Linear Time Solution to Prime Factorization by Tissue P Systems with Cell Division

Linea Time Solu ion o P ime Fac o iza ion by Tissue P Sys ems wi h Cell Di ision Xingyi Zhang1, Yunyun Niu2, Linqiang Pan2, Ma io J. P´e ez-Jim´enez3 1School o Compu e Science and Technology Anhui Uni e si y, 230039 He ei, China [email p o ec ed] 2Key Labo a o y o Image P ocessing and In elligen Con ol Depa men o Con ol Science and Enginee ing Huazhong Uni e si y o Science and Technology, 430074 Wuhan, China [email p o ec ed], [email p o ec ed] 3Depa men o Compu e Science and A i icial In elligence Uni e si y o Se illa, A da. Reina Me cedes s/n, 41012 Se illa, Spain [email p o ec ed] Summa y. P ime ac o iza ion is use ul and c ucial o public-key c yp og aphy, and i s applica ion in public-key c yp og aphy is possible only because p ime ac o iza ion has been p esumed o be di icul . A polynomial- ime algo i hm o p ime ac o iza ion on a quan um compu e is gi en by P. W. Sho in 1997. In his wo k, a linea - ime solu ion o p ime ac o iza ion is gi en on a kind o biochemical compu a ional de ices – issue P sys ems wi h cell di ision, ins ead o physical compu a ional de ices. 1 In oduc ion In ma h, p ime ac o iza ion is he b eaking down o a composi e numbe in o smalle p imes, which when mul iplied oge he equal he o iginal in ege . Cu - en ly, hough he p ime ac o iza ion p oblem is no known o be NP-hand, no e icien algo i hm is publicly known. I is gene ally conside ed in ac able. The p esumed compu a ional ha dness o his p oblem is a he hea o se e al algo- i hms in c yp og aphy such as RSA [15]. Many a eas o ma hema ics and compu e science ha e been b ough o bea on he p ime ac o iza ion p oblem, including ellip ic cu es, algeb aic numbe heo y, and quan um compu ing. A polynomial- ime algo i hm o p ime ac o iza ion on a quan um compu e is gi en by P. W. Sho in 1997 [16]. This will ha e signi ican implica ions o c yp og aphy i a la ge quan um compu e is e e buil . Howe e , be o e a p ac ical quan um compu e appea s, i is s ill o in e es o ind any easonable compu a ional de ices o sol ing p ime ac o iza ion p oblem. In his wo k, we shall gi e a linea - ime solu ion o p ime ac o iza ion on a class o 356 X. Zhang e al. biochemical compu a ional de ices – issue P sys ems wi h cell di ision, ins ead o physical compu a ional de ices. Tissue P sys ems wi h cell di ision is a class o compu a ional de ices in mem- b ane compu ing. Memb ane compu ing is an eme gen b anch o na u al com- pu ing, which is inspi ed by he s uc u e and he unc ioning o li ing cells, as well as he o ganiza ion o cells in issues, o gans, and o he highe o de s uc- u es. The de ices in memb ane compu ing, called P sys ems, p o ide dis ibu ed pa allel and non-de e minis ic compu ing models. Since Gh. P˘aun in oduced he i s P sys em in [12], his a ea is hea ily in es iga ed. Please e e o [13] o an in oduc ion o memb ane compu ing, and e e o [17] o u he bibliog aphy. In o mally, a P sys em consis s o a memb ane s uc u e, in he compa men s o which one places mul ise s o objec s which e ol e acco ding o gi en ules in a synch onous, non-de e minis ic, maximally pa allel manne . Tissue P sys ems a e a class o P sys ems, whe e memb anes a e placed in he nodes o a g aph. I is a ne o p ocesso s dealing wi h symbols and communica ing hese symbols along channels speci ied in ad ance. The communica ion among cells is based on sympo /an ipo ules, which was in oduced o P sys ems in [11]. Sympo ules mo e objec s ac oss a memb ane oge he in one di ec