attach to A at poition (x, y) and produce a new configuration A iff: (x, y) / A, and d D g(bd d(t), bd d (A(d(x, y)))) τ, and 4

5 (u, v) Z 2, (u, v) (x, y) A (u, v) = A(u, v), and A (x, y) = t. That i, a tile can attach to a configuration only in empty poition and only if the total trength of the appropriate binding domain on the tile in neighboring poition meet or exceed the temperature τ. For example, if for all σ, g(σ, σ) = and τ = 2 then a tile t can attach only at poition with matching binding domain on the tile in at leat two adjacent poition. Given a tile ytem S = T, g, τ, a et of tile Γ, and a eed configuration S : Z 2 Γ, if the above condition are atified, one may attach tile of T to S. Configuration produced by repeated attachment of tile from T are aid to be produced by S on S. If thi proce terminate, then the configuration achieved when no more attachment are poible i called the final configuration. At ome time, it may be poible for more than one tile to attach at a given poition, or there may be more than one poition where a tile can attach. If for all equence of tile attachment, all poible final configuration are identical, then S i aid to produce a unique final configuration on S. et S = T, g, τ, and let S be a eed configuration uch that S produce a unique final configuration F on S. et W 2 T Z2 be the et of all tilepoition pair t, (x, y) uch that t can attach to S at (x, y). et S be the configuration produced by adding all the element of W to S in one time tep. Define W, W 2, and S 2, S 3, imilarly. et n be the mallet natural number uch that S n F. Then n i the aembly time of S on S to produce F. I allow the codomain of S to be Γ, a et of tile which may be different from T. The reaon i that I will tudy ytem that compute function uing minimal et T ; but the eed, which ha to code for the input of the function, may contain more ditinct tile than there are in T. Therefore, I wih to keep the two et eparate. Note that, at any temperature τ, it take Θ(n) ditinct tile to aemble an arbitrary n-bit input uch that each tile code for exactly one of the bit. Winfree howed that the tile aembly model with τ = 2 i Turing-univeral [Win98a] by howing that a tile ytem can imulate Wang tile [Wan6], which Robinon howed to be univeral [Rob7]. Adleman et al. howed that the tile aembly model with τ = i Turing-univeral [AKKR2]. 2 Computing function et N = Z. Intuitively, in order for a tile ytem to determinitically compute a function f : N N, for ome eed that encode a number a N, the tile ytem hould produce a unique final configuration which encode f(a). To allow configuration to encode number, I will deignate each tile a a - or a -tile and define an ordering on the poition of the tile. The i th bit of a i iff the tile in the i th poition of the eed configuration i a -tile (note that empty will almot alway be a -tile). I will alo require that the eed contain 5

6 tile coding for bit of a and no more than a contant number of extraneou tile. The final configuration may have more than one poible ordering of the attachment. Formally, let Γ and T be et of tile. et be the et of all poible eed configuration S : Z 2 Γ, and let Ξ be the et of all poible finite configuration C : Z 2 Γ T. et v : Γ T {, } code each tile a a - or a -tile and let o, o f : N Z 2 be two injection. et the eed encoding function e : N map a eed S to a number uch that e (S) = i= 2i v(s(o (i))) if and only if for no more than a contant number of (x, y) not in the image of o, (x, y) S. et the anwer encoding function e f : Ξ N map a configuration F to a number uch that e f (F ) = i= 2i v(f (o f (i))). et S = T, g, τ be a tile ytem. I ay that S compute a function f : N N iff for all a N there exit a eed configuration S uch that S produce a unique final configuration F on S and e (S) = a and e f (F ) = f(a). I generalize the above definition for a larger et of function. et ˆm, ˆn N and f : N ˆm Nˆn be a function. For all m < ˆm and n < ˆn, let o m, o fn : N Z 2 be injection. et the eed encoding function e m : N map a eed S to ˆm number uch that e m (S) = i= 2i v(s(o m (i))) iff for no more than a contant number of (x, y) not in the union of the image of all o m, (x, y) S. et the anwer encoding function e fn : Ξ N each map a configuration F to a number uch that e fn (F ) = i= 2i v(f (o fn (i))). Then I ay that S compute the function f iff for all a N ˆm = a, a,, a ˆm, there exit a eed configuration S uch that S produce a unique final configuration F on S and e m (S) = a m and e f (F ), e f (F ),, e fˆn = f( a). 2. Nondeterminitic computation Until now, I have looked only at determinitic computation. In order for a ytem to compute a function, I required that the ytem produce unique final configuration. In ome implementation of the tile aembly model ytem, many aemblie happen in parallel. In fact, it i often almot impoible to create only a ingle aembly, and thu there i a parallelim that my previou definition did not take advantage of. Here, I define the notion of nondeterminitic computation in the tile aembly model. I have defined a tile ytem to produce a unique final configuration on a eed if for all equence of tile attachment, all poible final configuration are identical. If different equence of tile attachment attach different tile in the ame poition, the ytem i aid to be nondeterminitic. Intuitively, a ytem nondeterminitically compute a function iff at leat one of the poible equence of tile attachment produce a final configuration which code for the olution. Since a nondeterminitic computation may have unucceful equence of attachment, it i important to ditinguih the ucceful one. Further, in many implementation of the tile aembly model that would imulate all the nondeterminitic execution at once, it i ueful to be able to identify which execution ucceeded and which failed in a way that allow electing only the ucceful one. For ome problem, only an exponentially mall fraction of 6