ion, whe eas an ipo ules mo e objec s ac oss a memb ane in opposi e di ec ions. This model has wo bio- logical inspi a ions (see [9]): in e cellula communica ion and coope a ion be ween neu ons. In [14], issue P sys ems a e endowed wi h he abili y o ge ing new cells based on he mi osis o cellula di ision, hus ob aining he abili y o gene a ing an exponen ial amoun o wo kspace in polynomial ime. Such a ian o issue P sys ems is called issue P sys ems wi h cell di ision. Tissue P sys ems wi h cell di ision we e widely in es iga ed o sol ing NP- comple e p oblems. Some o hem deal wi h non-nume ical NP-comple e decision p oblems, such as SAT p oblem [14], 3-colo ing p oblem [2], e ex co e [4]. O he s deal wi h nume ical NP-comple e decision p oblems, ha is, decision p oblems whose ins ances consis o se s o sequences o in ege numbe s, such as subse sum [3], pa i ion p oblem [5]. Al hough p ime ac o iza ion we shall conside is a nume ical p oblem, i is nei he a decision p oblem no an op imiza ion p oblem. In his wo k, we shall cons uc a amily o issue P sys ems wi h cell di ision, which can decompose in ege numbe s in a linea ime wi h espec o he leng h o bina y ep esen a ion o he in ege o be ac o ed. As a esul o compu a ion, a p ime numbe is sen o a p e ixed ou pu memb ane, ins ead o yes o no. Up o now, besides he e a e wo polynomial- ime solu ions o p ime ac o - iza ion by P sys ems wi h ac i e memb anes [6, 10], one well known polynomial algo i hm ha sol es ac o iza ion p oblem is based on quan um compu e [16]. As he case o quan um compu e , he solu ion gi en in his wo k indica es how pow- e ul issue P sys ems wi h cell di ision can be, al hough a his momen nobody knows how o build a biochemical compu e . The pape is o ganized as ollows. In Sec ion 2, some p elimina ies a e ecalled. The o mal de ini ion o issue P sys ems wi h cell di ision is gi en in Sec ion 3. A amily o issue P sys ems ha uni o mly sol e he ac o iza ion p oblem is Fac o iza ion by Tissue P Sys ems wi h Cell Di ision 357 p esen ed in Sec ion 4, wi h a sho o e iew o he compu a ion and he necessa y esou ces. Conclusions and commen s a e p esen ed in Sec ion 5. 2 P elimina ies An alphabe Σis a non-emp y se , whose elemen s a e called symbols. An o de ed sequence o symbols is a s ing. The numbe o symbols in a s ing uis he leng h o he s ing, and i is deno ed by |u|. As usual, he emp y s ing (wi h leng h 0) will be deno ed by λ. The se o s ings o leng h nbuil wi h symbols om he alphabe Σis deno ed by Σnand Σ∗=∪n≥0Σn. A language o e Σis a subse om Σ∗. Amul ise mo e a se Ais a pai (A, ), whe e :A→Nis a mapping. I m= (A, ) is a mul ise , hen i s suppo is de ined as supp(m) = {x∈A| (x)> 0}and i s size is de ined as Px∈A (x). A mul ise is emp y ( esp. ini e) i i s suppo is he emp y se ( esp. ini e). I m= (A, ) is a ini e mul ise o e A, and supp(m) = {a1, . . . , ak}, hen i will be deno ed as m={{a (a1) 1, . . . , a (ak) k}}. Tha is, supe sc ip s indica e he mul iplici y o each elemen . I (x) = 0 o any x∈A, hen his elemen is omi ed. 3 Tissue P Sys ems wi h Cell Di ision In [8, 9], he i s de ini ion o he model o issue P sys ems was p oposed, whe e he memb ane s uc u e did no change along he compu a ion. We now shall in oduce a model o issue P sys ems wi h cell di ision based on he cell-like model o P sys ems wi h memb anes di ision [14]. The biological inspi a ion o his model is clea : ali e issues a e no s a ic ne wo k o cells, since new cells a e gene a ed by memb ane ission in a na u al way. The main ea u es o his model, om