7 the aemblie would repreent a olution, and finding uch an aembly would be difficult. For example, a DNA baed crytal growing ytem would create million of crytal, and only a few of them may repreent the correct anwer, while all other repreent failed computation. Finding a ucceful computation by ampling the crytal at random would require time exponential in the input. Thu it would be ueful to attach a pecial identifier tile to the crytal that ucceed o that the crytal may be filtered to find the olution quickly. It may alo be poible to attach the pecial identifier tile to olid upport o that the crytal repreenting ucceful computation may be extracted from the olution. I thu pecify one of the tile of a ytem a an identifier tile that only attache to a configuration that repreent a ucceful equence of attachment. et ˆm, ˆn N and let f : N ˆm Nˆn be a function. For all m < ˆm and n < ˆn, let o m, o fn : N Z 2 be injection. et the eed encoding function e m : N map a eed S to ˆm number uch that e m (S) = i= 2i v(s(o m (i))) iff for no more than a contant number of (x, y) not in the union of the image of all o m, (x, y) S. et the anwer encoding function e fn : Ξ N each map a configuration F to a number uch that e fn (F ) = i= 2i v(f (o fn (i))). et S be a tile ytem with T a it et of tile, and let r T. Then I ay that S nondeterminitically compute a function f with identifier tile r iff for all a N ˆm = a, a,, a ˆm there exit a eed configuration S uch that for all final configuration F that S produce on S, r F (Z 2 ) iff m < ˆm, e m (S) = a m and e f (F ), e f (F ),, e fˆn (F ) = f( a). I wa careful to pick the eed and anwer encoding function before the tile ytem becaue if you are allowed to tailor the encoding function to pecific ytem, you can eaily olve the halting problem by encoding all the complexity of the halting problem in the anwer encoding function (deign a ytem that on input n attache a ingle - and a ingle -tile to the eed and the anwer encoding function map to if the n th Turing machine halt on input n and to otherwie). While thi formalim doe not allow it, ometime it may be convenient to give yourelf ome freedom in defining the anwer encoding function and let it depend on certain apect of the aembly proce, but one ha to be extremely careful tepping on that lippery lope. In the main ytem decribed in thi paper, I will tep on exactly that lope, but I will argue carefully that the ytem I preent i equivalent to a lightly more complex one that fit the definition exactly a preented here. I have defined what it mean for a tile ytem to compute a function f, whether it i determinitically or nondeterminitically; however, I have not mentioned what happen if f i not defined on ome argument. Intuitively, the determinitic ytem hould not produce a unique final configuration, and no nondeterminitic aembly hould contain the identifier tile. The above definition formalize thi intuitive idea. In the remainder of thi paper I will examine ytem that factor poitive integer; that i, ytem that nondeterminitically compute, for all ζ 2, f(ζ) = α, β, uch that α, β 2 and αβ = ζ. factoring tile ytem. I refer to uch ytem a 7

8 γ γ γ γ * * Figure : There are 4 tile in Γ 4. The value in the middle of each tile t repreent that tile v(t) value and each tile name i written on it left. 3 Factoring tile ytem In thi ection, I will decribe two factoring tile ytem. To that end, I will introduce three tile ytem, building up factoring functionality one tep at a time. I will then combine thoe ytem to create a ytem that factor at temperature four, and then dicu implifying thi factoring ytem to work at temperature three. All the ytem ue Θ() ditinct tile. The factoring ytem, and the proof of their correctne, are baed in part on the multiplier ytem (one that determinitically compute f(α, β) = αβ) from [Bru]. Intuitively, thi ytem will nondeterminitically pick two number, multiply them, and then compare the reult to the input. If the reult and the input match, the aembly will include an identifier tile. The factoring tile ytem will ue a et of tile T 4. I will define thi et in three dijoint ubet: T 4 = T factor T T. The tile in T factor nondeterminitically gue two factor; T i identical to T defined in [Bru] and multiply the two factor; and the tile in T enure the computation i complete and compare the product of the factor to the input. Whenever conidering a number α N, I will refer to the ize of α, in bit, a n α. I will further refer to the i th bit of α a α i ; that i, for all i N, α i {, } uch that i α i2 i = α. The leat ignificant bit of α i α. Finally, I define λ α N to be the mallet poitive integer uch that α λα =. For example, let α = 2, then n α = 7, and λ α = 2 becaue the mallet poitive power of 2 in 2 i 2 2. Of coure, the ame definition extend to β, ζ and other variable. The tile ytem I will decribe will ue four type of tile to encode the input number the ytem i factoring. et the et of thoe tile be Γ 4 = {γ = null, null,, null, γ = null, null,, null, γ = null, null,, null, γ = null, null,, }. Figure how a graphical repreentation of Γ Gueing the factor I firt decribe a tile ytem that, given a eed coniting of jut a ingle tile will nondeterminitically pick two natural binary number, α and β, uch that α, β 2, and encode them uing tile. The ytem will ue the et of tile T factor. Figure 2(a) how the concept behind the tile in T factor. The concept include variable a and b. Each variable can take on a value the element of the et {, }. Figure 2(b) how the 3 actual tile type in T factor. The ymbol 8