he compu a ional poin o iew, a e ha cells a e no pola ized ( he con a y holds in he cell-like model o P sys ems wi h ac i e memb anes, see [13]); he cells ob ained by di ision ha e he same labels as he o iginal cell and i a cell is di ided, i s in e ac ion wi h o he cells o wi h he en i onmen is blocked du ing he di ision p ocess. In some sense, his means ha while a cell is di iding i closes i s communica ion channels wi h o he cells and wi h he en i onmen . Fo mally, a ( unc ion) compu ing issue P sys em wi h cell di ision o deg ee q≥1 and o de (m, n), m≥1, n ≥1, is a uple o he o m Π= (Γ, Σ, Λ, w1, . . . , wq,E,R, iin, iou ), whe e: 1. Γis he alphabe o objec s; 358 X. Zhang e al. 2. Σ={a1, . . . , am}is an o de ed inpu alphabe s ic ly con ained in Γ; 3. Λ={b1, . . . , bn}is an o de ed ou pu alphabe con ained in Γ; 4. w1, . . . , wqa e s ings o e Γ, desc ibing he ini ial mul ise s o objec s placed in he cells o he sys em a he beginning o he compu a ion; 5. E ⊆ Γis he se o objec s in he en i onmen in a bi a ily copies each; 6. Ris a ini e se o ules o he ollowing o ms: (a) (i, u/ , j), o i, j ∈ {0,1,2, . . . , q}, i 6=j,u, ∈Γ∗; Communica ion ules; 1,2,· · · , q iden i y he cells o he sys em, 0 is he en i onmen ; when applying a ule (i, u/ , j), he objec s o he mul ise ep esen ed by ua e sen om egion i o egion jand simul aneously he objec s o he mul ise a e sen om egion j o egion i(|u|+| |is called he leng h o he communica ion ule (i, u/ , j)); (b) [a]i→[b]i[c]i, whe e i∈ {1,2, . . . , q},a, b, c ∈Γ, and i6=iou ; Di ision ules; in eac ion wi h an objec a, he cell is di ided in o wo cells wi h he same label; all he objec s in he o iginal cells a e eplica ed and copies o hem a e placed in each o he new cells, wi h he excep ion o he objec a, which is eplaced by he objec bin he i s new cell and by cin he second one; he ou pu cell canno be di ided; 7. iin ∈ {1,2, . . . , q}is he inpu cell; 8. iou ∈ {0,1,2, . . . , q}is he ou pu cell. The ules o a sys em as abo e a e used in he non-de e minis ic maximally pa allel manne . In each s ep, all cells which can e ol e mus e ol e in a maximally pa allel way (in each s ep we apply a mul ise o ules which is maximal, no u he ule can be added). This way o applying ules has only one es ic ion when a cell is di ided, he di ision ule is he only one which is applied o ha cell in ha s ep; he objec s inside ha cell do no e ol e by means o communica ion ules. Thei labels p ecisely iden i y he ules which can be applied o hem. A con igu a ion o issue P sys em wi h cell di ision is desc ibed by all mul i- se s o objec s o e Γassocia ed wi h all he cells p esen in he sys em and he mul ise o objec s o e Γ− E associa ed wi h en i onmen . The ini ial con igu a- ion o he sys em Πwi h inpu w∈Σ∗is he uple (w1, w2, . . . , wiin w, . . . , wq; ∅); ha is, he co esponding con igu a ion a e adding he mul ise w o he con en o he inpu cell iin. The compu a ion s a s om he ini ial con igu a- ion and p oceeds as de ined abo e. When he e is no ule can be applied, he compu a ion s ops. Only hal ing compu a ions gi e a esul . I C={Ci}i< is a hal ing compu a ion, whe e Cia e con igu a ions, hen he esul o compu- a ion Ou pu (C)=(C −1 b1(iou ), C −1 b2(iou ), . . . , C −1 bn(iou )), whe e C −1 bj(iou ), 1≤j≤n, is he mul iplici y o objec bjin he egion iou in he hal ing con igu- a ion C −1. Fo a unc ion , we deno e he domain o by D( ) and he ange o by R( ). Fo a issue P sys em wi h cell di ision Πha ing o de ed inpu alphabe Σ={a1, a2, . . . , am}and o de ed ou pu alphabe Λ={b1, b2, . . . , bn}, and pa ial unc ion :Nm→Nn, unc ion is encoded in a una y no a ion in he ollowing Fac o iza ion by Tissue P Sys ems wi h Cell Di ision 359 way: (α1, . . . , αm)∈D( ) is exp essed by aα1 1aα2 2. . . aαm m; (β1, . . . , βn)∈R( ) is exp essed by bβ1 1bβ2 2. . . bβn n. De ini ion 1. We say ha a pa ial unc ion :Nm→Nnis compu ed in poly- nomial ime by a amily Π={Π( )| ∈N}o issue P sys ems wi h cell di ision in una y encoding i he ollowing holds: •The amily Πis polynomially uni o m by Tu ing machines, ha is, he e exis s a de e minis ic Tu ing machine wo king in polynomial ime which cons uc s he sys em Π( ) om ∈N. •The e exis a polynomial- ime compu able unc ion so e he domain D( )o unc ion such ha : − o each u= (α1, . . . , αm)∈D( ),s(u)is a na u al numbe and aα1 1. . . aαm m is an inpu mul ise o he sys em Π(s(u)); − he amily Πis polynomially bounded wi h ega d o ( , s), ha is, he e exis s a polynomial unc ion p, such ha o each u= (α1, . . . , αm)∈D( ) e e y compu a ion o Π(s(u)) wi h inpu aα1 1. . . aαm mis hal ing and, mo e- o e , i pe o ms a mos p(|u|)s eps; − he amily Πis sound wi h ega d o ( , s), ha is, o each u= (α1, . . . , αm)∈D( ), i he e exis s a compu a ion Co Π(s(u)) wi h inpu aα1 1. . . aαm msuch ha Ou pu (C) = (β1, . . . , βn), hen (u) = (β1,···, βn); − he amily Πis comple e wi h ega d o ( , s), ha is, o each u= (α1, . . . , αm)∈D( ), i (u)=(β1,· · · , βn), hen e e y compu a ion Co Π(s(u)) wi h inpu aα1 1. . . aαm mhas Ou pu (C) = {β1, . . . , βn}. In he De ini ion 1, he inpu and ou pu a e encoded in una y no a ion. How- e e , in classical complexi y heo y, based upon Tu ing machine, swi ching om bina y o una y encoding gene ally co esponds o simpli y he p oblem. In his wo k, bina y encoding is used o in ege ac o iza ion p oblem. In wha ollows, we will gi e he de ini ion ha a unc ion is compu ed by a amily o P sys ems wi h cell di ision in bina y encoding. In he case o bina y encoding, he inpu alphabe is no asked o be o de ed, and no ou pu alphabe is ixed. A( unc ion) compu ing issue P sys em wi h cell di ision wi h inpu o deg ee q≥1 is a uple o he o m Π= (Γ, Σ, w1, . . . , wq,E,R, iin, iou ), whe e: 1. Γis he alphabe o objec s; 2. Σis an (un-o de ed) inpu alphabe s ic ly con ained in Γ; 3. w1, . . . , wqa e s ings o e Γ, desc ibing he ini ial mul ise s o objec s placed in he cells o he sys em a he beginning o he compu a ion; 4. E ⊆ Γis he se o objec s in he en i onmen in a bi a ily copies each; 5. Ris a ini e se o ules o he ollowing o ms: (a) (i, u/ , j), o i, j ∈ {0,1,2, . . . , q}, i 6=j,u, ∈Γ∗; 360 X. Zhang e al. (b) [a]i→[b]i[c]i, whe e i∈ {1,2, . . . , q},a, b, c ∈Γ, and i6=iou ; 6. iin ∈ {1,2, . . . , q}is he inpu cell; 7. iou ∈ {0,1,2, . . . , q}is he ou pu cell. In seman ics, P sys ems ha ing un-o de ed alphabe s is he same wi h P sys- ems wi h o de ed inpu and ou pu alphabe s excep o he way o encoding inpu and ou pu . In he una y encoding, he sizes o o de ed inpu and ou pu alphabe s a e ela ed wi h he dimensions o domain and ange o unc ion ha is compu ed. Speci ically, an o de ed inpu alphabe {a1, . . . , am}and an o de ed ou pu alphabe {b1, . . . , bn}can encode each unc ion whose domain ( esp. ange) is a subse o Nm0(m0≤m) ( esp. Nn0(n0≤n)). In he bina y encoding, he size o alphabe is ela ed wi h bo h inpu alue and ou pu alue. Fo example, o he unc ion (x)=22x(x∈N) and an inpu n, he leng h o inpu nin bina y exp ession is blg nc+ 1, and he leng h o ou pu (n) in bina y exp essions is 2n+ 1, which is an exponen ial unc ion wi h espec o blg