9 * * b b b b * a a a a * (a) * f f R f btop f b f b f bbot * * f alef f alef f a f a f arig * f bbot f arig * * (b) Figure 2: The concept behind the tile in T factor include variable a and b (a). Each variable can take on a value the element of the et {, }. There are 3 actual tile type in T factor (b). The value in the middle of each tile t indicate it v(t) value (ome tile have no value in the middle becaue their v value will not be important and can be aumed to be ) and each tile name i written on it left. 9

10 on the ide of the tile indicate the binding domain of thoe ide. The value in the middle of each tile t indicate it v(t) value (ome tile have no value in the middle becaue their v value will not be important and can be aumed to be ). emma 3. et Σ factor = {,,,,,,, }. For all σ {,,, }, let g factor (σ, σ) = 4 (for all other σ Σ factor, g(σ, σ) i not important for now). et τ factor = 4. et T factor be a decribed in Figure 2(b). et S factor = T factor, g factor, τ factor. et S factor : Z 2 {γ } be a eed configuration with a ingle nonempty tile, uch that S(, ) = γ and x, y Z, (x, y) (, ) = S factor (x, y) = empty. Then for all α, β 2, z, S factor produce a final configuration F on S factor uch that:. F (, ) = γ 2. F (, ) = f btop 3. i {, 2,, n β 2}, F (, i) = f bβk, where k = n β i 4. F (, (n β )) = f bβbot 5. F (, n β ) = f R 6. F (, n β ) = f aαrig 7. i {, 2,, λ α }, F ( i, n β ) = f a 8. F ( λ α, n β ) = f a 9. i {λ α +, λ α + 2,, n α }, F ( i, n β ) = f aαilef. i {, 2,, z}, F ( (n α + i), n β ) = f alef. F ( (n α + z), n β ) = f And for all final configuration F that S factor produce on S factor, F correpond to ome choice of α, β 2 and z. Intuitively, S factor write out, in order, the bit of α and β, for all α, β 2, and pad α with z extra -tile, where z. Figure 3 how a ample final configuration that correpond to β = 3 = 2, α = 97 = 2, and z = 7. Proof: et α, β 2, let z. I firt how that tile may attach to S factor to produce a final configuration F that encode α and β and pad β with z -tile.. F and S factor mut agree at poition (, ), o F (, ) = γ. 2. Note that bd S (F (, )) = and the only tile type t with bd N (t) = i f btop, o F (, ) = f btop.

11 * * Figure 3: S factor produce a final configuration F on S factor uch that F encode two binary number. In thi example, F encode 3 = 2 and 97 = 2. Note that the number encoded in the lower row may have leading zero; in thi cae it ha 7 leading -tile. 3. Note that bd S (F (, )) =, and there are four type of tile t uch that bd N (t) = (f b, f b, f bbot, and f bbot ), o only thoe type of tile may attach below. Two of thoe tile type have bd S (t) = (f b and f b ), o again only thoe four tile type may attach below them. Therefore tile may attach uch that for all i {, 2,, n β 2}, F (, i) = f bβk, where k = n β i. 4. Until finally, a tile of type f bbot or f bbot attache o that F (, (n β )) = f bβbot. 5. Note that bd S (F (, (n β ))) =, and the only tile type t with bd N (t) = i f R, o F (, n β ) = f R. 6. Note that bd W (F (, n β )) =, and there are two type of tile t uch that bd E (t) = (f arig and f arig ), o only type of tile may attach at poition (, n β ). Thu the correct tile may attach uch that F (, n β ) = f aαrig. 7. Note that bd W (F (, n β )) =, and there are two type of tile t uch that bd E (t) = (f a and f a ), and only one of thoe two type ha bd W (t) = (f a ). Therefore, for all i {, 2,, λ α }, f a can attach uch that F ( i, n β ) = f a. 8. Until finally, a tile of type f a attache o that F ( λ α, n β ) = f a.

12 9. Note that bd W (F ( λ α, n β )) =, and there are three type of tile t uch that bd E (t) = (f alef, f alef, and f ). Two of thoe tile type have bd W (t) = (f alef and f alef ), o again only thoe three tile type may attach to the left of them. Therefore, tile may attach uch that for all i {λ α +, λ α + 2,, n α }, F ( i, n β ) = f aαilef.. Similarly, to the wet of the poition ( (n α, n β )), f alef may attach uch that for all i {, 2,, z}, F ( (n α + i), n β ) = f alef.. Until finally, a tile of type f attache o that F ( (n α + z), n β ) = f. et v : T factor {, } be uch that v(f b ) = v(f bbot ) = v(f alef ) = v(f a ) = v(f arig ) = v(f ) = v(null) = and v(f btop ) = v(f b ) = v(f bbot ) = v(f alef ) = v(f a ) = v(f arig ) =. For all other t T factor the v(t) value doe not matter and can be aumed to be either or. Note that: No more tile may attach to F. i= v(f (, (n β + i)))2 i = β. i= v(f ( i, n β))2 i = α. For all i {, 2,, z}, v(f ( (n α + i), n β )) =. Thu for all choice of α, β 2, and z, there exit a final configuration F produced by S factor on S factor that encode α and β and pad α with z -tile. Further note that for all final configuration F produced by S factor on S factor : F cannot encode β < 2 becaue v(f (, )) = implie that the mot ignificant bit of β i and F (, ) cannot be f R, thu β ha at leat 2 bit. F cannot encode α < 2 becaue the tile encoding α λα cannot be at poition (, ) and thu there exit a non-zero power of 2 in α. Thu all final configuration F produced by S factor on S factor encode ome α, β 2 and pad α with ome z -tile. For a ingle choice of α, β 2, there are everal final configuration that encode α and β. They differ in the choice of z. I am intereted only in two of thoe final configuration, for z = n β and for z = n β, becaue the product of α and β i either n α + n β or n α + n β bit long. emma 3.2 (Factor aembly time lemma) For all α, β 2, the aembly time of the final configuration F produced by S factor on S factor that encode α and β and pad α with n β or n β -tile i Θ(n α + n β ). Proof: For each tile in F to attach, a tile in a pecific location mut have attached before it (either to the north or to the eat). Thu there i no parallelim in thi aembly, and the aembly time equal the total number of tile that attach, which i Θ(n α + n β ). 2

13 emma 3.3 et each tile that may attach to a configuration at a certain poition attach there with a uniform probability ditribution. For all α, β 2, let δ = αβ, then the probability of aembling a particular final configuration encoding α and β and padding α with n δ n α -tile i at leat ( 4) nβ ( 3) nδ. Proof: I will calculate the probabilitie of each tile attaching in it proper poition and then multiply thoe probabilitie together to get the overall probability of a final configuration.. The eed automatically become part of the final configuration with probability p =. 2. There i only tile that may attach in poition (, ), o it attache with probability p 2 =. 3. For the next n β 2 poition, out of 4 poible tile that may attach, only i the correct one, o the probability that the next n β 2 tile attach correctly i p 3 = ( 4) nβ The tile repreenting the lat bit of β i alo out of the poible 4 o the probability of it attaching i p 4 = Only tile may attach below the th bit of β, o it probability of attaching i p 5 =. 6. There are 2 poible tile that may attach to repreent α th bit, and only i correct, o probability of it attaching i p 6 = The next λ α tile mut be of 2 poible tile, o the probability of all of them attaching correctly i p 7 = ( 2) λα. 8. The next tile encode the mallet poitive power of 2 in α and i of 2 poible tile, o it probability of attaching i p 8 = The ret of the bit of α can be encoded uing pecific tile from the 3 poibilitie, o the probability of all of them attaching correctly i p 9 = ( ) nα λ α 3.. The next n δ n α tile, depending on the deired final configuration, mut be of 3 poible tile, o the probability of all of them attaching correctly i p = ( 3) nδ n α.. Finally, the probability of the lat tile attaching i p = 3. The overall probability of a pecific final configuration i: i= p i ( nβ ( λα+ ( nα λ α+n δ n α ( 4) 2) 3) nβ ( ) nδ 4) 3 Corollary 3.4 et each tile that may attach to a configuration at a certain poition attach there with a uniform probability ditribution. For all α β 2, let δ = αβ, then the probability of aembling a particular final configuration encoding α and β and padding α with n δ n α -tile i at leat ( 6) nδ. 3

14 b a and(a,b) and(a,b) b c b b +and(b,c) c xor(b,c) xor(b,c) +or(b,c) c -xor(b,c) -xor(b,c) 2 a a c c a c c a c c a a b a b a b (a) (b) Figure 4: The concept behind the tile in T (a) include variable a, b, and c, each of which can take on a value the element of the et {, }. There are 28 actual tile type in T (b). Note that for each tile t, it v(t) value i indicated in the upper half of the middle of the tile. Proof: By emma 3.3, the probability of aembling a particular final configuration ( encoding α and β and padding α with n δ n α -tile i at leat ) nβ ( ) nδ 4 3. Becaue α β, n δ 2n β, o ( nβ ( ) nδ 4) 3 ( nδ ( ) nδ 2) 3 = ( nδ 6). 3.2 Multiplying the factor I have jut decribed a tile ytem that nondeterminitically aemble the repreentation of two binary number. I will now add tile to multiply thoe number. Figure 4(a) how the concept behind the tile in T. The concept include variable a, b, and c, each of which can take on a value the element of the et {, }. Figure 4(b) how the 28 actual tile type in T. Thee tile are identical 4