nc+ 1. Fo unc ions such as (x) = 22x, maybe, we need exponen ial (wi h espec o he inpu size) la ge alphabe o encode he unc ion in P sys ems, hence we canno cons uc a amily o P sys ems wi h cell di ision in polynomial ime by Tu ing machine o compu e unc ions such as (x)=22x. I depends on he p ope y o unc ion whe he a unc ion can be compu ed by issue P sys ems wi h cell di ision in bina y encoding. Fo p ime ac o iza ion p oblem, he ac o s a e less han he in ege o be ac o ed. In ac enables us o ind a easonable bina y encoding o p ime ac o - iza ion p oblem. Speci ically, we shall use he me hod om [7] o encode bina y numbe s by mul ise s o objec s. Le xk−1,· · · , x1, x0(wi h k≥1) be he bina y ep esen a ion o in ege x≥0, ha is, x=Pk−1 i=0 xi2i. We use he objec s om he ollowing alphabe Ak, o k≥1: Ak={hb, ji | b∈ {0,1}, j ∈ {1,2,· · · , k}}. Objec s hb, jiis used o ep esen bi bin posi ion jin he bina y encoding o an in ege numbe . Hence, o ep esen he abo e numbe xwe will use he ollowing mul ise (ac ually, a se ) o objec s: hxk−1, k −1i,· · · ,hx1,1i,hx0,0i. Le us ema k ha he alphabe Akdepends on he leng h o he bina y ep e- sen a ion o he numbe x. Mo eo e , i is clea ha wi h Akwe can ep esen all in ege numbe s in he ange 0,1,· · · ,2k−1. In o de o dis inguish be ween he objec s ha ep esen he bi s o di e en in ege s Aand B, a leading label A, B a e used o ma k each elemen in he mul ise . To his aim, he alphabe Akis modi ied as ollows: A0 k={hl, b, ji | l∈ {A, B}, b ∈ {0,1}, j ∈ {1,2,· · · , k}}. In his way, he i- h bi o A( ha is, ai) and he j- h bi o B( ha is, bj) a e ep esen ed by he objec s hA, ai, iiand hB, bj, ji, espec i ely. Fac o iza ion by Tissue P Sys ems wi h Cell Di ision 361 In gene al, we gi e he ollowing de ini ion ha a unc ion is compu ed by P sys ems wi h cell di ision in bina y encoding. De ini ion 2. We say ha a pa ial unc ion :N→Nis compu ed in polynomial ime by a amily Π={Π( )| ∈N}o issue P sys ems wi h cell di ision in bina y encoding i he ollowing holds: •The amily Πis polynomially uni o m by Tu ing machines, ha is, he e exis s a de e minis ic Tu ing machine wo king in polynomial ime which cons uc s he sys em Π( ) om ∈N. •The e exis s a pai (cod, s)o polynomial- ime compu able unc ions o e he domain D( )o unc ion such ha : − o each u∈D( ),s(u)is a na u al numbe and cod(u)is an inpu mul ise o he sys em Π(s(u)); − he amily Πis polynomially bounded wi h ega d o ( , cod, s), ha is, he e exis s a polynomial unc ion p, such ha o each u∈D( )e e y compu a ion o Π(s(u)) wi h inpu cod(u)is hal ing and, mo eo e , i pe - o ms a mos p(|u|)s eps; − he amily Πis sound wi h ega d o ( , cod, s), ha is, o each u∈D( ), i he e exis s a compu a ion Co Π(s(u)) wi h inpu cod(u)and he objec s in egion iou in he las con igu a ion o Cencode (β1,· · · , βq)∈Nq, hen (u) = (β1,· · · , βq); − he amily Πis comple e wi h ega d o ( , cod, s), ha is, o each u∈ D( ), i (u)=(β1,· · · , βq)∈Nq, hen in e e y compu a ion o Π(s(u)) wi h inpu cod(u), he objec s in egion iou in he las con igu a ion encode (β1,· · · , βq). 