15 to the multiplier tile from [Bru], thu I will reference Theorem 2.4 from [Bru]: Theorem 3.5 (Theorem 2.4 from [Bru]) et Σ = {,,,,,, 2, 2}, for all σ Σ, g (σ, σ) =, τ = 2, and T be a et of tile over Σ a decribed in Figure 4(b). Then S = T, g, τ compute the function f(α, β) = αβ. The reader may refer to [Bru] for the proof of Theorem 3.5. Corollary 3.6 For all σ Σ, let g (σ, σ) = 2 and τ = 4. Then S = T, g, τ compute the function f(α, β) = αβ. Proof: For all α, β N, S compute the function f(α, β) = αβ. Therefore, there exit ome eed S, which encode α and β, uch that S produce a unique final configuration F on S. Further, there exit at leat one equence of attachment W = t, (x, y ), t, (x, y ),, t k, (x k, y k ) uch that t attache at poition (x, y ) to S to produce S, t attache at poition (x, y ) to S to produce S 2, and o on to produce the final configuration S k+ = F. Since for all σ Σ, g (σ, σ) = and τ = 2, each tile t i mut have at leat two non-empty neighbor to poition (x i, y i ) in S i whoe appropriate binding domain match t i binding domain. By the unique final configuration corollary (Corollary 2. from [Bru]), S produce a unique final configuration on S. Since for all i, each tile t i ha at leat two non-empty neighbor to poition (x i, y i ) in S i whoe appropriate binding domain match t i binding domain, and for all σ Σ, g (σ, σ) = 2 and τ = 4, the tile t i can attach to S i to form S i+. Thu W i a valid equence of attachment for S to S and S mut produce the unique final configuration S k+ = F on S, and thu compute the function f(α, β) = αβ. Intuitively, Corollary 3.6 ay that doubling the trength of every binding domain and the temperature doe not alter the logic of aembly of S. To compute the product of two number α and β, S attache tile to a eed configuration which encode α and β, to reach a final configuration which encode αβ. The logic of the ytem dictate that every time a tile attache, it outh and eat neighbor have already attached, and it wet and north neighbor have not attached [Bru]. Thu by doubling the trength of every binding domain and the temperature, a tile t can attach at poition (x, y) in S iff t alo attached at (x, y) in S. Thu the final configuration in the two ytem will be identical, and therefore, S compute the function f(α, β) = αβ. Figure 5(a) how a ample eed configuration S which encode β = 3 = 2 and α = 97 = 2. Figure 5(b) how the final configuration F that S produce of S. Along the top row of F, the binary number 2 = 999 encode the product αβ = 97 3 = 999. It i not a coincidence that S look exactly like a final configuration that S factor produce on S factor. emma 3.7 et α, β 2, let S be the final configuration produced by S factor on S factor that encode α and β and ha n β padding -tile. et F be the unique 5

16 * * * * (a) * * * * (b) Figure 5: A ample eed configuration S that encode β = 3 = 2 and α = 97 = 2 (a). Note that S i identical to a final configuration produced by S factor on S factor. S produce a final configuration F on S which encode the product αβ = 97 3 = 999 = 2 along the top row (b). 6

17 final configuration that S produce on S. et v : T {, } be defined a in Figure 5. Then F encode the binary number δ = αβ = i= v(f ( i, ))2i. That i, the i th bit of δ i v(f ( i, )). Further, if αβ ha only n α + n β bit, then it i ufficient to pad S factor with n β -tile. The emma follow directly from the proof of Corollary 3.6 and Theorem 3.5. S form the ide of a rectangle that ha the ame binding domain internal to the rectangle a the eed that S ue in [Bru]. S produce, on S the final configuration F, which encode the product of the input on preciely the th row, with the i th bit of the product being v(f (, i)). Note that the product of two binary number α and β will alway have either n α + n β or n α + n β bit. emma 3.8 (Multiplication aembly time lemma) For all α, β 2, the aembly time for S on a eed encoding α and β and padding α with n β or n β -tile i Θ(n α + n β ). Thi emma follow directly from the aembly time corollary (Corollary 2.2 from [Bru]). 3.3 Checking the product I have decribed two ytem: one to produce two random number and another to multiply them. I now preent one more ytem with two job: to make ure the multiplication i complete and to compare an input to the reult of the multiplication. Figure 6(a) how the concept behind the tile in T. The concept include variable b, c and d, each of which can take on a value the element of the et {, }. Figure 6(b) how the 9 actual tile type in T. The tile with a check mark in the middle will erve a the identifier tile. emma 3.9 et Σ = {,,,,,,,,, }. For all σ {,, }, let g (σ, σ) = and for all σ {,,,,,, }, let g (σ, σ ) = 2. et τ = 4. et T be a decribed in Figure 6(b). For all α, β, ζ 2, let δ = αβ, let S be the final configuration produced by S factor on S factor that encode α and β and ha n δ n α padding -tile. et F be the unique final configuration that S produce on S. et S be uch that: x, y Z, F (x, y) empty = S (x, y) = F (x, y). S ( n ζ, 2) = γ. i {,,, n ζ }, S ( i, 2) = γ ζi et S = T, g, τ. Then S produce a unique final configuration F on S and αβ = ζ iff F ( n ζ, ) =. 7