4 A Linea Time Solu ion o he Fac o iza ion P oblem When we discuss he p ime ac o iza ion p oblem, i is necessa y o dis inguish wo di e en e sions o he p oblem: decision p oblem e sion and unc ion p oblem e sion. The decision p oblem e sion o p ime ac o iza ion can be o mula ed as “is na composi e numbe ?” (o equi alen ly: “is na p ime numbe ?”). This e sion is na u al and use ul because mos well-s udied complexi y classes a e de ined as classes o decision p oblems, no unc ion p oblems. Bu he decision p oblem e sion o p ime ac o iza ion is much easie han he p oblem o inding he ac o s o n. Speci ically, i can be sol ed in polynomial ime (wi h espec o he numbe o digi s o n) wi h he AKS p imali y es [1]. The unc ion p oblem e sion o p ime ac o iza ion: gi en an in ege n, ind an in ege dwi h 1 < d < n ha di ides n(o conclude ha nis p ime). I is i ially in he class FNP, bu we do no known whe he i lies in class FP o no . This e sion is gene ally conside ed in ac able, which means ha no polynomial- ime (wi h espec o he ins ance size) algo i hm is known ha sol es i on e e y 362 X. Zhang e al. ins ance; and i is he e sion sol ed by mos p ac ical implemen a ions. In his wo k, we shall conside a es ic ed e sion o p ime ac o iza ion p oblem, based on he ollowing wo ac s. (1) Gi en an algo i hm o in ege ac o iza ion, one can ac o any in ege down o i s cons i uen p ime ac o s by epea ed applica- ion o his algo i hm. (2) No all numbe s o a gi en leng h a e equally ha d o ac o . Semip imes ( he p oduc o wo p ime numbe s) a e belie ed as he ha des ins ances o in ege ac o iza ion o cu en ly known echniques. P oblem 1. NAME: ac o iza ion. – INSTANCE: a posi i e in ege numbe which is he p oduc o wo p ime num- be s. – OUTPUT: he p ime ac o ha is no g ea e han ano he one. Nex , we shall cons uc a amily {Π(k)}k∈No issue P sys ems wi h cell di ision o ac o in ege s, whe e each sys em Π(k) can decompose all numbe s o leng h kin bina y o m, p o ided ha an app op ia e inpu mul ise is gi en. The esolu ion is a b u e o ce algo i hm, which consis s o he ollowing s ages: •Gene a ion S age: By di ision, all he possible pai s o in ege numbe s o leng h kin bina y o m a e p oduced (one pai o each memb ane wi h label 2). •P e-checking S age: In his s age, he p oduc o each pai o in ege numbe s o leng h kis calcula ed. •Checking S age: The sys em checks whe he o no he e exis s a pai o in ege numbe s such ha hei p oduc equals o he numbe n o be composed. •Ou pu S age: The sys em sends o he ou pu egion a p ime numbe . Fo each k∈N, Π(k) = (Γ(k), Σ(k), w1, w2,R(k),E(k), iin, iou ), wi h he ollowing componen s: •Γ(k) = Σ(k)∪ {ai, bi,hX, 0, ii, i, gi|0≤i≤k−1}∪ {hA, j, ii,hB, j, ii,hA0, j, ii,hB0, j, ii | 0≤i≤k−1,0≤j≤1}∪ {hAj, l, ii,hBj, l, ii | 0≤i≤k−1,0≤j≤ dlg ke+ 1,0≤l≤1}∪ {ci|0≤i≤4k+dlg 2ke+dlg ke+ 5}∪{c0 i|1≤i≤ dlg ke+ 2k+ 3}∪ {hC, 0, ii,hC, 1, ii | 0≤i≤2k−1} ∪ {hi, ji | 0≤i, j ≤k−1}∪ {di| −1≤i≤k−2}∪{ei| −1≤i≤k−1}∪{z}. •Σ(k) = {hn, 0, ii,hn, 1, ii | 0≤i≤k−1}. •w1={{c0}}. •w2={{a0a1· · · ak−1b0b1···bk−1z}} ∪ {{hi, ji | 0≤i, j ≤k−1}}. • R(k) is he se o ules: 1. Di ision ule: 1,i ≡[ai]2→[hA, 0, ii]2[hA, 1, ii]2, o 0 ≤i≤k−1; 2,i ≡[bi]2→[hB, 0, ii]2[hB, 1, ii]2, o 0 ≤i≤k−1. Fac o iza ion by Tissue P Sys ems wi h Cell Di ision 363 2. Communica ion ules: 3,i ≡(1, ci/c2 i+1,0), o 0 ≤i≤2k−1; 4≡(1, c2k/z, 2); 5,i ≡(2, c2k+i/c2 2k+i+1,0), o 0 ≤i≤ dlg 2ke − 1; 6,i,j ≡(2, c2k+dlg 2kehA, j, ii/c2k+dlg 2ke+1hA0, j, ii,0), o 0 ≤i≤k−1, 0 ≤j≤1; 7,i,j ≡(2, c2k+dlg 2kehB, j, ii/c2k+dlg 2ke+1hB0, j, ii,0), o 0 ≤i≤k−1, 0 ≤j≤1; 8≡(2, c2k+dlg 2ke+1/c0 1c2k+dlg 2ke+2,0); 9,i ≡(2, c2k+dlg 2ke+i/c2k+dlg 2ke+i+1,0), o 2 ≤i≤ dlg ke+ 2k+ 4; 10,i ≡(2, c0 i/c0 i+1,0), o 1 ≤i≤ dlg ke+ 2k+ 2; 11,i,j,l ≡(2,hAj, l, ii/hAj+1, l, ii2,0), o 0 ≤i≤k−1, 0 ≤j≤ dlg ke, 0 ≤l≤1; 12,i,j,l ≡(2,hBj, l, ii/hBj+1, l, ii2,0), o 0 ≤i≤k−1, 0 ≤j≤ dlg ke, 0 ≤l≤1; 13,i,j ≡(2,hAdlg ke+1,0, iihBdlg ke+1,0, jihi, ji/hC, 0, i +ji,0), o 0 ≤i, j ≤k−1; 14,i,j ≡(2,hAdlg ke+1,0, iihBdlg ke+1,1, jihi, ji/hC, 0, i +ji,0), o 0 ≤i, j ≤k−1; 15,i,j ≡(2,hAdlg