18 c* b b dc (a) * * * * (b) Figure 6: The concept behind the tile in T (a) include variable b, c and d, each of which can take on a value the element of the et {, }. There are 9 actual tile type in T ; each tile name i written on it left. The tile with a check mark in the middle will erve a the identifier tile. Proof: Firt, oberve that if the tile i ever to attach, it mut attach in poition ( n ζ, ) becaue the um of the g value of the binding domain of i exactly 4, o it mut match it neighbor on all ide with non-null binding domain to attach at temperature 4, and becaue the only tile with a outh binding domain, matching north binding domain, i γ, and it can only occur in the eed, and only in poition ( n ζ, 2). Conider F. Working backward, for a tile to attach at poition ( n ζ, ), the tile directly outh, at poition ( n ζ, ) mut have a north binding domain. Therefore, that tile i one of the following four tile: {,,, }. For any one of thoe four tile to attach, it eat neighbor wet binding domain mut be,, or, meaning two thing: the firt bit of the binding domain cannot be 2, which implie that the carry bit of the tile to the eat i, and the econd bit of the binding domain cannot be, which implie that hifted α ha not run pat the wet bound of the computation. Together, thoe two propertie enure that the multiplication i proceeding correctly. In turn, the tile directly outh of that poition, at poition ( n ζ, ), by the ame reaoning, mut be one of thoe four tile and the tile to it eat ha the two above propertie. And o on, all the tile in the column n ζ mut be one of thoe four tile, until the poition ( n ζ, n β ), where the tile mut be f (by the definition, S ( n δ, n β ) = f ). Therefore, the tile may attach only if n δ = n ζ and every row of the multiplication ha a carry bit and ha not overhifted α. Working backward again, for a tile to attach at poition ( n ζ, ) in F, the tile directly eat mut have a wet binding domain, and thu mut be 8

19 one of the following four tile: {,,, }. For each of thoe tile, the firt digit of it north binding domain matche the econd digit of it outh binding domain. Therefore, v(f ( (n ζ ), )) mut equal v(f ( (n ζ ), 2)). The ame argument hold for the tile to the eat of that tile, and o on until poition (, ), where F (, ) = γ. By emma 3.7, if δ = αβ, then i N, v(s ( i, )) = δ i. Therefore, a tile may attach at poition ( n ζ, ) to F if and only if γ = αβ. Figure 7(a) how a ample eed S for β = 3 = 2, α = 97 = 2, and ζ = 999 = 2. Figure 7(b) how the final configuration F that S produce on S. Becaue ζ = αβ, F contain the tile. emma 3. (Checking aembly time lemma) For all α, β, ζ 2, if ζ = αβ then the aembly time of the final configuration F produced by S on S that encode α, β, and ζ and pad α with n β or n β -tile i Θ(n ζ ). Proof: Each tile in the th row of F that attache require one other pecific tile to attach firt. The ame i true for column n ζ. Thu there i no parallelim in each of thoe two aembling procee, and the aembly time for each proce equal the total number of tile that attach in that proce, thu the overall aembly time i Θ(max(n ζ, n β )) = Θ(n ζ ). 3.4 Factoring at temperature four I have defined three different ytem that perform the neceary piece of factoring a number. I now put them together into a ingle ytem S 4 and argue that S 4 i a factoring ytem. Theorem 3. (Temperature four factoring theorem) et Σ 4 = Σ factor Σ Σ. et T 4 = T factor T T. et g 4 be uch that g 4 agree with g factor, g, and g on their repective domain (note that for all element of the domain of more than one of thoe function, thoe function agree). et τ 4 = 4. Then the ytem S 4 = T 4, g 4, τ 4 i a factoring tile ytem. If a tile ytem i the combination of three ditinct tile ytem, the behavior of the large ytem need not be the combined behavior of the three maller ytem the tile from different ytem can interfere with each other. However, I have deigned S factor, S, and S to work together, without interfering. For the mot part, each ytem ue a dijoint et of binding domain, haring binding domain only where tile from the different ytem are deigned to interact. A a reult, tile from each ytem have a particular et of poition where they can attach: tile from T factor can only attach in column and row n β, tile from T can only attach in the rectangle defined by the row and (n β ) and column and (n ζ ), and tile from T can only attach in column n ζ and row, thu the tile do not interfere with each other. Proof:(of Theorem 3.). For all ζ 2, et S 4 be uch that 9

20 * * * * * * * * * * * * * * * * * * (a) * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * (b) Figure 7: A ample eed configuration S that encode β = 3 = 2, α = 97 = 2, and ζ = 999 = 2 (a). S produce a final configuration F on S, which include a tile iff ζ = αβ (b). 2