ke+1,1, iihBdlg ke+1,0, jihi, ji/hC, 0, i +ji,0), o 0 ≤i, j ≤k−1; 16,i,j ≡(2,hAdlg ke+1,1, iihBdlg ke+1,1, jihi, ji/hC, 1, i +ji,0), o 0 ≤i, j ≤k−1; 17,i ≡(2,hC, 0, iihC, 0, ii/hC, 0, ii,0), o 0 ≤i≤2k−2; 18,i ≡(2,hC, 0, iihC, 1, ii/hC, 1, ii,0), o 0 ≤i≤2k−2; 19,i ≡(2,hC, 1, iihC, 1, ii/hC, 0, iihC, 1, i + 1i,0), o 0 ≤i≤2k−2; 20,i,j ≡(2, c0 dlg ke+2k+3hC, 1, iihn, j, k −1i/λ, 0), o k≤i≤2k−2, 0 ≤j≤1; 21,i,j ≡(2, c4k+dlg 2ke+dlg ke+5hC, j, iihn, j, ii/hX, 0, ii,0), o 0 ≤i≤k−1, 0 ≤j≤1; 22 ≡(2,hX, 0, k −1i/dk−2,0); 23,i ≡(2, dihX, 0, ii/di−1,0), o 0 ≤i≤k−2; 24 ≡(2, d−1/ek−1,0); 25,i,j ≡(2,hAdlg ke+1, j, iihBdlg ke+1, j, iiei/hAdlg ke+1, j, ii hBdlg ke+1, j, iiei−1,0), o 0 ≤i≤k−1, 0 ≤j≤1; 26 ≡(2, e−1/ 0,0); 27,i ≡(2,hAdlg ke+1,1, iihBdlg ke+1,0, iiei/hAdlg ke+1,1, ii hBdlg ke+1,0, ii 0,0), o 0 ≤i≤k−1; 28,i,j ≡(2, ihBdlg ke+1, j, ii/ i+1hB0, j, ii,0), o 0 ≤i≤k−2, 0 ≤j≤1; 29,j ≡(2, k−1hBdlg ke+1, j, k −1i/hB0, j, k −1i,0), o 0 ≤j≤1; 30,i,j ≡(2,hB0, j, ii/λ, 3), o 0 ≤i≤k−1, 0 ≤j≤1; 31,i ≡(2,hAdlg ke+1,0, iihBdlg ke+1,1, iiei/hAdlg ke+1,0, ii hBdlg ke+1,1, iig0,0), o 0 ≤i≤k−1; 32,i,j ≡(2, gihAdlg ke+1, j, ii/gi+1hA0, j, ii,0), o 0 ≤i≤k−2, 0 ≤j≤1; 370 X. Zhang e al. 1 z 16 2 c 16 3 〈 C , 0,1 〉 〈 C , 0,2 〉 〈 n , 1, 1 〉 〈 X , 0, 0 〉 〈 A 2, 0,0 〉 2 2 c 16 3 〈 C , 0,1 〉 〈 C , 0,2 〉 〈 n , 1, 1 〉 〈 X , 0,0 〉 〈 A 2, 1, 0 〉 2 〈 A 2, 0,1 〉 2 〈 A 2, 0,1 〉 2 〈 B 2, 0,0 〉 2 〈 B 2, 0,1 〉 2 〈 B 2, 0,0 〉 2 〈 B 2, 0,1 〉 2 2 c 16 3 〈 C , 0,1 〉〈 C , 0,2 〉 〈 n, 1, 1 〉 〈 X , 0,0 〉 〈 A 2, 0,0 〉 2 2 c 16 3 〈 C , 0,1 〉 〈 C , 0,2 〉 〈 n, 1, 1 〉 〈 X , 0,0 〉 〈 A 2, 1,1 〉 2 〈 A 2, 1,1 〉 2 〈 A 2, 1, 0 〉 2 〈 B 2, 0,0 〉 2 〈 B 2, 0,1 〉 2 〈 B 2, 0,0 〉 2 〈 B 2, 0,1 〉 2 2 c 16 3 〈 C , 0,1 〉 〈 C , 0,2 〉 〈 n , 1, 1 〉 〈 X , 0, 0 〉 〈 A 2, 0,0 〉 2 2 c 16 4 〈 C , 1,0 〉 〈 C , 0,1 〉 〈 C , 0,2 〉 〈 n , 1, 1 〉 〈 n, 0,0 〉 〈 A 2, 0,1 〉 2 〈 A 2, 0,1 〉 2 〈 A 2, 1, 0 〉 2 〈 B 2, 0,1 〉 2 〈 B 2, 1, 0 〉 2 〈 B 2, 0,1 〉 2 〈 B 2, 1, 0 〉 2 2 c 16 2 〈 C , 0,2 〉 〈 X , 0, 1 〉〈 X , 0,0 〉 〈 A 2, 1, 1 〉 2 2 c 16 3 〈 C ,0,2 〉 〈 X , 0, 1 〉 〈 A 2, 1,1 〉 2 〈 A 2, 0,0 〉 2 〈 A 2, 1, 0 〉 2 〈 B 2, 0,1 〉 2 〈 B 2, 1, 0 〉 2 〈 B 2, 0,1 〉 2 〈 B 2, 1, 0 〉 2 2 c 16 3 〈 C , 0,2 〉 〈 X , 0, 0 〉 〈 A 2, 0,1 〉 2 2 c 16 2 〈 C , 0,2 〉 〈 X , 0, 1 〉 〈 X , 0, 0 〉 〈 A 2, 0,1 〉 2 〈 A 2, 0,0 〉 2 〈 A 2, 1, 0 〉 2 〈 B 2, 1,1 〉 2 〈 B 2, 0,0 〉 2 〈 B 2, 1, 1 〉 2 〈 B 2, 0,0 〉 2 2 c 16 3 〈 C , 0,1 〉 〈 X , 0,0 〉 〈 A 2, 1, 1 〉 2 2 c 16 3 〈 C , 1,1 〉 〈 X , 0, 0 〉 〈 A 2, 1,1 〉 2 〈 A 2, 0,0 〉 2 〈 A 2, 1, 0 〉 2 〈 B 2, 1,1 〉 2 〈 B 2, 0,0 〉 2 〈 B 2, 1, 1 〉 2 〈 B 2, 0,0 〉 2 2 c 16 3 〈 C , 0,1 〉 〈 C , 0,2 〉 〈 n , 1, 1 〉 〈 X , 0,0 〉 〈 A 2, 0,1 〉 2 2 c 16 3 〈 C , 1,0 〉 〈 C , 0,2 〉 〈 X , 0, 1 〉 〈 n, 0,0 〉 〈 A 2, 0,1 〉 2 〈 A 2, 0,0 〉 2 〈 A 2, 1, 0 〉 2 〈 B 2, 1, 1 〉 2 〈 B 2, 1, 0 〉 2 〈 B 2, 1, 1 〉 2 〈 B 2, 1, 0 〉 2 2 c 16 3 〈 C , 1,1 〉 〈 X , 0,0 〉 〈 A 2, 1, 1 〉 2 2 c 16 4 〈 C , 1,0 〉 〈 C , 0,1 〉 〈 C , 0,2 〉 〈 n , 0, 0 〉 〈 A 2, 1,1 〉 2 〈 A 2, 0,0 〉 2 〈 A 2, 1, 0 〉 2 〈 B 2, 1, 1 〉 2 〈 B 2, 1, 0 〉 2 〈 B 2, 1, 1 〉 2 〈 B 2, 1, 0 〉 2 c ' 8 c ' 8 c ' 8 c ' 8 c ' 8 c ' 8 c ' 8 c ' 8 c ' 8 c ' 8 c ' 8 〈 C ,1,0 〉 〈 n, 0,0 〉 〈 C , 0,1 〉 〈 n , 1, 1 〉 c ' 8 Fig. 5. The con igu a ion o sys em Π(2) o ac o ing in ege numbe 2 a s ep 19 5 Conclusions and Commen s P ime ac o iza ion p oblem is no in i sel widely use ul p oblem. I has become use ul only because i has been ound o be c ucial o public-key c yp og aphy, and his applica ion is in u n possible only because hey ha e been p esumed o be di icul . Cu en ly, no de e minis ic polynomial- ime algo i hm is known, which can be execu ed on Tu ing machines, ha sol es he p oblem o e e y possible ins ance. I is o in e es o explo e any possible and easonable way o sol e p ime ac o iza ion p oblem because o i s impo ance in public-key c yp og aphy. P ime ac o iza ion p oblem is nei he decision p oblem no op imiza ion p ob- lem. In his wo k, i is conside ed as a unc ion p oblem, and in he amewo k o issue P sys ems wi h cell di ision, a linea - ime solu ion o p ime ac o iza ion p oblem is gi en. The ini ial s uc u e o he sys ems is e y simple, which consis s o h ee cells. The sys em is ini ialized wi h inpu ing in o he ixed inpu cell he mul ise ha exp esses he in ege numbe n o be ac o ed. A e a linea ime wi h espec o he size o n(i. e., blg kc+ 1), we can ead ou one ac o o nin he ou pu cell. Fac o iza ion by Tissue P Sys ems wi h Cell Di ision 371 1 z 16 2 c 16 3 〈 C , 0,1 〉 〈 C , 0,2 〉 〈 n, 1, 1 〉 〈 X , 0, 0 〉 〈 A 2, 0,0 〉 2 2 c 16 3 〈 C , 0,1 〉 〈 C , 0,2 〉 〈 n, 1, 1 〉 〈 X , 0,0 〉 〈 A 2, 1, 0 〉 2 〈 A 2, 0,1 〉 2 〈 A 2, 0,1 〉 2 〈 B 2, 0,0 〉 2 〈 B 2, 0,1 〉 2 〈 B 2, 0,0 〉 2 〈 B 2, 0,1 〉 2 2 c 16 3 〈 C , 0,1 〉 〈 C , 0,2 〉 〈 n, 1, 1 〉 〈 X , 0,0 〉 〈 A 2, 0,0 〉 2 2 c 16 3 〈 C , 0,1 〉 〈 C , 0,2 〉 〈 n, 1, 1 〉 〈 X , 0,0 〉 〈 A 2, 1,1 〉 2 〈 A 2, 1,1 〉 2 〈 A 2, 1, 0 〉 2 〈 B 2, 0,0 〉 2 〈 B 2, 0,1 〉 2 〈 B 2, 0,0 〉 2 〈 B 2, 0,1 〉 2 2 c 16 3 〈 C , 0,1 〉 〈 C , 0,2 〉 〈 n, 1, 1 〉 〈 X , 0, 0 〉 〈 A 2, 0,0 〉 2 2 c 16 4 〈 C , 1,0 〉 〈 C , 0,1 〉 〈 C , 0,2 〉 〈 n, 1, 1 〉 〈 n, 0,0 〉 〈 A 2, 0,1 〉 2 〈 A 2, 0,1 〉 2 〈 A 2, 1, 0 〉 2 〈 B 2, 0,1 〉 2 〈 B 2, 1, 0 〉 2 〈 B 2, 0,1 〉 2 〈 B 2, 1, 0 〉 2 2 c 16 2 〈 C , 0,2 〉 〈 A 2, 1, 1 〉 2 2 c 16 3 〈 C ,0,2 〉 〈 A 2, 1,1 〉 2 〈 A 2, 0,0 〉 2 〈 A 2, 1, 0 〉 2 〈 B 2, 0,1 〉〈 B 2, 1, 0 〉〈 B 2, 0,1 〉 2 〈 B 2, 1, 0 〉 2 2 c 16 3 〈 C , 0,1 〉 〈 X , 0,0 〉 〈 A 2, 0,1 〉 2 2 c 16 2 〈 C , 0,2 〉 〈 A 2, 0,1 〉 〈 A 2, 0,0 〉 2 〈 A 2, 1, 0 〉 〈 B 2, 1,1 〉 2 〈 B 2, 0,0 〉 2 〈 B 2, 1, 1 〉 2 〈 B 2, 0,0 〉 2 2 c 16 3 〈 C , 0,1 〉 〈 X , 0, 0 〉 〈 A 2, 1, 1 〉 2 2 c 16 3 〈 C , 1,1 〉 〈 X , 0, 0 〉 〈 A 2, 1,1 〉 2 〈 A 2, 0,0 〉 2 〈 A 2, 1, 0 〉 2 〈 B 2, 1,1 〉 2 〈 B 2, 0,0 〉 2 〈 B 2, 1, 1 〉 2 〈 B 2, 0,0 〉 2 2 c 16 3 〈 C , 0,1 〉 〈 C , 0,2 〉 〈 n, 1, 1 〉 〈 X , 0,0 〉 〈 A 2, 0,1 〉 2 2 c 16 3 〈 C , 1,0 〉 〈 C , 0,2 〉 〈 n, 0,0 〉 〈 A 2, 0,1 〉 2 〈 A 2, 0,0 〉 2 〈 A 2, 1, 0 〉 2 〈 B 2, 1, 1 〉 2 〈 B 2, 1, 0 〉 2 〈 B 2, 1, 1 〉 2 〈 B 2, 1, 0 〉 2 2 c 16 3 〈 C , 1,1 〉 〈 X , 0,0 〉 〈 A 2, 1, 1 〉 2 2 c 16 4 〈 C , 1,0 〉 〈 C , 0,1 〉 〈 C , 0,2 〉 〈 n , 0, 0 〉 〈 A 2, 1,1 〉 2 〈 A 2, 0,0 〉 2 〈 A 2, 1, 0 〉 2 〈 B 2, 1, 1 〉 2 〈 B 2, 1, 0 〉 2 〈 B 2, 1, 1 〉 2 〈 B 2, 1, 0 〉 2 c ' 8 c ' 8 c ' 8 c ' 8 c ' 8 c ' 8 c ' 8 c ' 8 c ' 8 c ' 8 c ' 8 d 0 d 0 〈 C ,1,0 〉 〈 n, 0,0 〉 〈 B' , 0,1 〉 〈 B' , 1, 0 〉 〈 A' , 0,1 〉〈 A' , 1, 0 〉 〈 C , 0,2 〉 〈 n, 1, 1 〉 c ' 8 Fig. 6. The con igu a ion o sys em Π(2) o ac o ing in ege numbe 2 a s ep 25 P sys em is a highly dis ibu ed pa allel model o compu a ion. Cu en ly, nobody knows how o build a biochemical compu e /an a i icial issue-like com- pu e . P sys ems may be implemen ed using molecules, cells o a la ge compu e ne wo k such as he In e ne . Al hough i goes beyond he scope o his wo k o discuss he implemen a ion o P sys ems, clea ly, i is o pa icula in e es and i is a big challenging opic. Acknowledgemen s The wo k was suppo ed by Na ional Na u al Science Founda ion o China (61033003, 61003038 and 30870826), Ph.D. P og ams Founda ion o Minis y o Educa ion o China (20100142110072), Fundamen al Resea ch Funds o he Cen- al Uni e si ies (2010ZD001), and Na u al Science Founda ion o Hubei P o ince (2008CDB113 and 2008CDB180). Ma io J. 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