21 S 4 (, ) = γ. S 4 ( n ζ, 2) = γ. i N uch that i < n ζ, S 4 ( i, 2) = γ ζi For all binary number α, β 2, and for all z, tile from T 4 will attach to S 4 a follow: the tile from T factor nondeterminitically attach to encode two binary number: α in column and β in row n β, padding α with z -tile, a decribed in emma 3.. By emma 3.7 the tile from T attach, in the rectangle defined by the row and (n β ) and column and (n α +z ), to encode δ = αβ in the top row. Finally, tile from T attach, in column n ζ and row, uch that, by emma 3.9 the tile only attache if δ = ζ and z = n ζ n α {n β, n β }. et the identifier tile be. Thu only if there exit a choice of α, β 2 uch that αβ = ζ will the identifier tile attach. Further, if the identifier tile doe attach to a configuration F 4, then F 4 encode α and β a defined in emma 3.. Therefore, S 4 nondeterminitically and identifiably compute the function f(ζ) = α, β. Figure 8(a) how the eed configuration S 4 for ζ = 999 = 2. Figure 8(b), 8(c), and 8(d) how one poible progreion of tile attaching to produce a final configuration that S 4 produce on S 4. I mentioned in ection 2. that the factoring ytem I decribe doe not exactly fit the definition of a ytem computing a function nondeterminitically. The reaon i that the two factor are encoded along column and row n β. Thu the anwer encoding function depend on the ize of the final configuration. However, it would be imple to modify the factoring ytem lightly, leaving all the ame functionality, but alo copying α and β to fixed poition, ay in row. Thi modification would make the ytem fit the definition perfectly. I will not bother to formally define uch a ytem here. While it i poible for tile to attach in exactly the order hown in Figure 8, other order of attachment are alo poible. For example, ome tile in T may attach before ome other tile in T factor do, but they do not interfere with each other becaue they attach in dijoint et of poition. Figure 9 how an alternative poible progreion of tile attaching to the eed configuration to produce a final configuration that S 4 produce on S 4. I have hown the detail of the final configuration that find factor α and β of ζ. It i alo intereting to oberve what happen when the configuration encode an α and β whoe product doe not equal ζ or when α i padded with the wrong number of -tile. Figure (a) demontrate an attempted aembly on choice of α = 9 = 2 and β = 84 = 2. The multiplication find the product αβ = 7644 = 2 and becaue 7644 ζ, the tile along row do not attach and thu the tile cannot attach. Figure (b) encode the correct choice of α and β; however, it doe not pad the α with enough -tile. The multiplication cannot complete, and the tile along the wet column do not attach and thu the tile cannot attach. I now examine the aembly time of S 4 and the fraction of nondeterminitic aemblie that will produce the factor (or the probability of finding the 2

22 * * * * * * * * * * * * * * * * * * * * * * * * * * * * (b) * * * * * (d) Figure 8: S4 factor a number ζ encoded in the eed configuration, e.g. ζ = 999 = 2 (a). In one equence of poible attachment, encoding of two randomly elected number α and β attach nondeterminitically to the eed uch that α, β 2, e.g. α = 97 = 2, β = 3 = 2 (b). S4 then determinitically multiplie α and β to encode δ = αβ (c), and then enure the multiplication i complete and compare δ to ζ to check if they are equal. If they are equal, a pecial X tile attache to ignify that the factor have been found (d). 22 * * (c) * 2 * 2 * * * * * * * * * * * * * * * * * * * * * * * * (a) *

23 * * * * * * * * * * * * * * * * * * * * * * * * * * * (d) Figure 9: S4 factor a number ζ encoded in the eed configuration, e.g. ζ = 999 = 2 (a). In a equence of poible attachment alternate to that in Figure 8, ome tile from all 3 et Tfactor, T, and TX attach without waiting for all the attachment from the other et to complete (b) and (c). Each et tile have deignated area of attachment and cannot attach outide of thoe area, o they do not interfere with each other. Finally, for each choice of α, β 2 and z, all equence of attachment reult in the unique final configuration for thoe α, β, and z, containing a pecial X tile iff αβ = ζ and z = nζ nα (d) (c) * 2 * * * * * * * * * * * * * 2 * * * * * * * * * * * * * * * * (b) (a) 2 * * * * *

24 * * * * * * * * * * * * * * * * * * * * * no tile can attach (a) * * * * * * * * * * * * * * * * * * * * * * * * * * * no tile can attach (b) Figure : If αβ ζ or the number of -tile padding α i incorrect, the tile cannot attach. In (a), the product of 9 = 2 and 84 = 2 doe not equal 9988 = 2, o the tile along the th row do not attach. In (b), α i padded with too few -tile and the tile in the wet column do not attach. Note that both thee configuration are final configuration. 24

25 factor). emma 3.2 (Factoring aembly time lemma) For all α, β, ζ 2 uch that αβ = ζ, the aembly time for S 4 to produce a final configuration F that encode α and β i Θ(n ζ ). Proof: By the factor aembly time emma (lemma 3.2), the encoding of α and β will take Θ(n α + n β ) = Θ(n ζ ) tep. By the multiplication aembly time lemma (emma 3.8), multiplying α and β will take Θ(n α + n β ) = Θ(n ζ ) tep. By the checking aembly time lemma (emma 3.), checking if αβ = ζ will take Θ(n ζ ) tep. When working together, thee ytem do not affect each other peed (though they may work in parallel), o the aembly time for S 4 to produce a final configuration F that encode α and β i Θ(n ζ ). emma 3.3 (Probability of aembly lemma) For all ζ 2, given the eed S 4 encoding ζ, auming each tile that may attach to a configuration at a certain poition attache there with a uniform probability ditribution, the probability that a ingle nondeterminitic execution of S 4 find α, β 2 uch that αβ = ζ i at leat ( ) nζ 6 Proof: In the wort cae, a compoite number ζ ha only 2 prime factor (counting multiplicity). Either of the 2 factor could be α and the other β, but aume α β (by forcing the larger factor to be α, I underetimate the probability). By Corollary 3.4, the probability p of aembling a particular configuration ( encoding α and β and padding α with n ζ n α -tile i at leat ) nζ 6, a long a ζ = αβ. The implication of the probability of aembly lemma (emma 3.3) i that a parallel implementation of S 4, uch a a DNA implementation like thoe in [BRW5, RPW4], with 6 n ζ.5 chance of finding eed ha at leat a e ζ factor and one with (6 n ζ ) eed ha at leat a ( ) e chance. The emma provide a lower bound on the probability of a ucceful aembly achievable by a ditribution of tile attachment probabilitie. I can derive an upper bound on that probability, for large ζ, by arguing that a α and β become large, the dominant majority of the tile in T factor that mut attach to encode α are f alef and f alef and to encode β are f b and f b. Becaue the ditribution of and bit in a random number i even, and at leat one of α and β mut become large a ζ become large, for all probability ditribution of tile attachment, the probability that a ingle nondeterminitic execution of S 4 find α, β 2 uch that αβ = ζ i at mot ( 2) nζ. Note that T 4 contain 5 ditinct tile. 3.5 Factoring at temperature three I have decribed how S 4 factor at temperature four. It wa impler to explain how S 4 work at temperature four, motly becaue I could double the trength of every binding domain and the temperature of S from [Bru]; however, it i 25

26 actually poible to ue roughly the ame tile to compute at temperature three, by altering the glue trength function. The key obervation i that no tile need all four of it neighbor in a configuration in order to attach. The mot a tile need i three, e.g. the tile. I will need to differentiate ome of the eat-wet binding domain from their previouly identical north-outh binding domain counterpart. That i, uppoe that ome of the binding domain of the tile in T 4 were prefixed by v (for vertical) or h (for horizontal). Formally, let Σ 3 = {,,,,, v, v, v, v, v, v,,, h, h, h, h, h, h, h2, h2}. The trength of the binding domain need to be defined uch that each tile attache only under the ame condition under which it attached in S 4. The tile in T factor mut attach with only neighbor preent, o for all σ {,, }, let g 3 (σ, σ) = 3. The tile in T mut attach when 2 of their neighbor, the outh and the eat neighbor, are preent o for all σ {h, h, h, h, h, h, h2, h2}, let g 3 (σ, σ) = 2 and for all σ {v, v, v, v, v, v}, let g 3 (σ, σ ) =. (Note that the alternative making the horizontal binding domain have trength and vertical trength 2 would work for the tile in T, but would not work for the tile in T.) The tile,,, and mut attach when 2 of their neighbor are preent, and ince their eat binding domain have already been defined to be of trength 2, let g 3 (, ) =. Finally, the other tile in T only attach when 3 of their neighbor are preent, and all their binding domain o far have been defined to be of trength, o for all σ {,, }, let g 3 (σ, σ) =. Figure how the tile with the new binding domain labeled. Theorem 3.4 (Temperature three factoring theorem) et T 3 be a defined in Figure. et τ 3 = 3. Then T 3, g 3, τ 3 i a factoring tile ytem. Proof: et α, β, ζ 2, and let S3 be the eed that encode ζ a in Theorem 3.. Then ζ = αβ iff S 4 produce ome final configuration F on S3 that encode α and β and contain the identifier tile and all final configuration that contain encode factor of ζ. et W be a equence of attachment in S 4 that produce F on S3. W = t, (x, y ), t, (x, y ),, t k, (x k, y k ) uch that t attache at poition (x, y ) to S3 to produce S3, t attache at poition (x, y ) to S3 to produce S3 2, and o on to produce the final configuration S3 k+ = F. Remember that g 3 i deigned uch that every tile in T 3 attache in S 3, under and only under the ame condition a it counterpart tile in T 4 attached in S 4. Thu the equence of attachment W i alo a valid equence of attachment for S 3. Therefore, on eed S3, encoding ζ, for all final configuration F produced by S 3 on S3, F contain iff F encode α and β uch that ζ = αβ. emma 3.5 For all α, β, ζ 2 uch that αβ = ζ, the aembly time for S 4 to produce a final configuration F that encode α and β i Θ(n ζ ). emma 3.6 For all ζ 2, given the eed S encoding ζ, auming uniform ditribution of all tile, the probability that a ingle nondeterminitic execution of S 3 find α, β 2 uch that αβ = ζ i at leat ( ) nζ 6 26